MODELAÇÃO E ESTIMAÇÃO DO RISCO DE CRÉDITO

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  ❈❛t❛r✐♥❛ ❆❧❡①❛♥❞r❛ ▲♦♣❡s ❞♦ ❱❛❧❡ ▼❖❉❊▲❆➬➹❖ ❊ ❊❙❚■▼❆➬➹❖ ❉❖ ❘■❙❈❖ ❉❊ ❈❘➱❉■❚❖

  ❊❙❚❯❉❖ ❉❊ ❯▼❆ ❈❆❘❚❊■❘❆ ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ♥❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ◆♦✈❛ ❞❡ ▲✐s❜♦❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ❡ ❆♣❧✐❝❛çõ❡s ✲ ❆❝t✉❛r✐❛❞♦✱ ❊st❛tíst✐❝❛ ❡ ■♥✈❡st✐❣❛çã♦ ❖♣❡r❛❝✐♦♥❛❧✳

  ▲✐s❜♦❛ ✷✵✶✵

  ❈❛t❛r✐♥❛ ❆❧❡①❛♥❞r❛ ▲♦♣❡s ❞♦ ❱❛❧❡ ▼❖❉❊▲❆➬➹❖ ❊ ❊❙❚■▼❆➬➹❖ ❉❖ ❘■❙❈❖ ❉❊ ❈❘➱❉■❚❖

  ❊❙❚❯❉❖ ❉❊ ❯▼❆ ❈❆❘❚❊■❘❆ ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ♥❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ◆♦✈❛ ❞❡ ▲✐s❜♦❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ❡ ❆♣❧✐❝❛çõ❡s ✲ ❆❝t✉❛r✐❛❞♦✱ ❊st❛tíst✐❝❛ ❡ ■♥✈❡st✐❣❛çã♦ ❖♣❡r❛❝✐♦♥❛❧✳ ❉✐ss❡rt❛çã♦ ❖r✐❡♥t❛❞❛ ♣♦r✿ Pr♦❢❡ss♦r❛ ❉♦✉t♦r❛ ●r❛❝✐♥❞❛ ❘✐t❛ ❉✐♦❣♦ ●✉❡rr❡✐r♦ Pr♦❢❡ss♦r ❉♦✉t♦r ▼❛♥✉❡❧ ▲❡♦t❡ ❚❛✈❛r❡s ■♥❣❧ês ❊sq✉í✈❡❧

  ▲✐s❜♦❛ ✷✵✶✵

  ✐ ❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r s❡ ❞❡❞✐❝❛r ❝♦♠ t❛♥t♦ ❛✜♥❝♦ à ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ♣❡❧♦ s❡✉

❛♣♦✐♦ ✐♥❝❡ss❛♥t❡ ❡ ✐♥❝❡♥t✐✈♦✱ ♣❡❧♦ ❝❛r✐♥❤♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ♥♦s ❞✐❛s ♠❡♥♦s ❢á❝❡✐s ❡ ♣❡❧❛

❝♦♠♣r❡❡♥sã♦ ♣❡❧❛s ♠✐♥❤❛s tã♦ ❧♦♥❣❛s ❛✉sê♥❝✐❛s✳ ❆♦ ❏♦ã♦✱ ♣♦r t♦❞❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❡stí♠✉❧♦ tr❛♥s♠✐t✐❞♦✱ ♣❡❧♦ ❛♠♦r✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ♣❡❧♦s ✐♥t❡r♠✐♥á✈❡✐s ❞✐❛s ❡♠ q✉❡ ♠❡ ❛✉s❡♥t❡✐✳ ❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❋❈❚✱ ♣❡❧♦s ✈❛❧✐♦s♦s ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ♠❡ tr❛♥s♠✐t✐r❛♠ ❡ ✉♠

❡s♣❡❝✐❛❧ ❛❣r❛❞❡❝✐♠❡♥t♦ ❛♦s ♠❡✉s s✉♣❡r✈✐s♦r❡s Pr♦❢❡ss♦r ❉♦✉t♦r ▼❛♥✉❡❧ ▲❡♦t❡ ❊sq✉í✈❡❧

❡ Pr♦❢❡ss♦r❛ ❉♦✉t♦r❛ ●r❛❝✐♥❞❛ ❘✐t❛ ●✉❡rr❡✐r♦ ♣❡❧♦ s❡✉ ❛♣♦✐♦ ❡ ❡♥❝♦r❛❥❛♠❡♥t♦✱ ♣❡❧❛

❡①♣❡r✐ê♥❝✐❛✱ ♣r♦✜ss✐♦♥❛❧✐s♠♦ ❡ ❞❡❞✐❝❛çã♦✱ ♣❡❧❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✱ ❛♠✐③❛❞❡ ❡ ❝♦♥✜❛♥ç❛ ❡♠

♠✐♠✳

  P♦r ú❧t✐♠♦✱ ♠❛s ❝❡rt❛♠❡♥t❡ ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ ❛♦s ♠❡✉s ❛♠✐❣♦s ♣❡❧❛ ❛♠✐③❛❞❡✱

✐♥❝❡♥t✐✈♦ ❝♦♥st❛♥t❡✱ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❛♣♦✐♦ ❞✉r❛♥t❡ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛

✈✐❞❛ ❡ ❡♠ ❡s♣❡❝✐❛❧ ❞✉r❛♥t❡ ❡st❡ ♣❡r❝✉rs♦✳ ❖❜r✐❣❛❞♦✦

  ✐✐✐

❘❡s✉♠♦

❖s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ sã♦ ♠♦❞❡❧♦s q✉❛♥t✐t❛t✐✈♦s ✉t✐❧✐③❛❞♦s ❢r❡q✉❡♥t❡♠❡♥t❡

♥❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♠❡❞✐r ❡ ♣r❡✈❡r ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♣♦ss✉✐♥❞♦

  ✉♠❛ ❛✈✉❧t❛❞❛ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦✳

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❝❡♥tr❛✲s❡ ♥♦ ❡st✉❞♦ ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ à ❍❛❜✐t❛çã♦✳

❖ ♦❜❥❡❝t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❞❡t❡r♠✐♥❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❞❡ ✉♠ ❝❧✐❡♥t❡

❛q✉❛♥❞♦ ❞❡ ✉♠ ♥♦✈♦ ♣❡❞✐❞♦ ❞❡ ❝ré❞✐t♦✱ ❝♦♠ ❜❛s❡ ❡♠ ✐♥❢♦r♠❛çõ❡s ❢❛❝✉❧t❛❞❛s ♣❡❧♦

  ♠❡s♠♦✳ ❖s ❞❛❞♦s ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ♦ ♠♦❞❡❧♦ ❞❡

❈r❡❞✐t ❙❝♦r✐♥❣ ❞❡ ❛♣r♦✈❛çã♦ ❞❡ ❝ré❞✐t♦✱ ❛tr❛✈és ❞❡ ✉♠❛ té❝♥✐❝❛ ❡st❛tíst✐❝❛ ♠✉❧t✐✈❛r✐✲

❛❞❛✱ ❞❡s✐❣♥❛❞❛♠❡♥t❡ ❛ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛✳ ❖ ♠♦❞❡❧♦ ♦❜t✐❞♦ ❛❣r❡❣❛ ✈❛r✐á✈❡✐s ❝♦♠♦ ❛

✐❞❛❞❡ ❞♦ ❝❧✐❡♥t❡✱ ♦ ❡st❛❞♦ ❝✐✈✐❧✱ ♦ ✈❛❧♦r ❞♦ ❝ré❞✐t♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♣r❡st❛çõ❡s ❡ ♦ r❡♥❞✐✲

♠❡♥t♦ ❧íq✉✐❞♦✳ ❆❧❣✉♠❛s ❞❡st❛s ✈❛r✐á✈❡✐s r❡♣r❡s❡♥t❛♠ ❛tr✐❜✉t♦s q✉❡ ❝♦♥tr✐❜✉❡♠ ♣❛r❛

♦ ❛✉♠❡♥t♦ ❞❛ ✐♥❝❛♣❛❝✐❞❛❞❡ ❞❡ ❝✉♠♣r✐♠❡♥t♦ ❞❛s ♦❜r✐❣❛çõ❡s ❞❡ ❝ré❞✐t♦✱ ❡♥q✉❛♥t♦ q✉❡

♦✉tr❛s ❝♦♥tr✐❜✉❡♠ ♣❛r❛ ❛ ❞✐♠✐♥✉✐çã♦ ❞❡ss❛ ✐♥❝❛♣❛❝✐❞❛❞❡✳

  P❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ❛♣r♦✈❛çã♦ ❞❡ ❝ré❞✐t♦✱ é ❡ss❡♥❝✐❛❧ ❛ ❡st✐♠❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✳ ▼♦str❛✲s❡ q✉❡ ♦ ❝á❧❝✉❧♦ ❞♦ s♣r❡❛❞ ♠í♥✐♠♦ q✉❡ ❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ ❞❡✈❡ ❛♣❧✐❝❛r

❡♠ ❝❛❞❛ ♣❡❞✐❞♦ ❞❡ ❝ré❞✐t♦ é ❢✉♥çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛✲

  

çã♦✳ ❯♠ ❡st✉❞♦ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♣♦❞❡rá s❡r ❡❢❡❝t✉❛❞♦ ♣❛r❛

❡st✐♠❛r ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✳ P♦r ✜♠✱ ❢❛③✲s❡ ✉♠❛ ❜r❡✈❡ ❛✈❛❧✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❛♣r♦✈❛çã♦ ❞❡ ❝ré❞✐t♦ ♦❜t✐❞♦✱ ❡

❝♦♥❝❧✉✐✲s❡ q✉❡ ♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ ❛♣❧✐❝❛❞♦✱ t❡♠ ❝❛♣❛❝✐❞❛❞❡ ❞❡✱ ❛ ♣❛rt✐r

  

❞♦s ❞❛❞♦s ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✱ r❡❝✉♣❡r❛r ❛ ✐♥❢♦r♠❛çã♦ ❡ss❡♥❝✐❛❧ à ❡st✐♠❛çã♦ ❞❛ ♣r♦✲

❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✳ P❛❧❛✈r❛s ❝❤❛✈❡✿ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦✱ ▼♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✱ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❉❡❢❛✉❧t✱ ❚❛①❛ ❞❡ ❘❡❝✉♣❡r❛çã♦✱ ❙♣r❡❛❞✱ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛✱ ❈❧✐❡♥t❡ ■♥❝✉♠♣r✐❞♦r✳

  ✈

❆❜str❛❝t

❈r❡❞✐t ❙❝♦r✐♥❣ ♠♦❞❡❧s ❛r❡ q✉❛♥t✐t❛t✐✈❡ ♠♦❞❡❧s ♦❢t❡♥ ✉s❡❞ ✐♥ ✜♥❛♥❝✐❛❧ ✐♥st✐t✉t✐♦♥s

✐♥ ♦r❞❡r t♦ ♠❡❛s✉r❡ ❛♥❞ ♣r❡❞✐❝t t❤❡ ❝r❡❞✐t r✐s❦✱ ❤❛✈✐♥❣ ❛ ❜❧♦❛t❡❞ ✐♠♣♦rt❛♥❝❡ ✐♥ t❤❡

  ♣r♦❝❡ss ♦❢ ❣r❛♥t✐♥❣ ❝r❡❞✐t✳ ❚❤✐s ♣❛♣❡r ❢♦❝✉s❡s ♦♥ t❤❡ st✉❞② ♦❢ ❛ s✐♠✉❧❛t❡❞ ❤♦✉s✐♥❣ ❧♦❛♥ ♣♦rt❢♦❧✐♦✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❞❡❢❛✉❧t ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ❝❧✐❡♥t ✇❤❡♥ ♠❛❦❡s ❛ ♥❡✇ r❡q✉❡st ❢♦r ❝r❡❞✐t✱ ❜❛s❡❞ ♦♥ ✐♥❢♦r♠❛t✐♦♥ ♣r♦✈✐❞❡❞ ❜② ✐t✳ ❉❛t❛ ❢r♦♠ t❤❡ ❧♦❛♥ ♣♦rt❢♦❧✐♦ ✐s ✉s❡❞ t♦ ❞❡✈❡❧♦♣ ❛ ❈r❡❞✐t ❙❝♦r✐♥❣ ♠♦❞❡❧ ❢♦r ❝r❡❞✐t

❛♣♣r♦✈❛❧✱ t❤r♦✉❣❤ ❛ ♠✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝❛❧ t❡❝❤♥✐q✉❡✱ s✉❝❤ ❛s ❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥✳ ❚❤❡

  

♠♦❞❡❧ ❝♦♥t❛✐♥s ✈❛r✐❛❜❧❡s s✉❝❤ ❛s ❝❧✐❡♥t✬s ❛❣❡✱ ♠❛r✐t❛❧ st❛t✉s✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❝r❡❞✐t

❛♠♦✉♥t✱ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛②♠❡♥ts ❛♥❞ ♥❡t ✐♥❝♦♠❡✳ ❙♦♠❡ ♦❢ t❤❡s❡ ✈❛r✐❛❜❧❡s r❡♣r❡s❡♥t

❛ttr✐❜✉t❡s t❤❛t ❝♦♥tr✐❜✉t❡ t♦ t❤❡ ✐♥❝r❡❛s❡❞ ✐♥❛❜✐❧✐t② t♦ ❢✉❧✜❧❧ t❤❡ ♦❜❧✐❣❛t✐♦♥s ♦❢ ❝r❡❞✐t

✇❤✐❧❡ ♦t❤❡rs ❝♦♥tr✐❜✉t❡ t♦ t❤❡ ❞❡❝r❡❛s❡ ♦❢ t❤❛t ✐♥❝❛♣❛❝✐t②✳

  ❋♦r t❤❡ ♠♦❞❡❧ ♦❢ ❝r❡❞✐t ❛♣♣r♦✈❛❧✱ ✐t ✐s ❡ss❡♥t✐❛❧ t♦ ❡st✐♠❛t❡ t❤❡ ❞❡❢❛✉❧t ♣r♦❜❛❜✐❧✐t② ✳ ■t ✐s s❤♦✇♥ t❤❛t t❤❡ ♠✐♥✐♠✉♠ s♣r❡❛❞ t❤❛t t❤❡ ✜♥❛♥❝✐❛❧ ✐♥st✐t✉t✐♦♥ ♠✉st ❛♣♣❧② t♦

❡❛❝❤ ❝r❡❞✐t ❛♣♣❧✐❝❛t✐♦♥ ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❞❡❢❛✉❧t ♣r♦❜❛❜✐❧✐t② ❛♥❞ r❡❝♦✈❡r② r❛t❡✳ ❆♥

❡q✉✐✈❛❧❡♥t st✉❞② ❝♦✉❧❞ ❜❡ ♣❡r❢♦r♠❡❞ t♦ ❡st✐♠❛t❡ t❤❡ r❡❝♦✈❡r② r❛t❡✳

  ❋✐♥❛❧❧②✱ ✇❡ ♣r♦✈✐❞❡ ❛ ❜r✐❡❢ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❝r❡❞✐t ❛♣♣r♦✈❛❧ ♠♦❞❡❧ ♦❜t❛✐♥❡❞✱ ❛♥❞ ✐t

✐s ❝♦♥❝❧✉❞❡❞ t❤❛t t❤❡ ❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥ ♠♦❞❡❧✱ ✐s ❛❜❧❡ t♦✱ ❛❝❝♦r❞✐♥❣ t♦ ♣♦rt❢♦❧✐♦ ❞❛t❛✱

r❡tr✐❡✈❡ t❤❡ ❡ss❡♥t✐❛❧ ✐♥❢♦r♠❛t✐♦♥ ❢♦r ❡st✐♠❛t✐♥❣ t❤❡ ❞❡❢❛✉❧t ♣r♦❜❛❜✐❧✐t②✳ ❑❡②✇♦r❞s✿ ❈r❡❞✐t ❘✐s❦✱ ❈r❡❞✐t ❙❝♦r✐♥❣ ▼♦❞❡❧s✱ Pr♦❜❛❜✐❧✐t② ♦❢ ❉❡❢❛✉❧t✱ ❘❡❝♦✈❡r② ❘❛t❡✱ ❙♣r❡❛❞✱ ▲♦❣✐st✐❝ ❘❡❣r❡ss✐♦♥✱ ❉❡❢❛✉❧t✐♥❣ ❈❧✐❡♥t✳

  ❈❖◆❚❊Ú❉❖ ✈✐✐

  ❈♦♥t❡ú❞♦

  ✐ ✈✐✐

  ✐① ①✐

  ①✈ ①✈✐✐

  ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

  ✷✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

   ✸✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

   ✽✺ ✽✼ ✾✶

  ✶✶✺ ✶✶✺

  ❈❖◆❚❊Ú❉❖ ✐①

  ❚❛❜❡❧❛ ❞❡ ◆♦t❛çõ❡s ◆❖❚❆➬➹❖ ❉❊❙❈❘■➬➹❖ t

  ■♥st❛♥t❡ ♥♦ t❡♠♣♦✳ τ ▼♦♠❡♥t♦ ❡♠ q✉❡ ♦❝♦rr❡ ♦ ❞❡❢❛✉❧t✳ N (t) ■♥❞✐❝❛❞♦r ❞♦ ♣r♦❝❡ss♦ ❞❡ ❉❡❢❛✉❧t✳ EAD ❊①♣♦s✉r❡ ❛t ❉❡❢❛✉❧t✳ E(t) ▼♦♥t❛♥t❡ q✉❡ ♦ ❝r❡❞♦r ♣❡r❞❡✱ ❝❛s♦ ♦ ❞❡✈❡❞♦r ❡♥tr❡ ❡♠ ❞❡❢❛✉❧t ♥♦ t❡♠♣♦ t✳

  R(t) ♦✉ R ❚❛①❛ ❞❡ ❘❡❝✉♣❡r❛çã♦ ✭❘❡❝♦✈❡r② ❘❛t❡✮✳ L(t) ▲♦ss ●✐✈❡♥ ❉❡❢❛✉❧t✲▲●❉✳ T ▼❛t✉r✐❞❛❞❡✳ D(T ) ❉❡❢❛✉❧t ▲♦ss✳ P D i

  (t, T ) Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❞❡ ✉♠ ❞❡✈❡❞♦r i ❞❡st❡ t ❛té T ✳ P S(t, T ) Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡s❞❡ t ❛té T ✳ N ◆➸ ❞❡ ❞❡❢❛✉❧ts✱ ♥✉♠ ❝❡rt♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✳ λ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ ✭✶ ❛♥♦✮✳ E [D i

  (T )] P❡r❞❛ ❊s♣❡r❛❞❛ ■♥❞✐✈✐❞✉❛❧✳ U EL P❡r❞❛ ✐♥❡s♣❡r❛❞❛ ❞❡ ✉♠❛ ❝❛rt❡✐r❛

  ❯♥❡①♣❡❝t❡❞ ❉❡❢❛✉❧t ▲♦ss ✳ r

  ❚❛①❛ ❞❡ ❥✉r♦✳ s ❙♣r❡❛❞✳ P ▼❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ◆❛t✉r❛❧✳ p Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♥❛ ♠❡❞✐❞❛ ♥❛t✉r❛❧✳ Q ▼❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ▼❛rt✐♥❣❛❧❛✳ q Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♥❛ ♠❡❞✐❞❛ ♠❛rt✐♥❣❛❧❛✳

  X θ t

  Pr♦❝❡ss♦ ❊st♦❝ást✐❝♦✳ (X t

  ) t≥0

  ❱❛r✐á✈❡❧ q✉❡ ❞❡s❝r❡✈❡ ❛ ❡✈♦❧✉çã♦ ❞♦ ♣r❡ç♦ ❞❡ ✉♠ ❛❝t✐✈♦ ✲ Pr♦❝❡ss♦ ❞❡ ▼❛r❦♦✈✳ (B t

  ) t≥0

  Pr♦❝❡ss♦ ❇r♦✇♥✐❛♥♦✳ B(t, T ) Pr❡ç♦ ❡♠ t✱ ❞❡ ✉♠❛ ❞❡❢❛✉❧t✲❢r❡❡ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝♦♠ ✈❡♥❝✐♠❡♥t♦ ❡♠ T ✳

  ˜ B(t, T ) Pr❡ç♦ ❡♠ t✱ ❞❡ ✉♠❛ ❞❡❢❛✉❧t❛❜❧❡ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞✮ ❝♦♠ ✈❡♥❝✐♠❡♥t♦ ❡♠ T ✳

  ❊❉❊ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❊st♦❝ást✐❝❛✳ ■◆❊ ■♥st✐t✉t♦ ◆❛❝✐♦♥❛❧ ❞❡ ❊st❛tíst✐❝❛✳

  ❈❖◆❚❊Ú❉❖ ①✐

Pr❡❢á❝✐♦

  

❖ ❡st✉❞♦ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ t♦r♥♦✉✲s❡ ✉♠❛ t❛r❡❢❛ ❢✉♥✲

❞❛♠❡♥t❛❧ ♣❛r❛ ❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s q✉❡ ♥❡❝❡ss✐t❛♠ ❞❡ ❝♦♥tr♦❧❛r ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳

❈♦♠ ❛s ♠✉❞❛♥ç❛s ❞❛ ❡❝♦♥♦♠✐❛ ♠✉♥❞✐❛❧ ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s✱ ♦❝❛s✐♦♥❛❞❛s ❡ss❡♥❝✐❛❧✲

♠❡♥t❡ ♣❡❧❛ ❣❧♦❜❛❧✐③❛çã♦✱ t♦r♥♦✉✲s❡ ♥❡❝❡ssár✐♦ r❡❡str✉t✉r❛r ♦s ♠ét♦❞♦s ❞❡ ❛✈❛❧✐❛çã♦ ❞❡

r✐s❝♦✳ ❙ã♦ ✈ár✐❛s ❛s r❛③õ❡s q✉❡ ♦❝❛s✐♦♥❛r❛♠ ✉♠❛ ♥❡❝❡ss✐❞❛❞❡ ✉r❣❡♥t❡ ❞❡ ❣❡r✐r ♦ r✐s❝♦

❞❡ ❝ré❞✐t♦ ❞❡ ♠♦❞♦ ❡✜❝❛③✱ ♥♦♠❡❛❞❛♠❡♥t❡ ❛s ❛❧t❡r❛çõ❡s ♠❛❝r♦✲❡❝♦♥ó♠✐❝❛s ❞♦s ♣❛ís❡s✱

♦ ❛✉♠❡♥t♦ ❞❛ ❝♦♠♣❡t✐çã♦ ❜❛♥❝ár✐❛✱ ❛ ❡①♣❛♥sã♦ ❞♦s ♠❡r❝❛❞♦s ❞❡ ❝❛♣✐t❛✐s✱ ❛s ❛❧t❡r❛çõ❡s

❝♦♥st❛♥t❡s ❞❛ t❛①❛ ❞❡ ❥✉r♦ ♦✉ ❛s ♠❛r❣❡♥s ❞❡ s♣r❡❛❞s ♠❛✐s ❝♦♠♣❡t✐t✐✈❛s ❡ ♦s ✈❛❧♦r❡s

✈♦❧át❡✐s ❞❛s ❣❛r❛♥t✐❛s r❡❛✐s✳ P❡r❛♥t❡ ❡st❛ r❡❛❧✐❞❛❞❡✱ ❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s tê♠ ✈✐♥❞♦ ❛ ❞❡s❡♥✈♦❧✈❡r ♠♦❞❡❧♦s

✐♥t❡r♥♦s ❞❡ ❣❡stã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♦❢❡r❡❝❡r ❢❡rr❛♠❡♥t❛s ♠❛✐s ❡✜❝✐✲

❡♥t❡s ♣❛r❛ ❛ ✈❛❧♦r✐③❛çã♦ ❞❛ ❝❛rt❡✐r❛✱ ♠❡❞✐çã♦ ❞❡ r✐s❝♦s✱ ❛♣r❡ç❛♠❡♥t♦ ❞❡ ♥♦✈♦s ❡♠♣rés✲

t✐♠♦s✱ ♣♦t❡♥❝✐❛❧✐③❛♥❞♦ ♦s ❣❛♥❤♦s ❞♦s ❝❛♣✐t❛✐s ❡♠♣r❡st❛❞♦s ❛❞❡q✉❛♥❞♦✲♦s ❛♦ ♠♦♥t❛♥t❡

❞❡ ❝❛♣✐t❛❧ q✉❡ ❡st❡s ❞❡✈❡♠ ♠❛♥t❡r ❝♦♠♦ ♣❛rt❡ ❞❛ s✉❛ ❡str✉t✉r❛ ❞❡ ❝❛♣✐t❛❧✳

  ◆✉♠❛ ♥♦tí❝✐❛ ❞♦ ❏♦r♥❛❧ ❞❡ ◆❡❣ó❝✐♦s✱ ❛r❣✉♠❡♥t❛✿ ✏❆ ❝r✐s❡ ✜♥❛♥❝❡✐r❛ ❢♦✐ ✉♠ ✐♥✈❡♥t♦ ❞♦s ❜❛♥❝♦s ❛♥❣❧♦✲s❛①ó♥✐❝♦s ♠❛s ♦s ❜❛♥❝♦s ❡✉r♦♣❡✉s

❢♦r❛♠ ❜♦♥s ❛♣r❡♥❞✐③❡s ❞♦s ❡rr♦s ❛❧❤❡✐♦s✳ ❯♠ ❞♦s ❡rr♦s ❝❧❛♠♦r♦s♦s ❢♦✐ ♦ ❞❛ ♣rát✐❝❛ ❞♦s s♣r❡❛❞s

  ③❡r♦ ✈ír❣✉❧❛ q✉❛❧q✉❡r ❝♦✐s❛✱ ✉♠❛ ♥❡❣❛çã♦ ❞♦ r✐s❝♦ q✉❡ ❡stá ❤♦❥❡ ❛ s❛✐r ❝❛r❛✳ ❖ s♣r❡❛❞ q✉❡ s❡ s♦♠❛ às t❛①❛s ❊✉r✐❜♦r é s✐♠✉❧t❛♥❡❛♠❡♥t❡ ✉♠ ♠❡❞✐❞♦r ❞❡ r✐s❝♦ ❡ ♦ ❧✉❝r♦ ❞♦

❜❛♥❝♦✳ ❆♦ ❡❧✐♠✐♥á✲❧♦ q✉❛s❡ ❛té ❛♦ ③❡r♦✱ ♦s ❜❛♥❝♦s ❡st❛✈❛♠✱ ♣♦rt❛♥t♦✱ ❛ ❛❜❞✐❝❛r ❞❡ ♠❡❞✐r r✐s❝♦

♥♦ s❡✉ ❝❧✐❡♥t❡ ❡ ❞❛ s✉❛ ♠❛r❣❡♠ ❞❡ ❧✉❝r♦✳ ◆♦ ✐♥í❝✐♦✱ ♠✉✐t♦s ❜❛♥❝♦s ❞❛✈❛♠ ❝♦♠ ✉♠❛ ♠ã♦ ✭❛

t❛①❛ ❜❛✐①❛✮ ♦ q✉❡ t✐r❛✈❛♠ ❝♦♠ ❛ ♦✉tr❛ ✭❛s ❝♦♠✐ssõ❡s ❛❧t❛s✮✳ ▼❛s ❝♦♠♦ ♦s ❣♦✈❡r♥♦s ❧❤❡s ❢♦r❛♠

♣r♦✐❜✐♥❞♦ ❛s ❝♦♠✐ssõ❡s ❛❜✉s✐✈❛s✱ ❛s ❝❧á✉s✉❧❛s ❧❡♦♥✐♥❛s✱ ❛ ✐♠♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡

❝♦♥tr❛t♦s✱ ❡ss❛ ❢❛❝❡ ♦❝✉❧t❛ ❞♦ ❡♠♣rést✐♠♦ ❢♦✐ s❡♥❞♦ ❡①t✐♥t❛✳ ❋♦✐ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ♣r❡❞❛tór✐♦ ❝♦♥tr❛ s✐ ♠❡s♠♦s✿ ❤♦❥❡✱ ❣r❛♥❞❡ ♣❛rt❡ ❞♦ ❜❛❧❛♥ç♦ ❞♦s

❜❛♥❝♦s ❝♦♠❡r❝✐❛✐s ❡stá ❛♠❛rr❛❞♦ ❛ ❝ré❞✐t♦s à ❤❛❜✐t❛çã♦ ❞❡ ❞❡③❡♥❛s ❞❡ ❛♥♦s ❝♦♠ ✏s♣r❡❛❞s✑ ❜❛✐✲

①♦s ✲ ❡ ♥❛❞❛ ♣♦❞❡ ❢❛③❡r ♣❛r❛ ❝♦♥tr❛r✐á✲❧♦✳ ❖s ❝♦♥tr❛t♦s ❡stã♦ ❛ss✐♥❛❞♦s✱ ♦s ❞✐r❡✐t♦s ❛❞q✉✐r✐❞♦s

sã♦ ❞♦s ❝❧✐❡♥t❡s✳ ❊①❝❡♣t♦ ♦s ❝❧✐❡♥t❡s q✉❡ ♣r❡❝✐s❡♠ ❞❡ ❛❧t❡r❛r ❝♦♥❞✐çõ❡s ❝♦♥tr❛t✉❛✐s ❞♦ s❡✉ ❝ré❞✐t♦✳ ❯♠❛

♥❡❝❡ss✐❞❛❞❡ ❞♦ ❝❧✐❡♥t❡ é ✉♠ ♣r❡t❡①t♦ ♣❛r❛ ♦ ❜❛♥❝♦✿ ♦ s♣r❡❛❞ s♦❜❡✳ ❖ s♣r❡❛❞ s♦❜❡❄ ❊♥tã♦ ♦

❝❧✐❡♥t❡ ♥ã♦ ❛❧t❡r❛ ♥❛❞❛✳ ❈❡rt♦❄

  ❊rr❛❞♦✿ ❤á ❝❧✐❡♥t❡s q✉❡ ♥ã♦ tê♠ ❛❧t❡r♥❛t✐✈❛✳ ❆ r❡♣♦rt❛❣❡♠ ❞❡ ❤♦❥❡ ❞♦ ◆❡❣ó❝✐♦s ♠♦str❛

❝❛s♦s ❞❡ ❞❡s❡♠♣r❡❣♦ q✉❡ ♦❜r✐❣❛♠ ❛ ❞✐❧❛t❛r ♦ ♣r❛③♦ ❞♦ ❡♠♣rést✐♠♦✳ ❊ ❞❡ ❞✐✈ór❝✐♦s ❡♠ q✉❡

♦ ❝♦♥tr❛t♦ t❡♠ ❞❡ ♣❛ss❛r ❞❡ ❞♦✐s ♣❛r❛ ✉♠ t✐t✉❧❛r✳ ◆❡st❡ ❝❛s♦✱ ❛ t❛①❛ ❞❡ ❡s❢♦rç♦ ✭♣❛rt❡ ❞♦

  ①✐✐ ❈❖◆❚❊Ú❉❖

r❡♥❞✐♠❡♥t♦ ❛❢❡❝t❛ ❛♦ ♣❛❣❛♠❡♥t♦ ❞❛s ♣r❡st❛çõ❡s✮ ♣♦❞❡ ❛té ❝♦♥t✐♥✉❛r ❛ s❡r ❝✉♠♣r✐❞❛✱ ♠❛s

♠❡s♠♦ ❛ss✐♠ ❛ t❛①❛ s♦❜❡✳ ❆ ♦♣♦rt✉♥✐❞❛❞❡ t♦r♥❛✲s❡ ♦♣♦rt✉♥✐s♠♦✳✑

  ❆ ❝♦♥tr❛❞✐çã♦ é ❡✈✐❞❡♥t❡✱ é ❛♣❧✐❝❛❞♦ ✉♠ s♣r❡❛❞ ❡❧❡✈❛❞♦ ♣❛r❛ ❝❧✐❡♥t❡s ❝♦♠ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡s✳ ❉✐t♦ ✐st♦✱ ♦ ♣r♦♣ós✐t♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ s❡rá ❡st✉❞❛r ❡st❛t✐st✐❝❛♠❡♥t❡ ✉♠ ❝♦♥❥✉♥t♦

❞❡ ❝❛r❛❝t❡ríst✐❝❛s✱ q✉❡ ✐♥✢✉❡♥❝✐❛♠ ♦✉ ♥ã♦✱ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ✭❞❡❢❛✉❧t✮✳

  

❖ r❡s✉❧t❛❞♦ ❞❡st❡ ❡st✉❞♦ é ✉♠ ♠♦❞❡❧♦ ❝♦♥st✐t✉í❞♦ ♣❡❧❛s ❝❛r❛❝t❡ríst✐❝❛s q✉❡ ♠❡❧❤♦r

❡①♣❧✐❝❛♠ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✱ ♠❡❞✐♥❞♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❞❡ ❝❛❞❛ ❝❧✐❡♥t❡✳

◆♦ ❈❛♣ít✉❧♦ sã♦ ❛♣r❡s❡♥t❛❞♦s ❝♦♥❝❡✐t♦s✱ ♠❡❞✐❞❛s ❡ ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦

♣❛r❛ ❝❧✐❡♥t❡s ✐♥❞✐✈✐❞✉❛✐s ❡ ♣❛r❛ ❝❛rt❡✐r❛s ❞❡ ❝ré❞✐t♦ ❞❡ ✉♠❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ ❝♦♠

  

♦ ✐♥t✉✐t♦ ❞❡ ❢❛③❡r ✉♠ ❡♥q✉❛❞r❛♠❡♥t♦ ❣❡r❛❧ ❞♦s ♠♦❞❡❧♦s ❡ té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ♣❛r❛

♠♦❞❡❧❛çã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ◆♦ ❈❛♣ít✉❧♦ sã♦ ❞❡s❡♥✈♦❧✈✐❞❛s ❡①♣r❡ssõ❡s q✉❡ ❞❡t❡r♠✐♥❛♠ ♦ s♣r❡❛❞ ❛❞❡q✉❛❞♦ ❛

❝❛❞❛ ❝❧✐❡♥t❡✱ ❡♠ ❢✉♥çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❡st✐♠❛❞❛s✱

s❡♥❞♦ ❞❡s❝r✐t❛s ✈ár✐❛s ❛❜♦r❞❛❣❡♥s q✉❡ ❝♦♥❞✉③❡♠ ❛ r❡s✉❧t❛❞♦s ❝♦♠♣❛rá✈❡✐s✳ ❖ ❈❛♣ít✉❧♦ ♥♦ q✉❛❧ s❡ ✐♥s❡r❡ ❛ ♣❛rt❡ ♣rát✐❝❛ ❞❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐st❡ ♥❛ s✐♠✉❧❛çã♦

❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ à ❍❛❜✐t❛çã♦ ❡ ♥♦ ❡st✉❞♦ ❞❛ ♠❡s♠❛ ❛tr❛✈és ❞♦s ▼♦❞❡❧♦s

▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛tr❛✈és ❞❛ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛✳ ❋♦✐ ♥❡❝❡ssár✐❛

✉♠❛ ♣❡sq✉✐s❛ ♣ré✈✐❛ s♦❜r❡ ❛ ✐♥❢♦r♠❛çã♦ q✉❡ ❣❡r❛❧♠❡♥t❡ é ♣❡❞✐❞❛ ❛ ✉♠ s♦❧✐❝✐t❛♥t❡ ❞❡

❝ré❞✐t♦✱ ❝♦♠ ♦ ♦❜❥❡❝t✐✈♦ ❞❡ s❡ s✐♠✉❧❛r ❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❡✱ ❛tr❛✈és ❞❡ ♠ét♦❞♦s ❞❡

s❡❧❡❝çã♦ ❞❡ ✈❛r✐á✈❡✐s✱ ❡♥❝♦♥tr❛r ♦ ♠♦❞❡❧♦ q✉❡ ♠❡❧❤♦r ❡①♣❧✐❝❛ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✳

❉❡st❡ ♠♦❞♦✱ ❛tr❛✈és ❞❡st❡ ú❧t✐♠♦ ♠♦❞❡❧♦✱ é ♣♦ssí✈❡❧ ❡st✐♠❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t

❞❡ ✉♠ ♥♦✈♦ s♦❧✐❝✐t❛♥t❡ ❞❡ ❝ré❞✐t♦✳ ❋✐❝❛♠ ❡♠ ❛❜❡rt♦ ❛❧❣✉♠❛s q✉❡stõ❡s✱ q✉❡ ♥ã♦ ❢♦r❛♠ ❛♥❛❧✐s❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦✱ ♠❛s q✉❡ ♣♦❞❡♠ s❡r ♦❜❥❡❝t♦ ❞❡ ❡st✉❞♦s ❢✉t✉r♦s✿ ❼ ❊st✐♠❛çã♦ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✱ ♥❡❝❡ssár✐❛ ♣❛r❛ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ s♣r❡❛❞

  ♠í♥✐♠♦ ❛ ❛♣❧✐❝❛r✱ ♠❛s q✉❡ s❡ ❞❡❞✉③ s❡r ❡st✐♠❛❞❛ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t r❡❝♦rr❡♥❞♦✱ ♥♦ ❡♥t❛♥t♦✱ ❛ ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❞✐st✐♥t❛s✳ ❼ ❆♣❧✐❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❱órt✐❝❡s ❊st♦❝ást✐❝♦s✱ ✈❡r ❛ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❞✐♥â♠✐❝❛✱ ✐st♦ é✱ ❛❣r✉♣❛r ♦s ❝❧✐❡♥t❡s ❡♠ ✏❝❧❛ss❡s ❞❡ r✐s❝♦✑ ✭❝♦♠

  ❜❛s❡ ♥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡st✐♠❛❞❛✮✱ ❡st✐♠❛r ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐✲ çã♦ ❡♥tr❡ ❝❧❛ss❡s ❞❡ r✐s❝♦ ❡✱ ❛ss✉♠✐♥❞♦ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❛❜❡rt❛ ✭♣❡r♠✐t✐♥❞♦ ❛ s✉❜s❝r✐çã♦ ❞❡ ♥♦✈♦s ❝♦♥tr❛t♦s ❡ ❛ ❛♥✉❧❛çã♦ ❞❡ ❝♦♥tr❛t♦s ❡①✐st❡♥t❡s✮ ❛♥❛❧✐s❛r✱ ♥✉♠❛ ♣❡rs♣❡❝t✐✈❛ t❡♠♣♦r❛❧✱ ❛ ❡✈♦❧✉çã♦ ❞❛ ❞✐♠❡♥sã♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝❧❛ss❡s ❞❡ r✐s❝♦✱ ❡st✐♠❛♥❞♦ ♦ ♣❡s♦ r❡❧❛t✐✈♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝❧❛ss❡s ❞❡♥tr♦ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✳

  ◆❡st❡ tr❛❜❛❧❤♦✱ ♦♣t♦✉✲s❡ ♣♦r ✉t✐❧✐③❛r ❛❧❣✉♥s t❡r♠♦s té❝♥✐❝♦s ❡♠ ✐♥❣❧ês ♣♦✐s ❛✐♥❞❛

♥ã♦ ❡①✐st❡ ✉♠ ❝♦♥s❡♥s♦ ♣❛r❛ ❛ s✉❛ tr❛❞✉çã♦✳ ❊♠❜♦r❛ s❡❥❛ ❢❡✐t❛ ✉♠❛ tr❛❞✉çã♦ ❞♦s

t❡r♠♦s✱ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ✉t✐❧✐③❛✲s❡✱ ♣❛r❛ q✉❛s❡ t♦❞♦s✱ ❛ s✉❛ ✈❡rsã♦

  ❈❖◆❚❊Ú❉❖ ①✐✐✐ ◆♦ ✜♥❛❧ ❞❡st❡ tr❛❜❛❧❤♦ ❝♦♥st❛ ✉♠ ♣❡q✉❡♥♦ ❣❧♦ssár✐♦ ❝♦♠ ❛❧❣✉♥s ❞♦s t❡r♠♦s ✉t✐❧✐✲

  ③❛❞♦s✳

  ▲■❙❚❆ ❉❊ ❚❆❇❊▲❆❙ ①✈

  ▲✐st❛ ❞❡ ❚❛❜❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✳ ✳ ✳ ✳ ✳ ✼✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✳ ✳ ✳ ✳ ✳ ✼✾ ✳ ✳ ✳ ✳ ✳ ✽✵ ✳ ✽✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵

  ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①✈✐✐

  ▲✐st❛ ❞❡ ❋✐❣✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹

  ✶ ❈❛♣ít✉❧♦ ✶ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

◆❡st❡ ❈❛♣ít✉❧♦ sã♦ ❛♣r❡s❡♥t❛❞♦s ❝♦♥❝❡✐t♦s✱ ♠❡❞✐❞❛s ❡ ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣❛r❛

❝❧✐❡♥t❡s ✐♥❞✐✈✐❞✉❛✐s ❡ ♣❛r❛ ❝❛rt❡✐r❛s ❞❡ ❝ré❞✐t♦ ❞❡ ✉♠❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛✳ ✶✳✶ ▼♦❞❡❧♦s ❞❡ ❆♥á❧✐s❡ ❞❡ ❈ré❞✐t♦ ✲ ●❡♥❡r❛❧✐❞❛❞❡s ✶✳✶✳✶ ▼♦❞❡❧♦s ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

▼♦❞❡❧♦ é✱ ♣♦r ❞❡✜♥✐çã♦✱ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ s✐♠♣❧✐✜❝❛❞❛ ❞❡ ❛❧❣♦ r❡❛❧✳ ❉❡st❛ ❢♦r♠❛✱

❛❧❣♦r✐t♠♦s✱ ❢ór♠✉❧❛s✱ s✐st❡♠❛s ♦✉ r❡❣r❛s q✉❡ ✈✐s❛♠ r❡♣r❡s❡♥t❛r ♣r♦❝❡ss♦s ♦✉ ❛tr✐❜✉t♦s

r❡❛✐s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡

❝ré❞✐t♦✳

  ❉❡✈✐❞♦ às s✉❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ♦s ♠♦❞❡❧♦s ❢❛❝✐❧✐t❛♠ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ✉♠ ❢❡♥ó♠❡♥♦

❡✱ ❡✈❡♥t✉❛❧♠❡♥t❡✱ ❛ s✉❛ ❡①♣❧♦r❛çã♦✳ ❖s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♥ã♦ sã♦ ❡①❝❡♣çã♦✳

P❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❡①✐❣❡✱

❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛s ✈❛r✐á✈❡✐s q✉❡ ♣♦❞❡♠ ♣r♦✈♦❝❛r ❛ ♦❝♦rrê♥❝✐❛ ❞❡

  

✐♥❝✉♠♣r✐♠❡♥t♦✳ ❙❡❣✉❡✲s❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢❡rr❛♠❡♥t❛s ♣❛r❛ ❝♦♥str✉✐r

✉♠ ♠♦❞❡❧♦ ❢♦r♠❛❧✱ ❝♦♠ ❜❛s❡ ♥✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s q✉❡ r❡♣r❡s❡♥t❡♠ ❛ ❝❛rt❡✐r❛ ❞❡

❝ré❞✐t♦✳ ❋✐♥❛❧♠❡♥t❡✱ ❞❡✈❡♠ s❡r ❛♣❧✐❝❛❞♦s t❡st❡s ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ♦ ♠♦❞❡❧♦ t❡♠ ♦

❞❡s❡♠♣❡♥❤♦ ❡s♣❡r❛❞♦✳

  ❊♥tã♦✱ ♣♦❞❡ ❞✐③❡r✲s❡ q✉❡ ♦ t❡r♠♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ é ✉♠❛ ❞❡♥♦♠✐♥❛çã♦

❣❡♥ér✐❝❛ q✉❡✱ ♣❛r❛ ❛ s✉❛ ❝♦♥str✉çã♦✱ ♥❡❝❡ss✐t❛ ❞❡ ❞✐✈❡rs❛s ❢❡rr❛♠❡♥t❛s✳ ❉❡st❛s ❢❡r✲

r❛♠❡♥t❛s✱ ❝♦♥✈é♠ ❞❡st❛❝❛r ❛s té❝♥✐❝❛s ❡❝♦♥♦♠étr✐❝❛s✱ ❝♦♠♦ ❛ ❛♥á❧✐s❡ ▲♦❣✐t ❡ ❛♥á❧✐s❡

Pr♦❜✐t✱ r❡❞❡s ♥❡✉r♦♥❛✐s✱ ♠♦❞❡❧♦s ❞❡ ♦♣t✐♠✐③❛çã♦✱ s✐st❡♠❛s ❡s♣❡❝✐❛❧✐st❛s ♦✉ ❜❛s❡❛❞♦s ❡♠

r❡❣r❛s✱ s✐st❡♠❛s ❤í❜r✐❞♦s q✉❡ ✉t✐❧✐③❛♠ ❛ ❝♦♠♣✉t❛çã♦✱ ❡st✐♠❛t✐✈❛ ❡ s✐♠✉❧❛çã♦ ❞✐r❡❝t❛s✳

❙❡❣✉♥❞♦ ❛ ✜♥❛❧✐❞❛❞❡ ❞♦s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ é ❞✐✲

✈❡rs❛✳ ❆ ❛♣r♦✈❛çã♦ ❞❡ ❝ré❞✐t♦✱ ♦ ❛♣r❡ç❛♠❡♥t♦ ❞❡ ❝ré❞✐t♦✱ ❛ ❡str❛té❣✐❛ ❞❡ ❝♦❜r❛♥ç❛✱ ❛

❞❡t❡r♠✐♥❛çã♦ ❞❡ r❛t✐♥❣s ❞❡ ❝ré❞✐t♦✱ sã♦ ❛❧❣✉♥s ❞♦s ♦❜❥❡❝t✐✈♦s ✈✐s❛❞♦s ♣❡❧♦s ♠♦❞❡❧♦s

  ✷ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

❞❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❝❛s❤✲✢♦✇s✱ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❛❞❡q✉❛❞♦ ♣❛r❛ ✉♠ ❡♠♣rést✐♠♦ ♦✉ ✉♠

tít✉❧♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡st❡s ♠♦❞❡❧♦s ❢❛❝✐❧✐t❛♠ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦ ❞♦ ❛✈❛❧✐st❛ ♣♦✐s

❣❡r❛♠ ✐♥❢♦r♠❛çã♦ q✉❡ ♥ã♦ ❡st❛r✐❛ ❞✐s♣♦♥í✈❡❧ ❞❡ ♦✉tr❛ ❢♦r♠❛ ♦✉ q✉❡ t❡r✐❛ ❝✉st♦s ♠✉✐t♦

❡❧❡✈❛❞♦s✳ ◆❛ ❧✐t❡r❛t✉r❛ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r r❡❢❡rê♥❝✐❛s ❛ ❞✐❢❡r❡♥t❡s ❢♦r♠❛s ❞❡ ❝❧❛ss✐✜❝❛çã♦

♣❛r❛ ♦s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ❙❡❣✉♥❞♦ ♦s ♠♦❞❡❧♦s ❞❡

r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❡stã♦ ❞✐✈✐❞✐❞♦s ❡♠ s✐st❡♠❛s ❡s♣❡❝✐❛❧✐st❛s✱ ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✱

♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❘❛t✐♥❣ ❡ ♠♦❞❡❧♦s ❞❡ ♣♦rt❢ó❧✐♦✳ ❖s três ♣r✐♠❡✐r♦s✱ ♥♦ s❡✉ t♦❞♦✱

❢♦r♠❛♠ ✉♠ ❣r✉♣♦ ❞❡s✐❣♥❛❞♦ ▼♦❞❡❧♦s ❞❡ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ❘✐s❝♦ ✭❛❜♦r❞❛❣❡♠ t❡ór✐❝❛

tr❛❞✐❝✐♦♥❛❧✮ ❡ ♦ ú❧t✐♠♦ é ❜❛s❡❛❞♦ ♥❛ t❡♦r✐❛ ❞❡ ❝❛rt❡✐r❛s✱ ♣❡rt❡♥❝❡♥t❡ ❛ ✉♠❛ ❛❜♦r❞❛❣❡♠

♠❛✐s ❛❝t✉❛❧ ❞❡ ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ✶✳✶✳✷ ❈ré❞✐t♦ ❡ ❉❡❢❛✉❧t

❆ ♣❛❧❛✈r❛ ❝ré❞✐t♦ s✐❣♥✐✜❝❛ ❝♦♥✜❛♥ç❛✴s❡❣✉r❛♥ç❛✿ ❝ré❞✐t♦ ❂ ❝r❡r✱ ❞♦ ❧❛t✐♠ ❝r❡❞✐t✉♠✱

❝r❡❞❡r❡✳

  ❈ré❞✐t♦ é t❛♠❜é♠ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛ss✉♠✐r ❝♦♠♣r♦♠✐ss♦s✱ q✉❡r ♣❛r❛ ✜♥❛♥❝✐❛♠❡♥t♦s

q✉❡r ♣❛r❛ ❡♠♣rést✐♠♦s✱ ❥✉♥t♦ ❞♦ s✐st❡♠❛ ✜♥❛♥❝❡✐r♦✳ ❊ss❛ ❝❛♣❛❝✐❞❛❞❡ é ❛♥❛❧✐s❛❞❛ ❝♦♠

❜❛s❡ ❡♠ ♣❛râ♠❡tr♦s té❝♥✐❝♦s✳ ❆ ❝❡❞ê♥❝✐❛ ❞❡ ❝ré❞✐t♦ é ✉♠❛ r❡❧❛çã♦ ❝♦♥tr❛t✉❛❧ ♠❡❞✐❛♥t❡ ❛ q✉❛❧ ✉♠❛ ❞❛s ♣❛rt❡s✱ ❛

❝r❡❞♦r❛✱ ❡♥tr❡❣❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠♦♥t❛♥t❡ ❛ ♦✉tr❛ ♣❛rt❡✱ ❛ ❞❡✈❡❞♦r❛✱ s♦❜ ❛ ♣r♦♠❡ss❛

❞❡ ♣❛❣❛♠❡♥t♦ ❞♦ ✈❛❧♦r ❡♠ ❞í✈✐❞❛✳

  ❖ ❝r❡❞♦r é t♦❞❛ ❛ ♣❡ss♦❛ t✐t✉❧❛r ❞❡ ✉♠ ❝ré❞✐t♦✱ ♦✉✱ q✉❡ t❡♠ ❛ r❡❝❡❜❡r ❞❡ ♦✉tr❡♠

✉♠❛ ❝❡rt❛ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ❞✐♥❤❡✐r♦✳ Pr♦t❡❣✐❞♦ ♣❡❧❛ ❧❡✐✱ ♦ ❝r❡❞♦r ♣♦ss✉✐ ❛ ❢❛❝✉❧❞❛❞❡ ❞❡

❡①✐❣✐r ❞♦ ❞❡✈❡❞♦r ♦ ❝✉♠♣r✐♠❡♥t♦ ❞❛ ♦❜r✐❣❛çã♦ ♦✉ ♦ ♣❛❣❛♠❡♥t♦ ❞♦ ❝ré❞✐t♦✱ q✉❛♥❞♦ ❡st❡

s❡ t♦r♥❛ ❡①✐❣í✈❡❧✱ ✐st♦ é✱ s❡ ♦ ❝♦♥tr❛t♦ ♥ã♦ é ❝✉♠♣r✐❞♦ ♣❡❧♦ ❞❡✈❡❞♦r✳

  ❖ r✐s❝♦ ❞❛ ♣r♦♠❡ss❛ ❞❡ ♣❛❣❛♠❡♥t♦ ♥ã♦ s❡r ❝✉♠♣r✐❞❛ ♣♦r ♣❛rt❡ ❞♦ ❞❡✈❡❞♦r✱ ❞❡s✐❣♥❛✲

s❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ P♦❞❡ ❞✐③❡r✲s❡ q✉❡ ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ s✉r❣❡ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ♥♦ ❛❝t♦

❞❡ ❡♠♣r❡st❛r ✉♠❛ q✉❛♥t✐❛ ❛ ❛❧❣✉é♠✱ ✉♠❛ ✈❡③ q✉❡ ❡①✐st❡ s❡♠♣r❡ ❛ ✐♥❝❡rt❡③❛ ❡♠ r❡❧❛çã♦

❛♦ r❡t♦r♥♦ ❞❡ss❛ q✉❛♥t✐❛✳

  ❉❡❢❛✉❧t é ❛ ✐♥❝❛♣❛❝✐❞❛❞❡ ♣❛r❛ ❝✉♠♣r✐r ❛s ❝♦♥❞✐çõ❡s ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦ ♦✉ ❛❝♦r❞♦✱ ♦✉

s❡❥❛✱ é ♥ã♦ ❢❛③❡r ✉♠ ♣❛❣❛♠❡♥t♦ ❡♠ ❞í✈✐❞❛✳ ❆ ♣❛❧❛✈r❛ ❉❡❢❛✉❧t ♣♦❞❡ s❡r s✐♠♣❧❡s♠❡♥t❡✱

■♥❝✉♠♣r✐♠❡♥t♦✳ ✶✳✶✳✸ ❆♥á❧✐s❡ ❞❡ ❈ré❞✐t♦

❙❡❣✉♥❞♦ ❛ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦ ❝♦♥st✐t✉✐ ✉♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❛❝t✐✈✐❞❛❞❡s

❞❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s s❡♥❞♦ ❛ ♣r✐♥❝✐♣❛❧ ❝❛✉s❛ ❞❡ ✐♥s♦❧✈ê♥❝✐❛ ❞❛s ♠❡s♠❛s ❡✱ ✉♠❛

✈❡③ q✉❡ ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ é ✐♥❡r❡♥t❡ ❛ q✉❛❧q✉❡r ♦♣❡r❛çã♦ ✜♥❛♥❝❡✐r❛✱ é ❡ss❡♥❝✐❛❧ q✉❡

  

✶✳✶ ▼♦❞❡❧♦s ❞❡ ❆♥á❧✐s❡ ❞❡ ❈ré❞✐t♦ ✲ ●❡♥❡r❛❧✐❞❛❞❡s ✸

❯♠❛ ❜♦❛ ❣❡stã♦ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣♦r ♣❛rt❡ ❞❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s✱ é ✐♥❞✐s♣❡♥✲

sá✈❡❧ ♣❛r❛ q✉❡ s❡ ❡✈✐t❡ ❛ ✐♥s♦❧✈ê♥❝✐❛ ❞❛s ♠❡s♠❛s✳ ❆ ❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦ é ✉♠ ♣r♦❝❡ss♦

q✉❡ ❞❡✈❡ r❡✉♥✐r ✐♥❢♦r♠❛çõ❡s ❛ r❡s♣❡✐t♦ ❞♦ t♦♠❛❞♦r ❞❡ ❝ré❞✐t♦✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❛✈❛❧✐❛r

❛ s✉❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❝✉♠♣r✐r ❝♦♠ ❛s s✉❛s ♦❜r✐❣❛çõ❡s ❡ ❞❡✜♥✐r q✉❛♥t♦ à ❝♦♥❝❡ssã♦ ♦✉

♥ã♦ ❞♦ ❝ré❞✐t♦✳

  ❆ ❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦ ♣♦❞❡ s❡r tr❛t❛❞❛ t❡♥❞♦ ❡♠ ❝♦♥t❛ ❞✉❛s ♠❡t♦❞♦❧♦❣✐❛s✿ ❛ q✉❛❧✐✲

t❛t✐✈❛ ❡ ❛ q✉❛♥t✐t❛t✐✈❛✳ ❆ ❛♥á❧✐s❡ q✉❛❧✐t❛t✐✈❛ r❡♠❡t❡ ♣❛r❛ ❥✉❧❣❛♠❡♥t♦s s✉❜❥❡❝t✐✈♦s ♣♦r

♣❛rt❡ ❞♦ ❛♥❛❧✐st❛ ❞❡ ❝ré❞✐t♦✱ ❡♠ r❡❧❛çã♦ à ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♣❛❣❛♠❡♥t♦ ❞♦ t♦♠❛❞♦r ❞❡

❝ré❞✐t♦✳ ◆❡st❛ ❛❜♦r❞❛❣❡♠✱ ♣❡ss♦❛s ❡s♣❡❝✐❛❧✐③❛❞❛s sã♦ ❡♥❝❛rr❡❣✉❡s ❞❡ t♦♠❛r ❛ ❞❡❝✐sã♦

s♦❜r❡ ❛ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦✱ ✉t✐❧✐③❛♥❞♦ ❝r✐tér✐♦s q✉❛❧✐t❛t✐✈♦s ❡ s✉❜❥❡❝t✐✈♦s✳ ❆ ❛♥á❧✐s❡

q✉❛♥t✐t❛t✐✈❛ ✉t✐❧✐③❛ ✐♥❢♦r♠❛çõ❡s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ♠♦❞❡❧♦s ❡st❛tíst✐❝♦s ❡ ❡❝♦♥♦♠étr✐❝♦s✳

◆❡st❛ ú❧t✐♠❛ ❛❜♦r❞❛❣❡♠ ❝♦♥s✐❞❡r❛♠✲s❡ ♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❡ ♦s ♠♦❞❡❧♦s

❜❛s❡❛❞♦s ♥❛ t❡♦r✐❛ ❞❡ ❝❛rt❡✐r❛s✳

  ❆ ♣r✐♥❝✐♣❛❧ ✈❛♥t❛❣❡♠ ❞❛ ❛❜♦r❞❛❣❡♠ q✉❛❧✐t❛t✐✈❛ é ❛ ❡s♣❡❝✐✜❝✐❞❛❞❡ ❝♦♠ q✉❡ é tr❛t❛❞♦

❝❛❞❛ ❝❛s♦✳ ❆ ♣r✐♥❝✐♣❛❧ ❞❡s✈❛♥t❛❣❡♠ é ❛ s✉❛ ❞❡♣❡♥❞ê♥❝✐❛ ♥❛ ❡①♣❡r✐ê♥❝✐❛ ❞♦ ❛✈❛❧✐❛❞♦r✱

♦ ❜❛✐①♦ ✈♦❧✉♠❡ ❞❡ ♣r♦❞✉çã♦ ❡ ♦ ❡♥✈♦❧✈✐♠❡♥t♦ ♣❡ss♦❛❧ ❞♦ ❝♦♥❝❡❞❡♥t❡ ❞❡ ❝ré❞✐t♦✳ ❆s r❡❣r❛s ❜❡♠ ❞❡✜♥✐❞❛s ❡♠ r❡❧❛çã♦ às ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ❝❧✐❡♥t❡s ❡ às ♦♣❡r❛çõ❡s ❞❡

❝ré❞✐t♦✱ ❜❛s❡❛❞❛s✱ ❡♠ ❣❡r❛❧✱ ❡♠ ♠♦❞❡❧♦s ❡st❛tíst✐❝♦s✱ sã♦ ♦ ❢❛❝t♦r ♣♦s✐t✐✈♦ ❞❛ ❛♥á❧✐s❡

q✉❛♥t✐t❛t✐✈❛✳

  ❆ ❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦ ❝❧áss✐❝❛ ❡ ♠❛✐s ❛♥t✐❣❛ ❞❡♣❡♥❞❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞❛ ♦♣✐♥✐ã♦ ❞❡

❡s♣❡❝✐❛❧✐st❛s ❜❡♠ tr❡✐♥❛❞♦s✱ ♥♦ ❡♥t❛♥t♦✱ ❛❝t✉❛❧♠❡♥t❡✱ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♦❜t❡r ✉♠❛ ❛✈❛❧✐✲

❛çã♦ ❞❡ r✐s❝♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞❛ ❡ ♠❛✐s ❝♦rr❡❝t❛ ♦❜r✐❣♦✉ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ té❝♥✐❝❛s

❡st❛tíst✐❝❛s ❞❡ ♠♦❞♦ ❛ ❣❛r❛♥t✐r ✉♠❛ ❞✐♠✐♥✉✐çã♦ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❛s ✐♥st✐t✉✐çõ❡s ✜✲

♥❛♥❝❡✐r❛s✳ ❆s té❝♥✐❝❛s ❞❡ ♣❡sq✉✐s❛ ❡st❛tíst✐❝❛✱ t❛✐s ❝♦♠♦ ❛ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱

r❡❞❡s ♥❡✉r♦♥❛✐s✱ ♣r♦❣r❛♠❛çã♦ ♠❛t❡♠át✐❝❛ ❡ s✐♠✉❧❛çã♦ ♣r♦❜❛❜✐❧íst✐❝❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛

♦ ❛✈❛♥ç♦ ❞❛s té❝♥✐❝❛s ❞❡ ♠❡♥s✉r❛çã♦ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ✶✳✶✳✹ ❙✐st❡♠❛s ❊s♣❡❝✐❛❧✐st❛s ❞❡ ❆♥á❧✐s❡ ❙✉❜❥❡❝t✐✈❛

❖s s✐st❡♠❛s ❡s♣❡❝✐❛❧✐st❛s sã♦ ♠♦❞❡❧♦s ❝✉❥❛ ❢✉♥çã♦ é ❛♣♦✐❛r ❛ ❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦ ❝❧áss✐❝❛

r❡❢❡r✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡✳ ❊st❡s ♠♦❞❡❧♦s ❜❛s❡✐❛♠✲s❡ ♥♦ ❥✉❧❣❛♠❡♥t♦ s✉❜❥❡❝t✐✈♦ ❞❡ ♣r♦✜s✲

s✐♦♥❛✐s ❡s♣❡❝✐❛❧✐③❛❞♦s ♥❛ ár❡❛✱ ❝✉❥❛ ❞❡❝✐sã♦ ❞❡ ❝♦♥❝❡❞❡r ♦✉ ♥ã♦ ❝ré❞✐t♦ é ❛♣❡♥❛s ❞❛ s✉❛

r❡s♣♦♥s❛❜✐❧✐❞❛❞❡✳ ❙❡❣✉♥❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧✐③❛❞♦ ❞♦

❛✈❛❧✐st❛✱ ♦ s❡✉ ❥✉❧❣❛♠❡♥t♦ s✉❜❥❡❝t✐✈♦✱ ❡ ❛ ❛tr✐❜✉✐çã♦ ❞❡ ♣❡s♦s ❛ ❝❡rt♦s ❢❛❝t♦r❡s✲❝❤❛✈❡

sã♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛♥t❡s ♥❛ ❞❡❝✐sã♦ ❞❛ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦✳

  ❊①✐st❡ ✉♠❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ ❢❛❝t♦r❡s q✉❡ ♣♦❞❡♠ s❡r ❛♥❛❧✐s❛❞♦s✱ ♥♦ ❡♥t❛♥t♦✱ ❛

❛♥á❧✐s❡ ♠❛✐s ❝♦♠✉♠ r❡❣❡✲s❡ ♣❡❧❛s ♣r✐♥❝✐♣❛✐s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ♣r♦♣♦♥❡♥t❡ ❞❡ ❝ré❞✐t♦✳

❊st❛ ❛♥á❧✐s❡ é ❣✉✐❛❞❛ ♣❡❧❛ ❛✈❛❧✐❛çã♦ ❞♦s ❈✬s ❞♦ ❝ré❞✐t♦✿ ❈❛rá❝t❡r✱ ❈❛♣❛❝✐❞❛❞❡✱ ❈❛♣✐✲

t❛❧✱ ❈♦❧❛t❡r❛❧✱ ❈♦♥❞✐çõ❡s ❡ ❈♦♥❣❧♦♠❡r❛❞♦✳ ❆♣ós ❛ ❛♥á❧✐s❡ ❞❛s ✐♥❢♦r♠❛çõ❡s r❡❝♦❧❤✐❞❛s

r❡❧❛t✐✈❛s ❛ ❡st❡s ❢❛❝t♦r❡s✱ ♦ ❛♥❛❧✐st❛ ♣♦♥❞❡r❛✲♦s ❡ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ❝r✐tér✐♦s s✉❜❥❡❝t✐✈♦s

  ✹ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ♣r♦❜❧❡♠❛s ♣❛r❛ ❡st❡ t✐♣♦ ❞❡ ❛♥á❧✐s❡✿ ❝♦♥s✐stê♥❝✐❛ ❡ s✉❜❥❡❝t✐✈✐❞❛❞❡✳

  ❖s ❈✬s ❞♦ ❝ré❞✐t♦ ′ s

❖s ✺ C ❞♦ ❝ré❞✐t♦ ❢♦r❛♠ ❡♥✉♥❝✐❛❞♦s✱ ❡♠ ✶✾✼✷✱ ♣♦r ❲❡st♦♥ ❡ ❇r✐❣❤❛♠✳

  ′ s

❛❝r❡s❝❡♥t❛ ❛♦s ✺ C ✉♠ s❡①t♦ C✱ q✉❡ s❡ r❡❢❡r❡ ❛♦ ❢❛❝t♦r ✏❈♦♥❣❧♦♠❡r❛❞♦✑✱ q✉❡ ❛❝t✉❛❧✲

  ♠❡♥t❡ é ❝♦♥s✐❞❡r❛❞♦ ♥❛s ❛♥á❧✐s❡s ❞❡ ❝ré❞✐t♦ ❞❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s✳

✶✳ ❈❛rá❝t❡r✿ ■♥❢♦r♠❛çõ❡s r❡❢❡r❡♥t❡s à í♥❞♦❧❡✱ ✐❞♦♥❡✐❞❛❞❡ ❡ r❡♣✉t❛çã♦ ❞♦ ❝❧✐❡♥t❡❀

✷✳ ❈❛♣❛❝✐❞❛❞❡✿ ❊st❡ ❈ ❞❡✈❡ ❢♦r♥❡❝❡r ✐♥❢♦r♠❛çõ❡s q✉❡ ♣♦ss✐❜✐❧✐t❡♠ ❛✈❛❧✐❛r s❡ ❛s r❡❝❡✐t❛s ❡ ❞❡s♣❡s❛s ❞♦ ❝❧✐❡♥t❡ ♣❡r♠✐t❡♠ ♦ ❝✉♠♣r✐♠❡♥t♦ ❞❛s ♦❜r✐❣❛çõ❡s ❛ s❡r❡♠

  ❛ss✉♠✐❞❛s❀ ✸✳ ❈❛♣✐t❛❧✿ ■♥❢♦r♠❛çõ❡s r❡❢❡r❡♥t❡s à ❡str✉t✉r❛ ❞❡ ❝❛♣✐t❛❧✱ ❡♥❞✐✈✐❞❛♠❡♥t♦✱ ❧✐q✉✐❞❡③✱ ❧✉❝r♦ ❡ ♦✉tr♦s í♥❞✐❝❡s ✜♥❛♥❝❡✐r♦s ♦❜t✐❞♦s ♣♦r ♠❡✐♦ ❞♦s ❞❡♠♦♥str❛t✐✈♦s ✜♥❛♥❝❡✐r♦s ❞♦ ❝❧✐❡♥t❡❀

  ✹✳ ❈♦❧❛t❡r❛❧✿ ➱ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❛ ♣❡ss♦❛ ✐♥❞✐✈✐❞✉❛❧ ♦✉ ❝♦❧❡❝t✐✈❛ ♣❛r❛ ♦❢❡r❡❝❡r ❣❛r❛♥✲ t✐❛s ❛♦ ❡♠♣rést✐♠♦❀ ✺✳ ❈♦♥❞✐çõ❡s✿ ■♥❢♦r♠❛çõ❡s r❡❢❡r❡♥t❡s à ❝❛♣❛❝✐❞❛❞❡ ❞♦ ❝❧✐❡♥t❡ ♦✉ ❞♦s ❛❞♠✐♥✐str❛❞♦r❡s ❡♠♣r❡s❛r✐❛✐s ❞❡ s❡ ❛❞❛♣t❛r❡♠ ❛ s✐t✉❛çõ❡s ❝♦♥❥✉♥t✉r❛✐s✱ t❡r ❛❣✐❧✐❞❛❞❡ ❡ ✢❡①✐❜✐❧✐✲

  ❞❛❞❡ ❞❡ s❡ ❛❞❛♣t❛r ❡ ❝r✐❛r ♠❡❝❛♥✐s♠♦s ❞❡ ❞❡❢❡s❛❀ ✻✳ ❈♦♥❣❧♦♠❡r❛❞♦✿ ■♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ s✐t✉❛çã♦ ❞❡ ♦✉tr❛s ❡♠♣r❡s❛s s✐t✉❛❞❛s ♥♦ ♠❡s♠♦ ❣r✉♣♦ ❡❝♦♥ó♠✐❝♦ ❡ ❝♦♠♦ ♣♦❞❡rã♦ ❛❢❡❝t❛r ❛ ❡♠♣r❡s❛ ❡♠ ❡st✉❞♦✳

  ✶✳✶✳✺ ▼♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣

❖s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ♣❡r♠✐t❡♠✱ r❡❝♦rr❡♥❞♦ ❛ té❝♥✐❝❛s ❡st❛tíst✐❝❛s✱ ❡st❛❜❡❧❡❝❡r

✉♠ ♣r♦❝❡ss♦ ❞❡ ❛tr✐❜✉✐çã♦ ❞❡ ♣♦♥t✉❛çõ❡s às ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ ❞❡ ❝r❡❞✐t♦✳ ❊st❡

♣r♦❝❡ss♦ ♣❡r♠✐t❡ ❡st✐♠❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ❞❛❞♦ ❝❧✐❡♥t❡ s❡r ✉♠ ❝❧✐❡♥t❡ ❝✉♠♣r✐❞♦r

♦✉ ✐♥❝✉♠♣r✐❞♦r✳

  ❊st❡s ♠♦❞❡❧♦s ✈✐s❛♠ ♦ ✐s♦❧❛♠❡♥t♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s q✉❡ ♣❡r♠✐t❡♠ ❞✐st✐♥❣✉✐r ♦s ❜♦♥s ❝ré❞✐t♦s ❞♦s ♠❛✉s ❝ré❞✐t♦s✳ ❖s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ sã♦ s✐st❡♠❛s q✉❡ ♦s ❝r❡❞♦r❡s ✉t✐❧✐③❛♠ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ s✉❛ ❞❡❝✐sã♦ q✉❛♥t♦ à ❝♦♥❝❡ssã♦ ♦✉ ♥ã♦ ❞❡ ✉♠ ❝ré❞✐t♦ ❛ ✉♠ ❝❧✐❡♥t❡✳ P❛r❛ ♦s ♠♦❞❡❧♦s tr❛❞✐❝✐♦♥❛✐s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❛tr✐❜✉❡♠

♣❡s♦s✱ ❞❡t❡r♠✐♥❛❞♦s ❡st❛t✐st✐❝❛♠❡♥t❡✱ ❛ ❛❧❣✉♥s ❛tr✐❜✉t♦s ❞♦ s♦❧✐❝✐t❛♥t❡✱ ❞❡ ♠♦❞♦ ❛

  ❝r✐❛r ✉♠ s❝♦r❡ ❞❡ ❝ré❞✐t♦✳ ❙❡❣✉♥❞♦ ♦s s✐st❡♠❛s ❞❡ ♣♦♥t✉❛çã♦ ❞❡ ❝ré❞✐t♦ ❡♥❝♦♥tr❛♠✲s❡

❡♠ q✉❛s❡ t♦❞♦s ♦s t✐♣♦s ❞❡ ❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦✳ ❖ ♦❜❥❡❝t✐✈♦ é ❣❡r❛❧♠❡♥t❡ ♦ ♠❡s♠♦❀ ♣ré✲

✐❞❡♥t✐✜❝❛r ❢❛❝t♦r❡s✲❝❤❛✈❡ q✉❡ ❞❡t❡r♠✐♥❡♠ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✱ ❡ ❛ ❝♦♠❜✐♥❛çã♦

  

✶✳✶ ▼♦❞❡❧♦s ❞❡ ❆♥á❧✐s❡ ❞❡ ❈ré❞✐t♦ ✲ ●❡♥❡r❛❧✐❞❛❞❡s ✺

❆ ♠❡t♦❞♦❧♦❣✐❛ ❜ás✐❝❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❞❡✈❡ t❡r ❡♠ ❝♦♥t❛✱ s❡❣✉♥❞♦ ❛s s❡❣✉✐♥t❡s ❡t❛♣❛s✿ ❼ P❧❛♥❡❛♠❡♥t♦ ❡ ❞❡✜♥✐çõ❡s✿ ♠❡r❝❛❞♦s ❡ ♣r♦❞✉t♦s ❞❡ ❝ré❞✐t♦ ♣❛r❛ ♦s q✉❛✐s ♦ s✐st❡♠❛ s❡rá ❞❡s❡♥✈♦❧✈✐❞♦❀ ✜♥❛❧✐❞❛❞❡s ❞❡ ✉s♦❀ t✐♣♦s ❞❡ ❝❧✐❡♥t❡s❀ ❝♦♥❝❡✐t♦ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ❛ s❡r ❛❞♦♣t❛❞♦❀ ❤♦r✐③♦♥t❡ ❞❡ ♣r❡✈✐sã♦ ❞♦ ♠♦❞❡❧♦❀

  ❼ ■❞❡♥t✐✜❝❛çã♦ ❞❛s ✈❛r✐á✈❡✐s✿ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ ♣r♦♣♦♥❡♥t❡ ❛♦ ❝ré❞✐t♦❀ ❝❛r❛❝t❡r✐③❛✲ çã♦ ❞❛ ♦♣❡r❛çã♦❀ s❡❧❡❝çã♦ ❞❛s ✈❛r✐á✈❡✐s s✐❣♥✐✜❝❛t✐✈❛s ♣❛r❛ ♦ ♠♦❞❡❧♦❀ ❛♥á❧✐s❡ ❞❛s r❡str✐çõ❡s ❛ s❡r❡♠ ❝♦♥s✐❞❡r❛❞❛s ❡♠ r❡❧❛çã♦ às ✈❛r✐á✈❡✐s❀ ❼ P❧❛♥❡❛♠❡♥t♦ ❛♠♦str❛❧ ❡ ❝♦❧❡❝t❛ ❞❡ ❞❛❞♦s✿ s❡❧❡❝çã♦ ❡ ❞✐♠❡♥s✐♦♥❛♠❡♥t♦ ❞❛ ❛♠♦s✲ tr❛❀ ❝♦❧❡❝t❛ ❞♦s ❞❛❞♦s❀ ♠♦♥t❛❣❡♠ ❞❛ ❜❛s❡ ❞❡ ❞❛❞♦s❀ ❼ ❉❡t❡r♠✐♥❛çã♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❛tr❛✈és ❞❡ té❝♥✐❝❛s ❡st❛tíst✐❝❛s✿ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐♥❛♥t❡ ♦✉ ❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛❀ ❼ ❉❡t❡r♠✐♥❛çã♦ ❞♦ ♣♦♥t♦ ❞❡ ❝♦rt❡✱ ❛ ♣❛rt✐r ❞♦ q✉❛❧ ♦ ❝❧✐❡♥t❡ é ❝❧❛ss✐✜❝❛❞♦ ❝♦♠♦ ❝✉♠♣r✐❞♦r ♦✉ ❜♦♠ ♣❛❣❛❞♦r❀ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ é ♦ ♣♦♥t♦ ❛ ♣❛rt✐r ❞♦ q✉❛❧ ❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ ♣♦❞❡ ❛♣r♦✈❛r ❛ ❝♦♥❝❡ssã♦ ❞♦ ❝ré❞✐t♦❀

  ❆ ❞✐❢❡r❡♥ç❛ ♠❛✐s ❛❝❡♥t✉❛❞❛ ❡♥tr❡ ♦s ♠♦❞❡❧♦s s✉❜❥❡❝t✐✈♦s ❡ ♦ ♠♦❞❡❧♦ ❞❡ ❈r❡❞✐t

❙❝♦r✐♥❣ r❡s✐❞❡ ♥♦ ❢❛❝t♦ ❞❡✱ ♥❡ss❡ ú❧t✐♠♦✱ s❡ ✈❛❧♦r✐③❛r ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♠ét♦❞♦s ❡st❛tíst✐❝♦s

q✉❡ ♣❡r♠✐t❡♠ ❛ s❡❧❡❝çã♦ ❞♦s ❢❛❝t♦r❡s✲❝❤❛✈❡ ❡ ❞♦s r❡s♣❡❝t✐✈♦s ♣❡s♦s✱ ❛ ♣❛rt✐r ❞♦s q✉❛✐s

s❡ ♦❜té♠ ✉♠❛ ♣♦♥t✉❛çã♦ ♣❛r❛ ❝❛❞❛ ❝❧✐❡♥t❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s s✉❛s ❝❛r❛❝t❡ríst✐❝❛s✳ ❆

♣♦♥t✉❛çã♦ ❣❡r❛❞❛✱ ❢♦r♥❡❝❡ ✐♥❞✐❝❛❞♦r❡s q✉❛♥t✐t❛t✐✈♦s ❞❛s ❤✐♣ót❡s❡s ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦

❞❡ss❡ ❝❧✐❡♥t❡ ❡ r❡♣r❡s❡♥t❛ ♦ r✐s❝♦ ❞❡ ♣❡r❞❛✳ ❊st❛ ♣♦♥t✉❛çã♦ ♣♦❞❡ s❡r ❝♦♠♣❛r❛❞❛ ❝♦♠ ✉♠

♣♦♥t♦ ❞❡ ❝♦rt❡ ♦✉ ❝♦♠ ✉♠❛ ♣♦♥t✉❛çã♦ ♠í♥✐♠❛ ❛❝❡✐tá✈❡❧ ❛ ♣❛rt✐r ❞❛ q✉❛❧ ❛ ✐♥st✐t✉✐çã♦

✜♥❛♥❝❡✐r❛ ❛♣r♦✈❛ ♦✉ ♥ã♦ ❛ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦✳

  r❡❢❡r❡ q✉❡ ♦ ♣r✐♠❡✐r♦ ♠♦❞❡❧♦ ❡st❛tíst✐❝♦ ❞❡ ❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦ r❡♠♦♥t❛

❛♦ ❛♥♦ ❞❡ ✶✾✹✺✳ ❖s ♣r✐♠❡✐r♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❞❡st✐♥❛✈❛♠✲s❡ ❛♦ ❝ré❞✐t♦ ❛♦

❝♦♥s✉♠♦✳ ❆ ❡①♣❛♥sã♦ ❞♦ ♠❡r❝❛❞♦ ❞❡ ❝ré❞✐t♦ ♠❛ss✐✜❝❛❞♦ ♦❜r✐❣♦✉ ♦s ❛♥❛❧✐st❛s ❛ ✉♠❛

♠❛✐♦r r❛♣✐❞❡③ ❡ ❤♦♠♦❣❡♥❡✐❞❛❞❡ ♥♦ tr❛t❛♠❡♥t♦ ❞♦s s❡✉s ❝❧✐❡♥t❡s ❡✱ ♣♦r ✐ss♦✱ ❛ ✉♠ ❛✉✲

♠❡♥t♦ ❞❛ ✉t✐❧✐③❛çã♦ ❞❡st❡s ♠♦❞❡❧♦s✳ ❆ ❡✈♦❧✉çã♦ ❞♦s s✐st❡♠❛s ✐♥❢♦r♠át✐❝♦s ♣♦ss✐❜✐❧✐t♦✉

♦ tr❛t❛♠❡♥t♦ ❡st❛tíst✐❝♦ ❛❞❡q✉❛❞♦ ❛ ❡ss❡ ❛✉♠❡♥t♦ ❞❡ ❞❛❞♦s✳

  ❆✐♥❞❛ q✉❡ ♦ ✉s♦ ❞❡ ♠ét♦❞♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ s❡❥❛ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞✐r❡❝❝✐♦♥❛❞♦

♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s s♦❜r❡ ❛ ❝♦♥❝❡ssã♦ ♦✉ ♥ã♦ ❞❡ ✉♠ ❝ré❞✐t♦✱ ❛❧❣✉♠❛s ✐♥st✐t✉✐çõ❡s

t❛♠❜é♠ ♦s ✉s❛♠ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ♠♦♥t❛♥t❡ ❞♦ ❝ré❞✐t♦ ❛ s❡r ❝♦♥❝❡❞✐❞♦✱ ❝♦♠♦ r❡❢❡r❡

  ❆s ❝❧❛ss✐✜❝❛çõ❡s ❞❡st❡s ♠♦❞❡❧♦s sã♦ ❝❧❛ss✐✜❝❛çõ❡s ♥✉♠ér✐❝❛s ♦✉ q✉❛♥t✐t❛t✐✈❛s✳ ❆

♣♦♥t✉❛çã♦ ❞❡ ❝ré❞✐t♦ é ✉♠ ♥ú♠❡r♦ q✉❡ r❡s✉❧t❛ ❞❡ ✉♠ ❡st✉❞♦ ❜❛s❡❛❞♦ ♥♦ ❤✐stór✐❝♦ ❞❡

♣❛❣❛♠❡♥t♦s✱ ♥♦ ♣❡r✜❧ ❞❡ ❞í✈✐❞❛ ❞♦ ❞❡✈❡❞♦r ❡ ♥♦s ❞❛❞♦s ❡st❛tíst✐❝♦s ❞❡ ♦✉tr♦s ❞❡✈❡✲

❞♦r❡s q✉❡ ♦s ❝r❡❞♦r❡s ✉t✐❧✐③❛♠ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r❡♠ ❝❡rt♦s

  ✻ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

♣♦♥t✉❛çã♦ ❞❡ ❝ré❞✐t♦ t❡♠ ✉♠ ✐♠♣❛❝t♦ ✐♠♣♦rt❛♥t❡ ♥♦ s♣r❡❛❞ q✉❡ s❡rá ❛tr✐❜✉í❞♦ q✉❛♥❞♦

s❡ ❝♦♥❝❡❞❡ ♦ ❝ré❞✐t♦✳ ▼❛✐s r❡❝❡♥t❡♠❡♥t❡✱ ♦ ❈r❡❞✐t ❙❝♦r✐♥❣ s❡r✈❡ ❞❡ ❜❛s❡ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠♦❞❡❧♦s ♠❛✐s ❝♦♠♣❧❡①♦s ❜❛s❡❛❞♦s ♥❛ t❡♦r✐❛ ❞❡ ❝❛rt❡✐r❛s✳ ◆♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ s✉♣õ❡✲s❡ q✉❡ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ❝❧✐❡♥t❡s q✉❡ ❡♥✲

tr❛rã♦ ❡♠ ❞❡❢❛✉❧t ♥♦ ❢✉t✉r♦ sã♦ s❡♠❡❧❤❛♥t❡s às ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ❝❧✐❡♥t❡s q✉❡ ❡♥tr❛r❛♠

❡♠ ❞❡❢❛✉❧t ♥♦ ♣❛ss❛❞♦✳ ❚❡♥❞♦ ❡♠ ❝♦♥t❛ ❡st❡ ♣r❡ss✉♣♦st♦✱ é ❝♦♠✉♠ ✉t✐❧✐③❛r✲s❡ ❛♠♦s✲

tr❛s ❞❡ ❝❧✐❡♥t❡s ❝✉♠♣r✐❞♦r❡s ❡ ✐♥❝✉♠♣r✐❞♦r❡s ❞❛ ✐♥st✐t✉✐çã♦ ❜❛♥❝ár✐❛ ❡ ❛♣❧✐❝❛r té❝♥✐❝❛s

❡st❛tíst✐❝❛s ❛♣r♦♣r✐❛❞❛s ♣❛r❛ ✐♥❢❡r✐r s♦❜r❡ ♦s ✐♥❞í❝✐♦s ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❝❧✐❡♥t❡

❡♠ ♣❛rt✐❝✉❧❛r✳ ❊♥tr❡ ❛s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ ❞❡ ❝ré❞✐t♦ ❡s❝♦❧❤✐❞❛s✱ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉❡♠ ♣❛r❛ ♦ ❛✉♠❡♥t♦ ♦✉tr❛s ♣❛r❛ ❛ ❞✐♠✐♥✉✐çã♦ ❞♦ ♥í✈❡❧ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦✳ ◆♦ â♠❜✐t♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ♣❛r❛ ❛ ❛✈❛❧✐❛çã♦ ❞❡ r✐s❝♦

❞❡ ❝ré❞✐t♦ ❡ ✐♥s♦❧✈ê♥❝✐❛ ❞❡ ❡♠♣r❡s❛s✱ ❛s ✈❛r✐á✈❡✐s ♠❛✐s ✉t✐❧✐③❛❞❛s sã♦ ♦s í♥❞✐❝❡s ❞❡

❝♦♥t❛❜✐❧✐❞❛❞❡ ❡ ✜♥❛♥❝❡✐r♦s ❝♦♠♦ ❛ r❡♥t❛❜✐❧✐❞❛❞❡✱ ❛ ❧✐q✉✐❞❡③ ❡ ♦ ❡♥❞✐✈✐❞❛♠❡♥t♦✳

  r❡❢❡r❡ q✉❡ ♦ ♠♦❞❡❧♦ ❩✲❙❝♦r❡ é ✉♠ ❡①❡♠♣❧♦ ❝❧áss✐❝♦ ❞❡ ♠♦✲ ❞❡❧♦s ❞❡ ♣r❡✈✐sã♦ ❞❡ ✐♥s♦❧✈ê♥❝✐❛ ❞❡ ❡♠♣r❡s❛s✳ ❩✲❙❝♦r❡

❈♦♠ ❜❛s❡ ❡♠ s❛❜❡✲s❡ q✉❡ ❛ ❢ór♠✉❧❛ ❩✲❙❝♦r❡ ♣❛r❛ ♣r❡✈❡r ❛ ❢❛❧ê♥❝✐❛

❢♦✐ ♣✉❜❧✐❝❛❞❛ ❡♠ ✶✾✻✽ ♣♦r ❊❞✇❛r❞ ■✳ ❆❧t♠❛♥✱ q✉❡ ❡r❛✱ ♥❛ é♣♦❝❛✱ ✉♠ ♣r♦❢❡ss♦r ❛❞❥✉♥t♦

❞❡ ❋✐♥❛♥ç❛s ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ◆♦✈❛ ❨♦r❦✳ ❆ ❢ór♠✉❧❛ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ♣r❡✈❡r ❛

♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❡♠♣r❡s❛ ✐r à ❢❛❧ê♥❝✐❛ ♥♦ ♣r❛③♦ ❞❡ ❞♦✐s ❛♥♦s✳ ❖ ❩✲❙❝♦r❡ ✉t✐❧✐③❛ ♦ r❡♥❞✐♠❡♥t♦ ❞❛s s♦❝✐❡❞❛❞❡s ♠ú❧t✐♣❧❛s ❡ ✈❛❧♦r❡s ❞♦ ❜❛❧❛♥ç♦ ♣❛✲ tr✐♠♦♥✐❛❧ ♣❛r❛ ♠❡❞✐r ❛ s❛ú❞❡ ✜♥❛♥❝❡✐r❛ ❞❡ ✉♠❛ ❡♠♣r❡s❛✳ ❖ ❩✲❙❝♦r❡ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ q✉❛tr♦ ♦✉ ❝✐♥❝♦ rá❝✐♦s ✜♥❛♥❝❡✐r♦s ❝♦♠✉♥s✱

♣♦♥❞❡r❛❞❛ ♣♦r ❝♦❡✜❝✐❡♥t❡s✳ ❖s ❝♦❡✜❝✐❡♥t❡s ❢♦r❛♠ ❡st✐♠❛❞♦s ❛tr❛✈és ❞❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡

✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡♠♣r❡s❛s q✉❡ t✐♥❤❛ ❞❡❝❧❛r❛❞♦ ❢❛❧ê♥❝✐❛ ❡ ❞❡ ✉♠❛ ❛♠♦str❛ ❞❡ ❡♠♣r❡s❛s

q✉❡ s♦❜r❡✈✐✈❡r❛♠✱ r❡❧❛❝✐♦♥❛❞❛s ✐♥❞✉str✐❛❧♠❡♥t❡ ❡ ❞❡ t❛♠❛♥❤♦ ❛♣r♦①✐♠❛❞♦ ✭❛❝t✐✈♦s✮✳

◆♦ ♠♦❞❡❧♦ ❩✲❙❝♦r❡ ❝♦♥s✐❞❡r❛✲s❡✿

  1 ❼ X ✿ ❲♦r❦✐♥❣ ❈❛♣✐t❛❧✴❚♦t❛❧ ❞❡ ❆❝t✐✈♦s

  2 ❼ X ✿ ▲✉❝r♦ ❝♦♥s❡r✈❛❞♦ ✴ ❚♦t❛❧ ❞❡ ❆❝t✐✈♦s

  3 ❼ X ✿ ❊❇■❚ ✴ ❚♦t❛❧ ❞❡ ❆❝t✐✈♦s

  4 ❼ X ✿ ▼❡r❝❛❞♦ ✴ ❉í✈✐❞❛

  5 ❼ X ✿ ❱❡♥❞❛s ✴ ❚♦t❛❧ ❞❡ ❆❝t✐✈♦s ❛ ♣❛rt✐r ❞❛s q✉❛✐s s❡ ❡st❛❜❡❧❡❝❡ ❛ ❢✉♥çã♦ ✏❙❝♦r❡✑ ♣❛r❛ ❝❛❞❛ ❞❡✈❡❞♦r✿

  Z = 1.2X 1 + 1.4X 2 + 3.3X 3 + 0.6X 4 + 0.999X

  5

  

✶✳✶ ▼♦❞❡❧♦s ❞❡ ❆♥á❧✐s❡ ❞❡ ❈ré❞✐t♦ ✲ ●❡♥❡r❛❧✐❞❛❞❡s ✼

❡ ♣❛r❛ ❛ q✉❛❧ s❡ ❞❡t❡r♠✐♥❛♠ ❛s ③♦♥❛s ❞❡ ❞❡❝✐sã♦✿ ❼ Z > 2.99 ✲❩♦♥❛ ✏❙❡❣✉r❛✑ ❼ 1.8 < Z < 2.99 ✲ ❩♦♥❛ ✏■♥t❡r♠é❞✐❛✑ ❼ Z < 1.80 ✲❩♦♥❛ ❞❡ ✏❙tr❡ss✑ ❖s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❞✐✈✐❞❡♠✲s❡ ❛✐♥❞❛ ❡♠ ❞✉❛s ❝❛t❡❣♦r✐❛s✿ ♦s ▼♦❞❡❧♦s ❞❡

  

❆♣r♦✈❛çã♦ ❞❡ ❈ré❞✐t♦ ✭❈r❡❞✐t ❙❝♦r✐♥❣ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦✮ ❡ ♦s ▼♦❞❡❧♦s ❞❡ ❈❧❛ss✐✜❝❛çã♦

❈♦♠♣♦rt❛♠❡♥t❛❧✱ ❡st❡s ú❧t✐♠♦s ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❜❡❤❛✈✐♦✉r❛❧ s❝♦r✐♥❣✱ ❝♦♥❢♦r♠❡ r❡❢❡r❡

❇❡❤❛✈✐♦✉r❛❧ ❙❝♦r✐♥❣

❙❡❣✉♥❞♦ ♦ ❜❡❤❛✈✐♦✉r❛❧ s❝♦r✐♥❣ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ♣r❡✈✐sã♦

❞❡ ❡✈❡♥t♦s ❛ss♦❝✐❛❞♦s ❛♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ❝♦♠♦ ♦ ✐♥❝✉♠♣r✐♠❡♥t♦ ❡ ♦s ♣❛❣❛♠❡♥t♦s ❡♠

❞✐❛✱ ❡♥tr❡ ♦✉tr❛s ❝❛r❛❝t❡ríst✐❝❛s✳ ❖ ❜❡❤❛✈✐♦✉r❛❧ s❝♦r✐♥❣ t❡♠ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦s ❛s♣❡❝t♦s

❝♦♠♣♦rt❛♠❡♥t❛✐s ❡ ❛s ❛❝t✐✈✐❞❛❞❡s ❞♦s ❝❧✐❡♥t❡s ❞❛ ✐♥st✐t✉✐çã♦✱ ♦✉ s❡❥❛✱ ❛✉①✐❧✐❛♠ ❛ ❣❡stã♦

❞♦s ❝ré❞✐t♦s ❞♦s ❝❧✐❡♥t❡s q✉❡ ❥á ♣♦ss✉❡♠ ❝ré❞✐t♦s ♥❛ ✐♥st✐t✉✐çã♦✳ ◆♦s ♠♦❞❡❧♦s ❞❡ ❜❡❤❛✈✐♦✉r❛❧ s❝♦r✐♥❣ ❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ ❛♥❛❧✐s❛ ♦ ❝♦♠♣♦rt❛✲

♠❡♥t♦ ❞♦s ❝❧✐❡♥t❡s ❡♠ ♦♣❡r❛çõ❡s ❛♥t❡r✐♦r❡s✳ ❊st❡s ♠♦❞❡❧♦s ✈✐s❛♠ ❡st✐♠❛r ❛ ♣r♦❜❛❜✐✲

❧✐❞❛❞❡ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❝❧✐❡♥t❡ q✉❡ ❥á ♣♦ss✉✐ ✉♠ ♣r♦❞✉t♦ ♦✉ ✉♠ ❝ré❞✐t♦ ❝♦♠

❛ ♠❡s♠❛✱ ❡♥q✉❛♥t♦ q✉❡ ♥♦s ♠♦❞❡❧♦s ❞❡ ❛♣r♦✈❛çã♦ ❞❡ ❝ré❞✐t♦ ❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛

♥ã♦ t❡♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❝❧✐❡♥t❡✳ ❖ ♥ú♠❡r♦ ❞❡ ❛tr❛s♦s ♥♦ ú❧t✐♠♦

❛♥♦ ❡ ♦ ✈♦❧✉♠❡ ❞❡ tr❛♥s❛❝çõ❡s ❞♦ ❝❧✐❡♥t❡ ❝♦♥tr✐❜✉✐ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦

r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❝❧✐❡♥t❡✳

  ❖s ♠♦❞❡❧♦s ❞❡ ❛♣r♦✈❛çã♦ ❞❡ ❝ré❞✐t♦ sã♦ ♦r✐❡♥t❛❞♦s ❡ss❡♥❝✐❛❧♠❡♥t❡ ♣❛r❛ ❛ ❝♦♥❝❡ssã♦

❡ ✈♦❧✉♠❡ ❞❡ ❝ré❞✐t♦✱ ❡♥q✉❛♥t♦ q✉❡ ♦s ♠♦❞❡❧♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❝♦♠♣♦rt❛♠❡♥t❛❧ ✈✐s❛♠

♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ❣❡stã♦ ❞♦s ❧✐♠✐t❡s ❞❡ ❝ré❞✐t♦✱ ❛s ❝♦❜r❛♥ç❛s ♣r❡✈❡♥t✐✈❛s ❡ ♦✉tr❛s ❡str❛✲

té❣✐❛s✳

  ❆s té❝♥✐❝❛s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ sã♦ ❝♦♥❤❡❝✐❞❛s ❡ ✉t✐❧✐③❛❞❛s ❡♠ ♣r❛t✐❝❛♠❡♥t❡ t♦❞❛s ❛s

✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s✳ ❖ ✉s♦ ❞❛s t❡❝♥♦❧♦❣✐❛s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❛✈❛♥ç❛❞❛s ♣❡r♠✐t❡✱ ❛♦

✉s✉ár✐♦ ❝♦♠✉♠✱ ♦ ❛❝❡ss♦ ❛♦s ♣♦❞❡r♦s♦s s✐st❡♠❛s ❞❡ s✐♠✉❧❛çã♦✱ ❝❛♣❛③❡s ❞❡ ♠♦❞❡❧❛r ❛

❡①♣♦s✐çã♦ ❛♦ ❝ré❞✐t♦✳

  ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ✐♥❢♦r♠át✐❝♦ ♣❡r♠✐t✐✉ ♦ ✉s♦ ❞❡ ♦✉tr❛s té❝♥✐❝❛s ♥❛ ❝♦♥str✉çã♦ ❞❡

s✐st❡♠❛s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✳ ◆♦s ❛♥♦s ✽✵ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ❛s té❝♥✐❝❛s ❞❡ r❡❣r❡ssã♦

❧♦❣íst✐❝❛ ❡ r❡❣r❡ssã♦ ❧✐♥❡❛r ♠ú❧t✐♣❧❛ ❡ q✉❡ sã♦✱ ❛❝t✉❛❧♠❡♥t❡✱ ❛s ❞✉❛s ♣r✐♥❝✐♣❛✐s té❝♥✐❝❛s

✉t✐❧✐③❛❞❛s ♥❛ ❝♦♥str✉çã♦ ❞♦s ♠♦❞❡❧♦s✳ ▼❛✐s r❡❝❡♥t❡♠❡♥t❡✱ ❛s té❝♥✐❝❛s ❞❡ ✐♥t❡❧✐❣ê♥✲

❝✐❛ ❛rt✐✜❝✐❛❧✱ ❝♦♠♦ ❛s r❡❞❡s ♥❡✉r♦♥❛✐s ❛rt✐✜❝✐❛✐s✱ ❢♦r❛♠ t❛♠❜é♠ ✐♠♣❧❡♠❡♥t❛❞❛s ❝♦♠

s✉❝❡ss♦✳ ❱❛♥t❛❣❡♥s ❞♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣

  ✽ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ❼ ❈♦♥s✐stê♥❝✐❛✿ sã♦ ♠♦❞❡❧♦s ❜❡♠ ❡❧❛❜♦r❛❞♦s✱ q✉❡ ✉t✐❧✐③❛♠ ❛ ❡①♣❡r✐ê♥❝✐❛ ❞❛ ✐♥s✲ t✐t✉✐çã♦✱ ❡ ❛❥✉❞❛♠ ♥❛ ❣❡stã♦ ❞♦s ❝ré❞✐t♦s ❞❡ ❝❧✐❡♥t❡s ❥á ❡①✐st❡♥t❡s ❡ ❞❡ ♥♦✈♦s s♦❧✐❝✐t❛♥t❡s❀

  ❼ ❋❛❝✐❧✐❞❛❞❡✿ ♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ✈✐s❛♠ ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❡ ❛ ❢á❝✐❧ ✐♥t❡r♣r❡✲ t❛çã♦✱ ❝♦♠ ✐♥st❛❧❛çã♦ r❡❧❛t✐✈❛♠❡♥t❡ ❢á❝✐❧❀ ❼ ▼❡❧❤♦r ♦r❣❛♥✐③❛çã♦ ❞❛ ✐♥❢♦r♠❛çã♦ ❞❡ ❝ré❞✐t♦✿ ❛ s✐st❡♠❛t✐③❛çã♦ ❡ ♦r❣❛♥✐③❛çã♦ ❞❛s ✐♥❢♦r♠❛çõ❡s ❝♦♥tr✐❜✉❡♠ ♣❛r❛ ❛ ♠❡❧❤♦r✐❛ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦❀ ❼ ❘❡❞✉çã♦ ❞❛ ♠❡t♦❞♦❧♦❣✐❛ s✉❜❥❡❝t✐✈❛✿ ♦ ✉s♦ ❞♦ ♠ét♦❞♦ q✉❛♥t✐t❛t✐✈♦ ❝♦♠ r❡❣r❛s ❝❧❛r❛s ❡ ❜❡♠ ❞❡✜♥✐❞❛s ❝♦♥tr✐❜✉✐ ♣❛r❛ ❛ ❞✐♠✐♥✉✐çã♦ ❞❛ s✉❜❥❡❝t✐✈✐❞❛❞❡ ❞❛ ❛✈❛❧✐❛çã♦

  ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦❀ ❼ ▼❛✐♦r ❡✜❝✐ê♥❝✐❛ ❞♦ ♣r♦❝❡ss♦✿ ♦ ✉s♦ ❞❡ ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ♥❛ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦ ❞✐r❡❝❝✐♦♥❛ ♦s ❡s❢♦rç♦s ❞♦s ❛♥❛❧✐st❛s✱ tr❛③❡♥❞♦ r❡❞✉çã♦ ❞❡ t❡♠♣♦ ❡ ♠❛✐♦r ❡✜❝✐ê♥❝✐❛ ❛ ❡st❡ ♣r♦❝❡ss♦✳

  ❉❡s✈❛♥t❛❣❡♥s ❡ ❧✐♠✐t❛çõ❡s ❞♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣

P❛r❛ ❛ ♣r✐♥❝✐♣❛❧ ❧✐♠✐t❛çã♦ ❞♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ é ♦ ❛s♣❡❝t♦

t❡♠♣♦r❛❧ ❞❛ ❛♠♦str❛✿

  ✏❖ t❡♠♣♦ ✭❛ é♣♦❝❛✮ é ✉♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❧✐♠✐t❛çõ❡s ❛♣r❡s❡♥t❛❞❛s ♣❡❧♦s ♠♦❞❡❧♦s ❞❡s❡♥✈♦❧✲

✈✐❞♦s ❛ ♣❛rt✐r ❞♦ ✉s♦ ❞❡ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐♥❛♥t❡✳ ❈♦♠ ♦ ❞❡❝♦rr❡r ❞♦ t❡♠♣♦✱ t❛♥t♦ ❛s ✈❛r✐á✈❡✐s

q✉❛♥t♦ s❡✉s ♣❡s♦s r❡❧❛t✐✈♦s s♦❢r❡♠ ❛❧t❡r❛çõ❡s✳ ❆s ✈❛r✐á✈❡✐s q✉❡✱ s❡❣✉♥❞♦ ❛ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐✲

♥❛♥t❡✱ sã♦ ❛s q✉❡ ♠❡❧❤♦r ❝❧❛ss✐✜❝❛♠ s♦❜ ❞❡t❡r♠✐♥❛❞❛ ❝♦♥❥✉♥t✉r❛ ❡❝♦♥ó♠✐❝❛✱ ♣♦❞❡♠ ♥ã♦ ♦ s❡r

❡♠ ♦✉tr❛ s✐t✉❛çã♦✳✑

  t❛♠❜é♠ r❡❢❡r❡♠ ♦ ❛s♣❡❝t♦ t❡♠♣♦r❛❧ ❝♦♠♦ ❛ ♣r✐♥❝✐♣❛❧ ❧✐♠✐✲ t❛çã♦✿ ✏❯♠ ♠♦❞❡❧♦ ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ♣♦❞❡ ❞❡❣r❛❞❛r✲s❡ ❝♦♠ t❡♠♣♦ s❡ ❛ ♣♦♣✉❧❛çã♦ ❡♠ q✉❡ ❡❧❡ é ❛♣❧✐❝❛❞♦ ❞✐✈❡r❣❡ ❞❛ ♣♦♣✉❧❛çã♦ ♦r✐❣✐♥❛❧ q✉❡ ❢♦✐ ✉s❛❞❛ ♣❛r❛ ❝♦♥str✉✐r ♦ ♠♦❞❡❧♦✑✳ t❛♠❜é♠ ❢❛❧❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❞❡s✈❛♥t❛❣❡♥s ❞♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦✲ r✐♥❣✿ ❼ ❈✉st♦ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✿ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ s✐st❡♠❛s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❛❝❛r✲ r❡t❛ ❝✉st♦s ❛♦ ♥í✈❡❧ ❞❛ s✉❛ ❝♦♥str✉çã♦ ❡ ♠❛♥✉t❡♥çã♦✳ ❼ ❊①❝❡ss♦ ❞❡ ❝♦♥✜❛♥ç❛ ♥♦s ♠♦❞❡❧♦s✿ ❛❧❣✉♠❛s ❡st❛tíst✐❝❛s ♣♦❞❡♠ ❡st✐♠❛r ♣♦r ✈❛✲ ❧♦r❡s s✉♣❡r✐♦r❡s ❛ ❡✜❝á❝✐❛ ❞♦s ♠♦❞❡❧♦s✱ ♣r♦✈♦❝❛♥❞♦ ✉♠ ❡①❝❡ss♦ ❞❡ ❝♦♥✜❛♥ç❛ ♥♦s

  ♠❡s♠♦s ♣♦r ♣❛rt❡ ❞♦s ✉t✐❧✐③❛❞♦r❡s✱ ♣♦✐s ♦s ♠❡♥♦s ❡①♣❡r✐❡♥t❡s ❝♦♥s✐❞❡r❛♠✲♥♦s ♣❡r❢❡✐t♦s ❡ ♥ã♦ ♣õ❡♠ ❡♠ ❝❛✉s❛ ♦ s❡✉ r❡s✉❧t❛❞♦❀ ❼ ❋❛❧t❛ ❞❡ ❞❛❞♦s ❛❞❡q✉❛❞♦s✿ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❞❛❞♦s ♥ã♦ ❢❛❝✉❧t❛❞♦s ♣♦❞❡ ♦r✐❣✐♥❛r ♣r♦❜❧❡♠❛s ♥❛ ✉t✐❧✐③❛çã♦ ❞♦s ♠♦❞❡❧♦s ❡ ❣❡r❛r r❡s✉❧t❛❞♦s ❞✐❢❡r❡♥t❡s ❞♦s ❡s♣❡r❛❞♦s✳

  ➱ ♥❡❝❡ssár✐♦ ❛♥❛❧✐s❛r ❛ q✉❛❧✐❞❛❞❡ ❡ ❛ ✜❞❡❞✐❣♥✐❞❛❞❡ ❞❛s ✐♥❢♦r♠❛çõ❡s ❞✐s♣♦♥✐❜✐❧✐✲

  

✶✳✶ ▼♦❞❡❧♦s ❞❡ ❆♥á❧✐s❡ ❞❡ ❈ré❞✐t♦ ✲ ●❡♥❡r❛❧✐❞❛❞❡s ✾

❼ ■♥t❡r♣r❡t❛çã♦ ❡q✉✐✈♦❝❛❞❛ ❞❛s ❝❧❛ss✐✜❝❛çõ❡s✿ ♦ ✉s♦ ✐♥❛❞❡q✉❛❞♦ ❞♦ s✐st❡♠❛✱ ❞❡✲ ✈✐❞♦ à ❢❛❧t❛ ❞❡ tr❡✐♥♦ ❡ ❢❛❧t❛ ❞❡ ❢♦r♠❛çã♦ s♦❜r❡ ❛ s✉❛ ✉t✐❧✐③❛çã♦✱ ♣♦❞❡ ♣r♦✈♦❝❛r ♣r♦❜❧❡♠❛s sér✐♦s à ✐♥st✐t✉✐çã♦✳

  ✶✳✶✳✻ ▼♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❘❛t✐♥❣

❙❡❣✉♥❞♦ ❛ ❈♦♠♣❛♥❤✐❛ P♦rt✉❣✉❡s❛ ❞❡ ❘❛t✐♥❣✱ ♦ r❛t✐♥❣ é ✉♠❛ ♦♣✐♥✐ã♦ s♦❜r❡ ❛ ❝❛♣❛❝✐❞❛❞❡

❡ ✈♦♥t❛❞❡ ❞❡ ✉♠❛ ❡♥t✐❞❛❞❡ ✈✐r ❛ ❝✉♠♣r✐r✱ ❞❡ ❢♦r♠❛ ❛t❡♠♣❛❞❛ ❡ ♥❛ í♥t❡❣r❛✱ ❞❡t❡r♠✐♥❛❞❛s

r❡s♣♦♥s❛❜✐❧✐❞❛❞❡s✳ ❖s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❘❛t✐♥❣ ❝♦♥s✐st❡♠ ♥✉♠❛ ❝❧❛ss✐✜❝❛çã♦✱ ♣✉❜❧✐❝❛❞❛ ♣♦r ❛❣ê♥❝✐❛s

❞❡ ❝ré❞✐t♦✱ ❜❛s❡❛❞❛ ♥✉♠❛ ❛♥á❧✐s❡ ❞❡t❛❧❤❛❞❛ ❞❡ ✉♠ ❤✐stór✐❝♦ ✜♥❛♥❝❡✐r♦✱ r❡❢❡r❡♥t❡ ❡ss❡♥✲

❝✐❛❧♠❡♥t❡ à ♣ró♣r✐❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✉♠ ❝❧✐❡♥t❡ ❝✉♠♣r✐r ❝♦♠ ❛s ♦❜r✐❣❛çõ❡s ❞❡ ❞í✈✐❞❛✳ ❆

❝❧❛ss✐✜❝❛çã♦ ♠❛✐s ❛❧t❛ é ❣❡r❛❧♠❡♥t❡ ❞❡s✐❣♥❛❞❛ ♣♦r ❆❆❆ ❡ ❛ ♠❡♥♦r ♣♦r ❉✱ ❝♦♠♦ s❡ ✈❡rá

❛❞✐❛♥t❡✳

  ❈♦♥❢♦r♠❡ ♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❘❛t✐♥❣ ❜❛s❡✐❛♠✲s❡ ♥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡

❞❡ ♣❡r❞❛ ♣❛r❛ ❝❧❛ss✐✜❝❛r ♦s ❡♠♣rést✐♠♦s ❡♠ ❝❛t❡❣♦r✐❛s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ❛tr✐❜✉✐♥❞♦✲

❧❤❡s ✈❛❧♦r❡s ❡ ❛❣r✉♣❛♥❞♦ ♦s ❝ré❞✐t♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣♦♥t✉❛çã♦ ♦❜t✐❞❛✳ ❆ ❛tr✐❜✉✐çã♦

❞❡ ♣♦♥t✉❛çã♦ ❛♦s ❝❧✐❡♥t❡s ❢♦r♥❡❝❡ ✉♠❛ ✈✐sã♦ ❣❡r❛❧ ❞❛ ❝❛♣❛❝✐❞❛❞❡ ✜♥❛♥❝❡✐r❛ ❡ ❞♦ ♥í✈❡❧

❞❡ r✐s❝♦ ❞♦ ❝❧✐❡♥t❡✳ ❆ ❛✈❛❧✐❛çã♦ ❞♦s ❢❛❝t♦r❡s q✉❛❧✐t❛t✐✈♦s ❡ q✉❛♥t✐t❛t✐✈♦s r❡❧❛t✐✈♦s ❛ ✉♠ ❝❧✐❡♥t❡ t❡♠

❝♦♠♦ r❡s✉❧t❛❞♦ ♦ r❛t✐♥❣ ❞❡ss❡ ❝❧✐❡♥t❡✳ ❙❡❣✉♥❞♦ ♣❛r❛ ♦❜t❡r ♦ r❛t✐♥❣

❞❡ ✉♠ ❝❧✐❡♥t❡ ❞❡✈❡♠ s❡r ❛♥❛❧✐s❛❞❛s ❛s ❝♦♥❞✐çõ❡s ❞♦ ♣r♦♣♦♥❡♥t❡ ✭❛ ✐♥❞ústr✐❛ ❡♠ q✉❡

❛❝t✉❛✱ ❛ s✉❛ ♣♦s✐çã♦ ♥❛ ✐♥❞ústr✐❛✱ ❞❡♠♦♥str❛çõ❡s ✜♥❛♥❝❡✐r❛s✱ r❡str✐çõ❡s ❝❛❞❛str❛✐s✱

❡t❝✮✱ ❛ ♦♣❡r❛çã♦ ❞❡ ❝ré❞✐t♦ ✭♣r❛③♦✱ ❣❛r❛♥t✐❛✱ ❧✐q✉✐❞❡③ ❞❛ ❣❛r❛♥t✐❛✮ ❡ ♦ ❝❡♥ár✐♦ ♠❛❝r♦✲

❡❝♦♥ó♠✐❝♦ ♥♦ ❣❡r❛❧✳ ❖ ❛❣r✉♣❛♠❡♥t♦ ❡♠ ❝❧❛ss❡s ❡ ❛ ♣♦♥❞❡r❛çã♦ ❞❡ss❡s ❢❛❝t♦r❡s ❝♦♥st✐t✉✐

♦ r❛t✐♥❣ ❞♦ s♦❧✐❝✐t❛♥t❡ ❞❡ ✉♠ ❡♠♣rést✐♠♦ ♦✉ ❞♦ ❡♠✐ss♦r ❞❡ ✉♠ tít✉❧♦✳

  ❆❝t✉❛❧♠❡♥t❡ ❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s ❞❡s❡♥✈♦❧✈❡♠ ✐♥t❡r♥❛♠❡♥t❡ ♦s ♣ró♣r✐♦s ♠♦✲

❞❡❧♦s ❞❡ ❈r❡❞✐t ❘❛t✐♥❣ ♦✉ ✉t✐❧✐③❛♠ ♦s q✉❡ sã♦ ❢❛❝✉❧t❛❞♦s ♣♦r ❛❣ê♥❝✐❛s ❞❡ r❛t✐♥❣✳ ❊st❛s

❛❣ê♥❝✐❛s sã♦ ❡♠♣r❡s❛s ❡s♣❡❝✐❛❧✐③❛❞❛s ♥❛ ❛✈❛❧✐❛çã♦ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ❆s ❛❣ê♥❝✐❛s ❞❡

r❛t✐♥❣ ♠❛✐s ❝♦♥❤❡❝✐❞❛s sã♦ ❛ ▼♦♦❞②✬s✱ ❛ ❙t❛♥❞❛r❞ ❛♥❞ P♦♦r✬s ❡ ❛ ❋✐t❝❤✳

  ◆❛ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❛ ♠❛✐♦r✐❛ ❞♦s s✐st❡♠❛s ❞❡ r❛t✐♥❣ ❝♦♥s✐❞❡r❛ ❛té ❞❡③ ❝❛t❡❣♦r✐❛s

❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❝ré❞✐t♦✱ ♥❛s q✉❛✐s é ❡st❛❜❡❧❡❝✐❞❛✱ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡✱ ❛ ♣❡r❝❡♥t❛❣❡♠

❡s♣❡r❛❞❛ ❞❡ ❞❡❢❛✉❧t q✉❡ ✐rá ♣❡r♠✐t✐r ❛ ♠❡♥s✉r❛çã♦ ❞❛ ♣❡r❞❛ t♦t❛❧ ❡s♣❡r❛❞❛✳ ❖s s✐st❡♠❛s ❞❡ r❛t✐♥❣✱ ❛❧é♠ ❞❛ s✉❛ ✉t✐❧✐❞❛❞❡ ♥❛s ♦♣❡r❛çõ❡s ❞❡ ❝ré❞✐t♦✱ sã♦ t❛♠❜é♠

♥❡❝❡ssár✐♦s ❞❡✈✐❞♦ às ❡①✐❣ê♥❝✐❛s ❞♦ ❇❛♥❝♦ ❞❡ P♦rt✉❣❛❧✱ q✉❡ ❡♥❢❛t✐③❛♠ ❛ ❝❧❛ss✐✜❝❛çã♦

❞❡ r✐s❝♦ ❝♦♠♦ ❢♦r♠❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❞❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛✳ ❆❣❡♥❝② ❈r❡❞✐t ❘❛t✐♥❣s ❈❧❛ss❡s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♦✉ ❘❛t✐♥❣

  ❆ ▼♦♦❞②✬s✱ ❛ ❋✐t❝❤ ❡ ❛ ❙t❛♥❞❛r❞ ❛♥❞ P♦♦r✬s sã♦ ❛❣ê♥❝✐❛s q✉❡ ♣r♦❞✉③❡♠ ❝❧❛ss✐✜❝❛✲

çõ❡s ❞❡ ❝ré❞✐t♦ ✭❝r❡❞✐t r❛t✐♥❣s✮ ♣❛r❛ ❡♠✐t❡♥t❡s ❞❡ ✐♥str✉♠❡♥t♦s ❞❡ ❞í✈✐❞❛✳ ❆s ❛❣ê♥❝✐❛s

  ✶✵ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

❡♠✐t❡♥t❡ ♥✉♠❛ ❞❛s ✈ár✐❛s ❝❧❛ss❡s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❝ré❞✐t♦✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❙✫P ❡ ❛

  ▼♦♦❞②✬s ✉t✐❧✐③❛♠ ❛ s❡❣✉✐♥t❡ ❝❧❛ss✐✜❝❛çã♦✿

❆✈❛❧✐❛çã♦ ❞❡ ❖❜r✐❣❛çõ❡s ❞❡ ▲♦♥❣♦ Pr❛③♦

❙✫P ▼♦♦❞②✬s ❙✐❣♥✐✜❝❛❞♦

  ❆❆❆ ❆❛❛ ❆❧tíss✐♠❛ ◗✉❛❧✐❞❛❞❡✱ ❘✐s❝♦ ❞❡ ❝ré❞✐t♦ ♠í♥✐♠♦✳ ❆❆ ❆❛ ❆❧t❛ ◗✉❛❧✐❞❛❞❡✱ ❘✐s❝♦ ❞❡ ❝ré❞✐t♦ ♠✉✐t♦ ❜❛✐①♦✳ ❆ ❆ ◗✉❛❧✐❞❛❞❡ ✏▼é❞✐❛ ❙✉♣❡r✐♦r✑✱ ❘✐s❝♦ ❞❡ ❝ré❞✐t♦ ❜❛✐①♦✳ ❇❇❇ ❇❛❛ Pr♦t❡❝çã♦ ❛❞❡q✉❛❞❛✱ ❘✐s❝♦ ❞❡ ❝ré❞✐t♦ ♠♦❞❡r❛❞♦✳ ❇❇ ❇❛ Pr♦♣❡♥s♦ ❛ ♣❛❣❛r✱ ♠❛s ❝♦♠ ❛❧❣✉♠❛ ✐♥❝❡rt❡③❛✳ ❇ ❇ ❊s♣❡❝✉❧❛t✐✈❛s✱ ❘✐s❝♦ ❞❡ ❝ré❞✐t♦ ❛❧t♦✳

❈❈❈ ❈❛❛ ❱✉❧♥❡r❛❜✐❧✐❞❛❞❡ ♣❛r❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❉❡❢❛✉❧t✱ ❘✐s❝♦ ❞❡ ❝ré❞✐t♦ ❛❧t♦✳

❈❈ ❈❛ ❆❧t❛♠❡♥t❡ ❊s♣❡❝✉❧❛t✐✈❛s✱ ❘✐s❝♦ ❜❛st❛♥t❡ ❛❧t♦✱ ❝♦♠ ❛❧❣✉♠❛s

  ♣❡rs♣❡❝t✐✈❛s ❞❡ r❡❝✉♣❡r❛çã♦✳

❈ ❈ ▼✉✐t♦ Pr♦♣❡♥s♦ ❛ ♥ã♦✲♣❛❣❛♠❡♥t♦❀ ❈❧❛ss❡ ♥♦♠✐♥❛❧ ♠❛✐s ❜❛✐①❛ ❡ sã♦

❝♦♥s✐❞❡r❛❞❛s ❡♠ ❉❡❢❛✉❧t ♣❡❧❛ ▼♦♦❞②✬s✳ ❉ ✲ ❉❡❢❛✉❧t

  ◆❘ ✲ ◆♦♥✲❘❛t❡❞ ❖s ❞❡✈❡❞♦r❡s ❝♦♠ ❝❧❛ss✐✜❝❛çã♦ BBB ♦✉ s✉♣❡r✐♦r sã♦ ✉s✉❛❧♠❡♥t❡ ❝❤❛♠❛❞♦s ❞❡

✏✐♥✈❡st♠❡♥t✲❣r❛❞❡✑ ❡ ♦s ✐♥✈❡st✐❞♦r❡s ❝♦♠ ❝❧❛ss✐✜❝❛çã♦ BB ♦✉ ✐♥❢❡r✐♦r sã♦ ❞❡s✐❣♥❛❞♦s

  

❝♦♠♦ ✏s♣❡❝✉❧❛t✐✈❡✲❣r❛❞❡✑✳ ❈❛❞❛ ❝❛t❡❣♦r✐❛ ♣♦❞❡ ❛✐♥❞❛ s❡r s❡❣✉✐❞❛ ❞❡ ✉♠ s✐♥❛❧ ❞❡ + ♦✉

− ♥❛ ❡s❝❛❧❛ ❞❛ ❙✫P ♦✉ ❞♦s ♥ú♠❡r♦s 1✱2 ♦✉ 3 ♥❛ ▼♦♦❞②✬s✱ ♣❛r❛ ♠♦str❛r ❛ ❞✉r❛çã♦ r❡❧❛t✐✈❛ ❞❡♥tr♦ ❞❛ ❝❛t❡❣♦r✐❛✳

  ❆ ❛✈❛❧✐❛çã♦ ❞❡ ❝ré❞✐t♦ é ✉t✐❧✐③❛❞❛ ❝♦♠♦ ✉♠ ❝❡rt✐✜❝❛❞♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❝❛♣❛❝✐❞❛❞❡

❞♦ ❡♠✐ss♦r r❡❡♠❜♦❧s❛r ❛s s✉❛s ♦❜r✐❣❛çõ❡s ❡ ❡♠♣rést✐♠♦s✳ ❆ ❝❧❛ss✐✜❝❛çã♦ é ✉♠ ❢❛❝t♦r

♠✉✐t♦ út✐❧ ♥❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦ ❞❡ ✈❡♥❞❛✴❝♦♠♣r❛ ❞❡ ♦❜r✐❣❛çõ❡s ❛ ✉♠ ❣r❛♥❞❡ ❣r✉♣♦

❞❡ ♣❡q✉❡♥♦s ✐♥✈❡st✐❞♦r❡s✳

  ❆s ❝❧❛ss✐✜❝❛çõ❡s ❞❡ r❛t✐♥❣ sã♦ r❡✈✐st❛s r❡❣✉❧❛r♠❡♥t❡ s❡♥❞♦✱ ♥♦ ❡♥t❛♥t♦✱ ♥❡❝❡ssár✐♦ ❤❛✈❡r ✉♠ ❡q✉✐❧í❜r✐♦ ❡♥tr❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡ ❛ ♣r❡❝✐sã♦ ❞❛s ❝❧❛ss✐✜❝❛çõ❡s✳ ✶✳✷ ▼❡❞✐❞❛s ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✶✳✷✳✶ ❉❡❢❛✉❧t ▲♦ss

❛✜r♠❛♠ q✉❡ ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ é ❛ ♠❛✐s ❛♥t✐❣❛ ❢♦r♠❛ ❞❡ r✐s❝♦

♥♦s ♠❡r❝❛❞♦s ✜♥❛♥❝❡✐r♦s✱ tã♦ ❛♥t✐❣♦ q✉❛♥t♦ ♦s ♣ró♣r✐♦s ❡♠♣rést✐♠♦s✱ r❡♠♦♥t❛♥❞♦ ❛

♣❡❧♦ ♠❡♥♦s ✶✽✵✵ ❛✳❈✳✳ ❖ r✐s❝♦ ❞❡ ❝ré❞✐t♦ é ✉♠ ❞♦s r✐s❝♦s ♠❛✐s ❝♦♠✉♥s ♥✉♠❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛✱ ❞❛❞♦ q✉❡ ❛ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦ é ❛ s✉❛ ♣r✐♥❝✐♣❛❧ ❛❝t✐✈✐❞❛❞❡✳ ❙❡❣✉♥❞♦ ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ é ♦ r✐s❝♦ ❞❡ ♣❡r❞❛ ❞❡✈✐❞♦ ❛ ✉♠❛ ❢❛❧❤❛ ❞❡

  ✶✳✷ ▼❡❞✐❞❛s ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✶✶

❞❡ ❝ré❞✐t♦✱ s❡❥❛ ♥❛ ár❡❛ ❝♦♠❡r❝✐❛❧✱ ✐♥❞✉str✐❛❧✱ ♠❛s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛s ✐♥st✐t✉✐çõ❡s ✜✲

  

♥❛♥❝❡✐r❛s✳ P♦r ❡ss❛ r❛③ã♦✱ é ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ❛ ❛♥á❧✐s❡ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ❝♦♠

✐♥t✉✐t♦ ❞❡ ✐❞❡♥t✐✜❝❛r ♦ ♥í✈❡❧ ❞❡ r✐s❝♦ ♣r❡s❡♥t❡ ♥✉♠❛ ♦♣❡r❛çã♦ ❞❡ ❝ré❞✐t♦✳ ❉❡✜♥✐çã♦ ✶✳✶✳ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

❉❡✜♥❡✲s❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✭♦✉ ❈r❡❞✐t ❘✐s❦✮ ❝♦♠♦ s❡♥❞♦ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ ❝♦♥tr❛♣❛rt❡

❢❛❧❤❛r ✉♠ ♣❛❣❛♠❡♥t♦✳ ◆♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❡ss❛ s✐t✉❛çã♦ ♦❝♦rr❡ ❞✐③✲s❡ q✉❡ ♦ ❞❡✈❡❞♦r

♦r✐❣✐♥♦✉ ✉♠ ❞❡❢❛✉❧t ✭✐♥❝✉♠♣r✐♠❡♥t♦✮✳

  P❛r❛ ❛♥❛❧✐s❛r ❛s ♣❡r❞❛s ❡♠ ❞❡t❛❧❤❡✱ ❝♦♥s✐❞❡r❡✲s❡ ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦ ♣❛r❛ ❝❛❞❛ ❞❡✲ ✈❡❞♦r✿ ❼ τ ✲ ■♥st❛♥t❡ ❡♠ q✉❡ ♦❝♦rr❡ ♦ ❞❡❢❛✉❧t❀ ❼ t ✲ ■♥st❛♥t❡ ♥♦ t❡♠♣♦ q✉❡ ♦❝♦rr❡ ❛♥t❡s ❞❛ ♠❛t✉r✐❞❛❞❡ ♦✉ ❞❛t❛ ❞❡ ✈❡♥❝✐♠❡♥t♦✱ ♦✉ s❡❥❛✱ t ∈ [0, T ]❀

  {τ ≤t} ❼ ■♥❞✐❝❛❞♦r ❞♦ ♣r♦❝❡ss♦ ❞❡ ❉❡❢❛✉❧t ✲ N(t) = ✶■ ❀ ❼ Pr♦❝❡ss♦ ❞❡ ❊①♣♦s✐çã♦ ✭❊①♣♦s✉r❡ ❛t ❉❡❢❛✉❧t ✲ ❊❆❉✮

  E(t) ✲ ▼♦♥t❛♥t❡ q✉❡ ♦ ❝r❡❞♦r ♣❡r❞❡✱ ❝❛s♦ ♦ ❞❡✈❡❞♦r ❡♥tr❡ ❡♠ ❞❡❢❛✉❧t ♥♦ t❡♠♣♦ t ✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ r❡❝✉♣❡r❛çã♦ ✐❣✉❛❧ ❛ ③❡r♦❀ ▼❡❞✐r ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣❛ss❛ t❛♠❜é♠ ♣♦r✱ ✐♠♣❧✐❝✐t❛♠❡♥t❡✱ ♠❡❞✐r ♦ ♠♦♥t❛♥t❡ ❞♦ ❡♠♣rést✐♠♦ ♥ã♦ r❡❝✉♣❡rá✈❡❧✱ ❡♠ ❝❛s♦ ❞❡ ❞❡❢❛✉❧t ♣♦r ♣❛rt❡ ❞♦ ❞❡✈❡❞♦r✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ❤❛✈❡rá ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❡st✐♠❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ❞❡✈❡❞♦r ❢❛❧❤❛r ♦ ❝✉♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦✱ ❜❡♠ ❝♦♠♦ ❡st✐♠❛r ♦ ♠♦♥t❛♥t❡ q✉❡ ❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ ♣❡r❞❡rá✱ ❡♠ ❝❛s♦ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✳ ❼ R(t) ✲ ❚❛①❛ ❞❡ ❘❡❝✉♣❡r❛çã♦ ✭♦✉ ❘❡❝♦✈❡r② ❘❛t❡✮ ➱ ❛ ♣❡r❝❡♥t❛❣❡♠ ❞♦ ♠♦♥t❛♥t❡ ❞❡ ❝ré❞✐t♦ ❝♦♥❝❡❞✐❞♦ q✉❡ ❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ r❡❝✉♣❡r❛✱ ❡♠ ❝❛s♦ ❞❡ ♦❝♦rr❡r ❞❡❢❛✉❧t ♣♦r ♣❛rt❡ ❞♦ ❞❡✈❡❞♦r✳ ◆ã♦ é s✐♠♣❧❡s ❡st✐♠❛r ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✱ ♣♦✐s ♦s ❞❛❞♦s s♦❜r❡ ❛s ♠❡s♠❛s sã♦✱ ❣❡r❛❧♠❡♥t❡✱ ❢r❛❣♠❡♥t❛❞♦s ❡ ❞❡♣❡♥❞❡♠ ❜❛st❛♥t❡ ❞♦s ♣r♦❝❡ss♦s ❞❡ ❢❛❧ê♥❝✐❛ ❡ ❞❛ ❝♦♥❥✉♥t✉r❛ ❡❝♦♥ó♠✐❝❛ à ❞❛t❛ ❞❛ ♦❝♦rrê♥❝✐❛ ❞♦ ❞❡❢❛✉❧t✳ ❼ L(t) ✲ ▲♦ss ●✐✈❡♥ ❉❡❢❛✉❧t ✭▲●❉✮ ❘❡♣r❡s❡♥t❛ ❛ ♣❡r❝❡♥t❛❣❡♠ ❞♦ ♠♦♥t❛♥t❡ ❞❡ ❝ré❞✐t♦ ❝♦♥❝❡❞✐❞♦ q✉❡ ❛ ✐♥st✐t✉✐çã♦

✜♥❛♥❝❡✐r❛ ♣❡r❞❡ ❡♠ ❝❛s♦ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❞❡✈❡❞♦r✱ ♥♦ ✐♥st❛♥t❡ t✳

❆ ▲♦ss ●✐✈❡♥ ❉❡❢❛✉❧t✱ L(t)✱ ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ t❛♠❜é♠ ♣❡❧❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✱ ❘❡❝♦✈❡r② ❘❛t❡✱ R(t) = 1 − L(t)✱ ❡ é ✉♠ ❢❛❝t♦r ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❉❡❢❛✉❧t ▲♦ss✱ D(t) ✱ q✉❡ s❡ ❞❡✜♥❡ ❛❞✐❛♥t❡✳

  ❊♠ ❣❡r❛❧✭✈❡r ❡s♣❡r❛✲s❡ q✉❡ ❛s t❛①❛s ❞❡ r❡❝✉♣❡r❛çã♦ s♦❜r❡ ❛ ❞í✈✐❞❛ s❡❥❛♠ ✐♥✢✉❡♥❝✐❛❞❛s ♣❡❧♦s s❡❣✉✐♥t❡s ❢❛❝t♦r❡s✿

  ✶✷ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ❼ ▲❡❣✐s❧❛çã♦ ❧♦❝❛❧ ♦♥❞❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❢❛❧ê♥❝✐❛ ❞❡❝♦rr❡❀

  ❼ ●r✉♣♦ ✐♥❞✉str✐❛❧ ❞♦ ❞❡✈❡❞♦r❀ ❼ ❈❧❛ss✐✜❝❛çã♦ ❞♦ ❉❡✈❡❞♦r ❛♥t❡s ❞♦ ❞❡❢❛✉❧t✳ ✭❛s t❛①❛s ❞❡ r❡❝✉♣❡r❛çã♦ sã♦ ♠❡♥♦r❡s ❡♠ ❛❧t✉r❛s ❞❡ r❡❝❡ssã♦✱ ♦✉

  ❼ ❈✐❝❧♦ ❊❝♦♥ó♠✐❝♦ s❡❥❛✱ ♠❛✐s ❞❡❢❛✉❧ts ❡ ♠❡♥♦r❡s t❛①❛s ❞❡ r❡❝✉♣❡r❛çã♦✮✳

  ❆s t❛①❛s ❞❡ r❡❝✉♣❡r❛çã♦ ❞❡♣❡♥❞❡♠ s✐st❡♠❛t✐❝❛♠❡♥t❡ ❞♦ ❈✐❝❧♦ ❊❝♦♥ó♠✐❝♦✱ ❤á ♣♦✉✲

❝❛s ❤✐♣ót❡s❡s ♣❛r❛ ❛ ❞✐✈❡rs✐✜❝❛çã♦ ❞❛ ❡st✐♠❛çã♦ ❞❡ ❡rr♦s✳ ➱ ❝♦♠♣❧✐❝❛❞♦ ❞✐③❡r✱ ❝♦♠

❜❛st❛♥t❡ s❡❣✉r❛♥ç❛✱ q✉❛✐s ❛s t❛①❛s ❞❡ r❡❝✉♣❡r❛çã♦ q✉❡ s❡ ❞❡✈❡♠ ✉t✐❧✐③❛r ♥♦s ♠♦❞❡❧♦s

❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳

  ❉❡✜♥❛✲s❡ ❛❣♦r❛ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❉❡❢❛✉❧t ▲♦ss ✭✈❡r ❉❡✜♥✐çã♦ ✶✳✷✳ ❉❡❢❛✉❧t ▲♦ss

P❛r❛ ❝❛❞❛ ❞❡✈❡❞♦r✱ ❛ ❉❡❢❛✉❧t ▲♦ss é ♦ ♠♦♥t❛♥t❡ ♣❡r❞✐❞♦ ♣❡❧♦ ❝r❡❞♦r ❞❡✈✐❞♦ à ♦❝♦rrê♥❝✐❛

❞❡ ❞❡❢❛✉❧t ❡ é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ s❡❣✉✐♥t❡✿

  

D(T ) := N (T ) × E(τ i ) × L(τ i )

✭✶✳✷✮ ♦♥❞❡ T é ❛ ♠❛t✉r✐❞❛❞❡ ✭❝♦♥s✐❞❡r❛✲s❡✱ ♣♦r ❡①❡♠♣❧♦✱ ✶ ❛♥♦✮ ❡ τ ♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ♦❝♦rr❡ ♦ ❞❡❢❛✉❧t ♣❛r❛ ♦ ❞❡✈❡❞♦r i✳

  ✶✳✷✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❉❡❢❛✉❧t

❙❡❣✉♥❞♦ ❧♦❣♦✱ ♣♦❞❡ ❞✐③❡r✲s❡ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t é ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s

❞❛ ❉❡❢❛✉❧t ▲♦ss✳ (t, T ) i

  ❙❡❥❛ P D ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❞❡ ✉♠ ❞❡✈❡❞♦r i✱ ❞❡st❡ ♦ ✐♥st❛♥t❡ t ❛té T ✳ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ár✈♦r❡ ❜✐♥♦♠✐❛❧✳

  ❈♦♥s✐❞❡r❛♥❞♦ ❛ ár✈♦r❡ ❜✐♥♦♠✐❛❧ ❛ ✉♠ ♣❡rí♦❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❞❡✈❡❞♦r ♣♦❞❡ ❡st❛r )

  1

  1

❡♠ ❞♦✐s ❡st❛❞♦s ♣♦ssí✈❡✐s✿ ❊st❛❞♦ ❉❡❢❛✉❧t (D ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ p ✱ ♦✉ ❡st❛❞♦ ❞❡

) s♦❜r❡✈✐✈ê♥❝✐❛ (S 1 ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ 1 − p 1 ✳ P❛r❛ ♦ ❝❛s♦ ❛ n✲♣❡rí♦❞♦s ♣r♦❝❡❞❡✲s❡ ❞❡

❢♦r♠❛ ❛♥á❧♦❣❛✳ ◆❡st❡ ❝❛s♦✱ ♣♦❞❡ ♦❜s❡r✈❛r✲s❡ ♥❛ ❋✐❣✉r❛ q✉❡ ❡①✐st❡♠ n ❝❛♠✐♥❤♦s

❞✐st✐♥t♦s q✉❡ ❧❡✈❛♠ ❛♦ ❞❡❢❛✉❧t✱ ♠❛s ❛♣❡♥❛s ✉♠ q✉❡ ❧❡✈❛ ❛♦ ♥♦♥✲❞❡❢❛✉❧t✱ ♦✉ s♦❜r❡✈✐✈ê♥❝✐❛✳

❉❡♥♦t❡✲s❡ P S(t, T ) ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡s❞❡ ♦ ✐♥st❛♥t❡ t ❛té T ✳

  

❼ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝✉♠✉❧❛t✐✈❛ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡s❞❡ t = 0 ❛té ❛♦ ♣❡rí♦❞♦ n é✿

P S(0, n) = P [ 1 )(1 − p 2 ) . . . (1 − p n ).

s♦❜r❡✈✐✈❡r ❞❡ 0 ❛ n] = (1 − p

  ✶✳✷ ▼❡❞✐❞❛s ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✶✸ ❼ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝✉♠✉❧❛t✐✈❛ ❞❡ ❞❡❢❛✉❧t ❛té ❛♦ ♣❡rí♦❞♦ n é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡

  ❡♥tr❛r ❡♠ ✐♥❝✉♠♣r✐♠❡♥t♦ ❡♠ q✉❛❧q✉❡r ✐♥st❛♥t❡ ♥♦ t❡♠♣♦ ❛♥t❡s ❞♦ ♣❡rí♦❞♦ n✱ ❡ é ❞❛❞♦ ♣♦r✿ P D(0, n) = 1 − P S(0, n).

  

❋✐❣✉r❛ ✶✳✶✿ ➪r✈♦r❡ ▼✉❧t✐✲P❡rí♦❞♦s

P❡r❞❛s ❊s♣❡r❛❞❛s i (0, T )

  ❙❡❥❛ P D ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❞♦ ❞❡✈❡❞♦r i✳ ❚❡♠♦s✱ ❝♦♥❢♦r♠❡ r❡❢❡r❡ q✉❡✿

E [N (T )] = P D (0, T ).

i i

  ❉❡✜♥✐çã♦ ✶✳✸✳ P❡r❞❛ ❊s♣❡r❛❞❛ ■♥❞✐✈✐❞✉❛❧ i i

  

❙❡❥❛♠ E ❡ L ❝♦♥❤❡❝✐❞♦s ❡ ❝♦♥st❛♥t❡s✳ ❆ ♣❡r❞❛ ❡s♣❡r❛❞❛ ❞❡ ✉♠ ❞❡✈❡❞♦r ❞✉r❛♥t❡ ♦

t❡♠♣♦ ❚ é✿

E [D (T )] := P D (0, T ) × E × L .

i i i i

  ✭✶✳✸✮ ◆♦t❡✲s❡ q✉❡ ❡♠❜♦r❛ ❛ ♣❡r❞❛ ❡s♣❡r❛❞❛ ✐♥❞✐✈✐❞✉❛❧ ♣♦ss❛ s❡r ♣❡q✉❡♥❛✱ ❛s ♣❡r❞❛s ❛❝✉✲

♠✉❧❛❞❛s✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ✈ár✐♦s ❞❡✈❡❞♦r❡s✱ ♣♦❞❡ s❡r ♠✉✐tíss✐♠♦

❣r❛♥❞❡✳ ✶✳✷✳✸ ▼❛tr✐③❡s ❞❡ ❚r❛♥s✐çã♦ ❡ ▼✐❣r❛çã♦ ❞❡ ❈ré❞✐t♦ ❚r❛♥s✐çã♦ ❛ ✶✲P❡rí♦❞♦ ❙❡❣✉♥❞♦ ✱ ❡♠ ♠♦❞❡❧♦s q✉❛♥t✐t❛t✐✈♦s ❞❡ ❣❡stã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱

❛ ✉t✐❧✐③❛çã♦ ❞❛s ❝❧❛ss✐✜❝❛çõ❡s r❡❢❡r✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡ ♥❛ ❙❡❝çã♦ ❡stá s✉❥❡✐t❛ ❛

❛❧t❡r❛çõ❡s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ é ♥❡❝❡ssár✐♦ ❢❛③❡r ❝♦rr❡s♣♦♥❞❡r ❛s ❝❧❛ss✐✜❝❛çõ❡s ❛❧❢❛❜é✲

t✐❝❛s ❡♠ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ♠✐❣r❛çã♦ ❞❡ ❝ré❞✐t♦✳ ■st♦ é✱ é ♥❡❝❡ssár✐♦ ❛tr✐❜✉✐r ✈❛❧♦r❡s

  ✶✹ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

❞❡❢❛✉❧t✳ ❆s ❛❣ê♥❝✐❛s ❞❡ ❝ré❞✐t♦ t❛♠❜é♠ ♣✉❜❧✐❝❛♠ ❡st❡ t✐♣♦ ❞❡ ✐♥❢♦r♠❛çã♦ ❡♠ ♠❛tr✐③❡s

  ❞❡ tr❛♥s✐çã♦✳ ❆ ▼♦♦❞②✬s ♣✉❜❧✐❝♦✉ ❛ ♠❛tr✐③ ✐❧✉str❛❞❛ ♥❛ ❚❛❜❡❧❛ ■♥✐t✐❛❧ ❘❛t✐♥❣ ❛t ②❡❛r✲❡♥❞ ✭✪✮ ❘❛t✐♥❣

  ❆❛❛ ❆❛ ❆ ❇❛❛ ❇❛ ❇ ❈❛❛ ❉❡❢❛✉❧t ❆❛❛ ✾✸✳✹✵ ✺✳✾✹ ✵✳✻✹ ✳✵✵✵ ✵✳✵✷ ✵✳✵✵ ✵✳✵✵ ✵✳✵✵ ❆❛ ✶✳✻✶ ✾✵✳✺✺ ✼✳✹✻ ✵✳✷✻ ✵✳✵✾ ✵✳✵✶ ✵✳✵✵ ✵✳✵✷ ❆ ✵✳✵✼ ✷✳✷✽ ✾✷✳✹✹ ✹✳✻✸ ✵✳✹✺ ✵✳✶✷ ✵✳✵✶ ✵✳✵✵

  ❇❛❛ ✵✳✵✺ ✵✳✷✻ ✺✳✺✶ ✽✽✳✹✽ ✹✳✼✻ ✵✳✼✶ ✵✳✵✽ ✵✳✶✺ ❇❛ ✵✳✵✷ ✵✳✵✺ ✵✳✹✷ ✺✳✶✻ ✽✻✳✾✶ ✺✳✾✶ ✵✳✷✹ ✶✳✷✾ ❇ ✵✳✵✵ ✵✳✵✹ ✵✳✶✸ ✵✳✺✹ ✻✳✸✺ ✽✹✳✷✷ ✶✳✾✶ ✻✳✽✶ ❈❛❛ ✵✳✵✵ ✵✳✵✵ ✵✳✵✵ ✵✳✻✷ ✷✳✵✺ ✹✳✵✽ ✻✾✳✷✵ ✷✹✳✵✻

  

❏✳P✳▼♦r❣❛♥✱✭✶✾✾✼✱ t❛❜❡❧❛ ✻✳✶✮

❚❛❜❡❧❛ ✶✳✶✿ ▼❛tr✐③ ❞❡ ❚r❛♥s✐çã♦

❈❛❞❛ ❡♥tr❛❞❛ ❞❛ ♠❛tr✐③ r❡♣r❡s❡♥t❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♠✐❣r❛çã♦ ❞❛ ❝❧❛ss❡✲❧✐♥❤❛ ♣❛r❛

  ❛ ❝❧❛ss❡✲❝♦❧✉♥❛✳ ◆♦t❡✲s❡ q✉❡ ❛ s♦♠❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❝❛❞❛ ❧✐♥❤❛ s♦♠❛ ✶✱ ♣♦✐s ✉♠❛ ❡♠♣r❡s❛ q✉❡

❡st❡❥❛ ❝♦♠ ❝❧❛ss✐✜❝❛çã♦ Baa✱ ♣♦r ❡①❡♠♣❧♦✱ ❡st❛rá ❝❡rt❛♠❡♥t❡ ❡♠ ❛❧❣✉♠❛ ❞❛s ❝❧❛ss❡s

♣♦ssí✈❡✐s ♥♦ ♣ró①✐♠♦ ❛♥♦✳

  ❖s ❡❧❡✈❛❞♦s ✈❛❧♦r❡s ♥❛ ❞✐❛❣♦♥❛❧ ❞❛ ♠❛tr✐③ r❡✢❡❝t❡♠ ❛ ✏❡st❛❜✐❧✐❞❛❞❡ ❞❛ ❝❧❛ss✐✜❝❛çã♦✑ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛s ❛❣ê♥❝✐❛s ❞❡ ❝❧❛ss✐✜❝❛çã♦✳ ❈❛s♦ s❡ ❛❝r❡s❝❡♥t❡ ✉♠❛ ❧✐♥❤❛ à ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦✱ ❞❡st✐♥❛❞❛ ❛♦ ❡st❛❞♦ ❞❡ ❞❡✲

❢❛✉❧t✱ ❡st❛ ❧✐♥❤❛ só t❡r✐❛ ③❡r♦s✱ ❡①❝❡♣t✉❛♥❞♦ ♥❛ ú❧t✐♠❛ ❝♦❧✉♥❛ q✉❡ t❡r✐❛ ♦ ✈❛❧♦r 100%✳

  ❆ss✉♠❡✲s❡✱ ♣♦rt❛♥t♦✱ q✉❡ ♦ ❡st❛❞♦ ✏❞❡❢❛✉❧t✑ é ✉♠ ❡st❛❞♦ ❛❜s♦r✈❡♥t❡✳ ❚r❛♥s✐çã♦ ❛ ◆✲P❡rí♦❞♦s ❈♦♥s✐❞❡r❛♥❞♦ ❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ❛ ✶✲♣❡rí♦❞♦✱ ♣♦❞❡ ✐r✲s❡ ♠❛✐s ❛❧é♠ ❡ ♣r♦❞✉③✐r

✉♠❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ❛ n✲♣❡rí♦❞♦s✳ P❛r❛ t❛❧✱ ✭✈❡r ❝♦♥s✐❞❡r❛♠✲s❡

  ❞✉❛s s✉♣♦s✐çõ❡s✿ ❼ ❚❡♠♣♦ ❞❡ ✐♥✈❛r✐â♥❝✐❛✳ ❆ss✉♠❡✲s❡ q✉❡ ❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ❛ ✶✲♣❡rí♦❞♦ é ❝♦♥s✲ t❛♥t❡✱ ♦✉ s❡❥❛✱ ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦ ❞❡ ❝❛❧❡♥❞ár✐♦✳ ❼ Pr♦♣r✐❡❞❛❞❡ ❞❡ ▼❛r❦♦✈✳ Pr♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ♠✐❣r❛çã♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ só ❞❡♣❡♥✲ ❞❡♠ ❞❛ ❛✈❛❧✐❛çã♦ ❛❝t✉❛❧✳ ❆ ❝❧❛ss✐✜❝❛çã♦ ❤✐stór✐❝❛ ✭✏✉♣❣r❛❞❡✑✴ ✏❞♦✇♥❣r❛❞❡✑✮ é ✐rr❡❧❡✈❛♥t❡ ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛r ❡st❛❞♦s ❢✉t✉r♦s✳

  ❊st❛s ❞✉❛s s✉♣♦s✐çõ❡s ❡stã♦ s✉❥❡✐t❛s ❛ ❛❧❣✉♠❛ ❝rít✐❝❛✳ Pr✐♠❡✐r♦✱ ❡①✐st❡ ❛❧❣✉♠❛

❡✈✐❞ê♥❝✐❛ q✉❡ ♦ ✏❞♦✇♥❣r❛❞✐♥❣✑ é ♠❛✐s ♣r♦✈á✈❡❧ ❡♠ s✐t✉❛çõ❡s ❞❡ r❡❝❡ssã♦ ❞♦ q✉❡ ❡♠

❢❛s❡s ❞❡ ✏❜♦♦♠✑ ❞♦ ❝✐❝❧♦ ❡❝♦♥ó♠✐❝♦✳ ❊♠ s❡❣✉♥❞♦ ❧✉❣❛r✱ ❡①✐st❡ ❡✈✐❞ê♥❝✐❛ ❞❡ q✉❡ ♦

  ✶✳✸ ▼♦❞❡❧♦s ♣❛r❛ ❈❛rt❡✐r❛s ✶✺

♣r♦✈á✈❡❧ ✉♠ ♥♦✈♦ ✏❞♦✇♥❣r❛❞❡✑ ♣❛r❛ ✉♠❛ ♠❡s♠❛ ❡♠♣r❡s❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❞♦ q✉❡ ♣❛r❛

  

✉♠❛ q✉❡ s❡ ♠❛♥t❡♥❤❛ ♥❛ ♠❡s♠❛ ❡s❝❛❧❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❤á ❛❧❣✉♠ t❡♠♣♦✳ ◆♦ ❡♥t❛♥t♦✱

❡st❛s ❞✉❛s s✉♣♦s✐çõ❡s sã♦ ❝♦♥s✐❞❡r❛❞❛s ❞❡s❞❡ q✉❡ ♣❡r♠✐t❛♠ ❛ s✐♠♣❧✐✜❝❛çã♦ ❞♦s ♠♦❞❡❧♦s

❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ❊ss❛s s✉♣♦s✐çõ❡s ♣❡r♠✐t❡♠ q✉❡ ❛ ❛♥á❧✐s❡ ❞❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ ❝❧❛ss✐✜❝❛çã♦✱ ❡♠

t❡r♠♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♣♦ss❛ s❡r tr❛t❛❞❛ r❡❝♦rr❡♥❞♦ à s♦❜❡❥❛♠❡♥t❡ ❝♦♥❤❡❝✐❞❛ ❚❡♦✲

r✐❛ ❞❛s ❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈✳

  P❛r❛ s❡ ♦❜t❡r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ tr❛♥s✐t❛r ❞♦ ❡st❛❞♦ i ♣❛r❛ ♦ ❡st❛❞♦ j✱ ❡♠ ❞♦✐s

♣❡rí♦❞♦s✱ ❤á q✉❡ ❝♦♥s✐❞❡r❛r t♦❞♦s ♦s ❝❛♠✐♥❤♦s ♣♦ssí✈❡✐s q✉❡ ❝♦♥❞✉③❡♠ ❞❡ i ❛ j✳ ❆

(2)

  P K = p ik p kj

♣r♦❜❛❜✐❧✐❞❛❞❡ é ❛ s♦♠❛ ❞❡ t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣♦ssí✈❡✐s✱ p ✱ ♦♥❞❡

ij k=1

  K = 8 ♣❛r❛ ❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ❞❛ ▼♦♦❞②✬s ♥❛ t❛❜❡❧❛ ❛♥t❡r✐♦r✳ ❊♥tã♦ ❛ ♠❛tr✐③ ❞❡ (2)

  = P × P tr❛♥s✐çã♦ ❛ ✷✲♣❡rí♦❞♦s s❡rá P ✳ ❊♠ ❣❡r❛❧✱ ❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ❛ n✲P❡r✐♦❞♦s é ❞❛❞❛ ♣♦r✿

  

(n) n

P = P

✭✶✳✹✮

  ❆s ❝❧❛ss✐✜❝❛çõ❡s r❡❢❡r✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡ ❡stã♦ ♥♦r♠❛❧♠❡♥t❡ ❞✐s♣♦♥í✈❡✐s ❛♣❡♥❛s ♣❛r❛

❣r❛♥❞❡s ❡♠♣r❡s❛s✳ ◆♦ ❡♥t❛♥t♦✱ ✉♠ ❜❛♥❝♦ t❡♠ ♣❡q✉❡♥❛s ❡ ♠é❞✐❛s ❡♠♣r❡s❛s ♥❛ s✉❛

❝❛rt❡✐r❛✳ P❛r❛ ❛ ❛✈❛❧✐❛çã♦ ❞❡ ♣❡q✉❡♥❛s ❡♠♣r❡s❛s✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ✈ár✐♦s ♠ét♦❞♦s

❡st❛tíst✐❝♦s✳ ❖s ♠♦❞❡❧♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❝ré❞✐t♦ ♣♦❞❡♠ ✉t✐❧✐③❛r ❛ s❡❣✉✐♥t❡ ✐♥❢♦r♠❛çã♦

♣❛r❛ ♣r❡✈❡r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✿ ❼ ❉❛❞♦s ❝♦♠ ❜❛❧❛♥ç♦ s♦❜r❡ ❡♥❞✐✈✐❞❛♠❡♥t♦ ❞♦ ❞❡✈❡❞♦r❀

  ❼ ❉❛❞♦s ❝♦♠ ❜❛❧❛♥ç♦ s♦❜r❡ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♣❛❣❛♠❡♥t♦❀ ❼ ●r❛✉ ❞❡ r✐s❝♦ ❞♦ ♥❡❣ó❝✐♦✳ ❉❛❞♦s s♦❜r❡ ❛ ✈♦❧❛t✐❧✐❞❛❞❡❀ ❼ ❆ ❝❛♣✐t❛❧✐③❛çã♦ ❞❡ ♠❡r❝❛❞♦ ❞❛ ❡♠♣r❡s❛❀ ❼ ❉❛❞♦s ♠❛❝r♦✲❡❝♦♥ó♠✐❝♦s✳

  ✶✳✸ ▼♦❞❡❧♦s ♣❛r❛ ❈❛rt❡✐r❛s

❆ ❡st✐♠❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♣❛r❛ ✉♠ ❝❧✐❡♥t❡ ✐♥❞✐✈✐❞✉❛❧ é ✉♠❛ ❛♥á❧✐s❡

✐♠♣♦rt❛♥t❡ ❛ ❡❢❡❝t✉❛r ♣♦r ♣❛rt❡ ❞❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛✱ ♠❛s s❡rá ♠❛✐s r❡❧❡✈❛♥t❡ ❞♦

♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❝♦♥❝❡ssã♦✱ ♦✉ ♥ã♦✱ ❞♦ ❝ré❞✐t♦ s♦❧✐❝✐t❛❞♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ✐♠♣♦rtâ♥❝✐❛

❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ✈❛✐ ♠✉✐t♦ ❛❧é♠ ❞❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦✳

  ❆s ✐♠♣❧✐❝❛çõ❡s ❢✉t✉r❛s ❞❡ ✉♠ ❡✈❡♥t✉❛❧ ❞❡❢❛✉❧t r❡✢❡❝t✐r✲s❡✲á ♥♦ ❞❡s❡♠♣❡♥❤♦ ❡ ❛♥á❧✐s❡

❞❛ ❝❛rt❡✐r❛ ❞❡ ❝❧✐❡♥t❡s ❞❛ ✐♥st✐t✉✐çã♦✳ ❆ss✐♠ s❡♥❞♦✱ ❛s ■♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s ♥❡❝❡s✲

s✐t❛♠ ❞❡ ✐♥str✉♠❡♥t♦s q✉❡ ❛♥❛❧✐s❡♠ ❛ ❝❛rt❡✐r❛ ❞❡ ❝❧✐❡♥t❡s ♥♦ s❡✉ t♦❞♦✳ P♦r ❡①❡♠♣❧♦✱

é ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❝♦♥❥✉♥t❛ ❞♦s ❝❧✐❡♥t❡s q✉❡ s❡rã♦ ❡st✐♠❛❞❛s ❛s r❡s❡r✈❛s ❞❡ ❝❛♣✐t❛❧

♥❡❝❡ssár✐❛s ♣❛r❛ ❢❛③❡r ❢❛❝❡ ❛♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❛ ❝❛rt❡✐r❛✳ ◆❛ s✐t✉❛çã♦ q✉❡ s❡rá tr❛t❛❞❛ ♥❡st❛ s❡❝çã♦✱ ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❞❡ ✐♥t❡r❡ss❡ s❡rá ❛

  ✶✻ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

  ✶✳✸✳✶ ❉❡❢❛✉❧t ▲♦ss ❞❡ ✉♠❛ ❈❛rt❡✐r❛ ❉❡✜♥✐çã♦ ✶✳✹✳ P♦rt❢♦❧✐♦ ❉❡❢❛✉❧t ▲♦ss

❆ ❉❡❢❛✉❧t ▲♦ss✱ ♥✉♠❛ ❝❛rt❡✐r❛ ❝♦♠ J ❞❡✈❡❞♦r❡s✱ é ❞❡✜♥✐❞❛ ♣♦r ✭✈❡r

  J J

  X X D(T ) := D (T ) = N (T ) × E (τ ) × L (τ ), j j i j j j ✭✶✳✺✮ j=1 j=1

  (T ) (T ) (τ ) (τ ) j j i j j j

  ❝♦♠ D ✱ N ✱ E ❡ L ❞❡✜♥✐❞♦s ❝♦♠♦ ♥❛ ❙✉❜s❡❝çã♦ ❉✐str✐❜✉✐çã♦ ❞❛ ❉❡❢❛✉❧t ▲♦ss ❞❡ ✉♠❛ ❝❛rt❡✐r❛

❆ ❞✐str✐❜✉✐çã♦ ❞♦ ❚♦t❛❧ ❞❛s ♣❡r❞❛s ❞❡ ❝ré❞✐t♦ ❞❛ ❝❛rt❡✐r❛ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r

F (x) := P [D(T ) ≤ x]

  ✳ ◆❛ ❣❡♥❡r❛❧✐❞❛❞❡ ❞❛s s✐t✉❛çõ❡s✱ ❛ ❛ss✉♠♣çã♦ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛

❡♥tr❡ ❛s ♣❡r❞❛s ✐♥❞✐✈✐❞✉❛✐s r❡✢❡❝t❡✲s❡ ♥✉♠❛ s✉❜✲❛✈❛❧✐❛çã♦ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❛ ❝❛rt❡✐r❛

✭✈❡r

  

❋✐❣✉r❛ ✶✳✷✿ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❚í♣✐❝❛ ❞❛ ❉❡❢❛✉❧t ▲♦ss ❞❡ ❯♠❛ ❈❛rt❡✐r❛

❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❛ ❉❡❢❛✉❧t ▲♦ss ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❡♠♣rést✐♠♦s ✉s✉❛✐s t❡♠✱ r❡❣r❛ ❣❡r❛❧✱ ❛s s❡❣✉✐♥t❡s ❝❛r❛❝t❡ríst✐❝❛s✿ ❼ ◆ã♦ s✐♠étr✐❝♦✿ ❆s ♣❡r❞❛s sã♦ tr✉♥❝❛❞❛s ❡♠ ③❡r♦✱ D ≥ 0✱ ✭♦ ♠❡❧❤♦r ❝❛s♦ é q✉❛♥❞♦

  ♥ã♦ ❤á ♣❡r❞❛s✮ ❡✱ ♥♦ ❡♥t❛♥t♦✱ ❛ ♣❡r❞❛ ♣♦❞❡ s❡r ❡①tr❡♠❛♠❡♥t❡ ❣r❛♥❞❡✳ ❼ ▼✉✐t♦ ✐♥❝❧✐♥❛❞❛ ♣♦s✐t✐✈❛♠❡♥t❡✿ ❆ ♠❛✐♦r ♠❛ss❛ ❡stá ♥❛ ár❡❛ ❞❡ ♣❡q✉❡♥❛s ♣❡r❞❛s✱

  ✶✳✸ ▼♦❞❡❧♦s ♣❛r❛ ❈❛rt❡✐r❛s ✶✼ ❼ ❈❛✉❞❛ ❞✐r❡✐t❛ ♣❡s❛❞❛✿ ❛s ❣r❛♥❞❡s ♣❡r❞❛s sã♦ ❛✐♥❞❛ ♣r♦✈á✈❡✐s✳ ❖s q✉❛♥t✐s ❞♦

  ✶

❱❛❧✉❡✲❛t✲❘✐s❦ ✭❱❛❘✮ ✉s✉❛❧ sã♦ ❛ss✐♠ ❛ss♦❝✐❛❞❛s ❛ ♣❡r❞❛s ♠✉✐t♦ ❣r❛♥❞❡s✳

■♥t❡♥s✐❞❛❞❡ ❞❡ ❉❡❢❛✉❧t

  

❙❡❣✉♥❞♦ ✉♠❛ ❝❧❛ss❡ ✐♠♣♦rt❛♥t❡ ❞❡ ♠♦❞❡❧♦s ❞❡ ❝ré❞✐t♦ ❛ss✉♠❡ q✉❡ ❛

♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t é ✉♠ ❛❝♦♥t❡❝✐♠❡♥t♦ ❛❧❡❛tór✐♦✳ ❆s ❤✐♣ót❡s❡s ✉s✉❛✐s sã♦ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❞❡❢❛✉❧ts ✭N✮✱ ♥✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✱ ❞✉r❛♥t❡ ✉♠ ❝❡rt♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥✳ ❙❡❥❛ λ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ ✭❣❡r❛❧♠❡♥t❡ ✶ ❛♥♦✮✳ ❊st❛

q✉❛♥t✐❞❛❞❡ é ❞❡s✐❣♥❛❞❛ ♣♦r t❛①❛ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ✭✏❞❡❢❛✉❧t r❛t❡✑✱ ✏❞❡❢❛✉❧t ✐♥t❡♥s✐t②✑

  ♦✉ ✏❞❡❢❛✉❧t ❤❛③❛r❞ r❛t❡✑✮✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥ t❡♠♦s q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r❡♠ k ❡✈❡♥t♦s ❡♠ ∆t é✿

  

−λ∆t k

e (λ∆t)

P [N = k] = .

  ♣❛r❛ k ∈ N

k!

❆ss✐♠✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ é ❞❛❞❛ ♣❡❧❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♥ã♦ ♦❝♦rr❡r ❞❡❢❛✉❧t✱ ✐✳❡✳✿

  −λ∆t

P S(t, t + ∆t) = P [N = 0] = e .

  ❊♠ ❣❡r❛❧✱ ♣♦❞❡♠♦s t❡r ❞✐❢❡r❡♥t❡s ✐♥t❡♥s✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ♣❛r❛ ♦s ❞✐❢❡r❡♥t❡s ♣❡rí♦❞♦s✳

❙❡ ♦s ♠♦❞❡❧♦s ❞❡ ❝ré❞✐t♦ sã♦ ❢♦r♠✉❧❛❞♦s ❡♠ t❡♠♣♦ ❝♦♥tí♥✉♦ s❡rá ✈❛♥t❛❥♦s♦ ❞❡✜♥✐r ✉♠❛

♣r♦❜❛❜✐❧✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ ❞❡ ❞❡❢❛✉❧t✳ ❙❡❥❛ λ(t)✱ ♣❛r❛ t ≥ 0✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ ✭✶ ❛♥♦✱

❣❡r❛❧♠❡♥t❡✮ ♣❛r❛ ♦ ✐♥st❛♥t❡ t✳ ❈♦♥s✐❞❡r❛✲s❡ ♦ ❧✐♠✐t❡ ❡♠ t❡♠♣♦ ❝♦♥tí♥✉♦ ❢❛③❡♥❞♦ ∆t →

✳ r❡❢❡r❡ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❞❡s❞❡ t ❛té T ✱ é ❞❛❞❛

♣❡❧♦ Pr♦❝❡ss♦ ❞❡ P♦✐ss♦♥✿ Z

  T

P S(t, T ) = exp{− λ(s)ds}.

t

  P❡r❞❛s ❊s♣❡r❛❞❛s ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❆ ♣❡r❞❛ ❡s♣❡r❛❞❛ ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ♣♦❞❡ s❡r ❞❡✜♥✐❞❛✱ s❡❣✉♥❞♦ ❝♦♠♦✿ ❉❡✜♥✐çã♦ ✶✳✺✳ P❡r❞❛ ❊s♣❡r❛❞❛ ❞❡ ✉♠❛ ❈❛rt❡✐r❛

  

I

X E [D(T )] := E [D (T )]. i

  ✭✶✳✻✮

i=1

  ◗✉❛♥❞♦ s❡ ❝♦♥❝❡❞❡ ✉♠ ❡♠♣rést✐♠♦✱ ♦s ❣❛♥❤♦s ❡s♣❡r❛❞♦s ❞❡✈❡♠ ❝♦❜r✐r ❛ ♣❡r❞❛

❡s♣❡r❛❞❛ ❞♦ r❡s♣❡❝t✐✈♦ ❡♠♣rést✐♠♦✳ ❖ ♣r✐♥❝í♣✐♦ é ✐❞ê♥t✐❝♦ ❛♦ ♣r✐♥❝í♣✐♦ q✉❡ r❡❣❡ ❛

❆❝t✐✈✐❞❛❞❡ ❙❡❣✉r❛❞♦r❛✳

  ✶ ❱❡r ●❧♦ssár✐♦

  ✶✽ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ P❡r❞❛s ■♥❡s♣❡r❛❞❛s ❞❡ ✉♠❛ ❈❛rt❡✐r❛

  

P❛r❛ ❛s ♣❡r❞❛s ✐♥❡s♣❡r❛❞❛s sã♦ ♠❡❞✐❞❛s ❡♠ r❡❧❛çã♦ ❛♦ q✉❛♥t✐❧ ❱❛❘ ❡

sã♦ ❝♦♠♦ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ✉♠ ❝❡♥ár✐♦ ✐♠♣r❡✈✐sí✈❡❧ ✭❝❡♥ár✐♦ ❞❡ str❡ss✮✳ k

  ❙❡❥❛ D ♦ q✉❛♥t✐❧ k ❞♦ ❱❛❘ ❞❛ ❝❛rt❡✐r❛✱ ✐st♦ é✱ ❛ ♣❡r❞❛ ♠á①✐♠❛ ❝♦♠ ♥í✈❡❧ ❞❡ ❝♦♥✜❛♥ç❛ k✳ ❉❡✜♥✐çã♦ ✶✳✻✳ P❡r❞❛ ■♥❡s♣❡r❛❞❛ ❞❡ ✉♠❛ ❈❛rt❡✐r❛

❆ ♣❡r❞❛ ✐♥❡s♣❡r❛❞❛ ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ✭❯♥❡①♣❡❝t❡❞ P♦rt❢♦❧✐♦ ▲♦ss ✲ ❯❊▲✮ é ❛ ❞✐❢❡r❡♥ç❛

❡♥tr❡ ♦ q✉❛♥t✐❧ ❞♦ ❱❛❘ ❡ ❛ ♣❡r❞❛ ❡s♣❡r❛❞❛ ❞❛ ♠❡s♠❛ ❝❛rt❡✐r❛✿

U EL := D − E[D(T )]

k

  ✭✶✳✼✮ ❊st❛ ♥ã♦ é✱ ♥♦ ❡♥t❛♥t♦✱ ❛ ♣❡r❞❛ ♠á①✐♠❛ ♣♦ssí✈❡❧✳ ❆ ♣❡r❞❛ ♠á①✐♠❛ ♣♦ssí✈❡❧ ♦❝♦rr❡

q✉❛♥❞♦ ♦ t♦t❛❧ ❞❛ ❝❛rt❡✐r❛ s❡ ♣❡r❞❡ ❡ ♥ã♦ ❤á ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ r❡❝✉♣❡r❛çã♦✳ ❆♦ ❝♦♥trár✐♦

❞❛s ♣❡r❞❛s ❡s♣❡r❛❞❛s✱ ❛s ✐♥❡s♣❡r❛❞❛s ♥ã♦ sã♦ ❛❞✐t✐✈❛s ❡♠ r❡❧❛çã♦ às ♣❡r❞❛s ✐♥❞✐✈✐❞✉❛✐s✳

❆ ♣❡r❞❛ ✐♥❡s♣❡r❛❞❛ é ✉t✐❧✐③❛❞❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ r❡s❡r✈❛ ❞❡ ❝❛♣✐t❛❧

♥❡❝❡ssár✐❛ ♣❛r❛ ❝♦❜r✐r ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❛ ❝❛rt❡✐r❛✳

  P♦❞❡♠♦s ❝♦♥❝❧✉✐r ❞❡st❡s r❡s✉❧t❛❞♦s q✉❡✿ ❼ P❡r❞❛s ✐♥❢❡r✐♦r❡s à ♣❡r❞❛ ❡s♣❡r❛❞❛ sã♦ s✉♣♦rt❛❞❛s ♣♦r ❧✐♥❤❛s ✐♥❞✐✈✐❞✉❛✐s ❞❡ ❝ré✲ ❞✐t♦✳ ❼ P❡r❞❛s ❛❞✐❝✐♦♥❛✐s s✉♣❡r✐♦r❡s à ♣❡r❞❛ ❡s♣❡r❛❞❛✱ ♠❛s ✐♥❢❡r✐♦r❡s à ♣❡r❞❛ ✐♥❡s♣❡r❛❞❛✱ sã♦ s✉♣♦rt❛❞❛s ♣❡❧❛s r❡s❡r✈❛s ❞❡ ❝❛♣✐t❛❧✳ ❼ ❋✐♥❛❧♠❡♥t❡✱ ❛s ♣❡r❞❛s s✉♣❡r✐♦r❡s à ♣❡r❞❛ ✐♥❡s♣❡r❛❞❛ ♣♦❞❡♠ r❡s✉❧t❛r ❡♠ ❢❛❧ê♥❝✐❛ ❞❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❡ t✐♣♦ ❞❡ ❛❝♦♥t❡❝✐♠❡♥t♦ ♣♦❞❡ s❡r ❝♦♥✲ tr♦❧❛❞♦✱ ❞❡✜♥✐♥❞♦ ♦ ♥í✈❡❧ ♦r✐❣✐♥❛❧ ❞♦ ❱❛❘✱ ♦✉ s❡❥❛ ❛❧t❡r❛r ♦ ✈❛❧♦r ❞❛ ♣✐♦r ♣❡r❞❛ ❡s♣❡r❛❞❛ ❞❡ ♠♦❞♦ ❛ q✉❡ s❡❥❛♠ t♦♠❛❞❛s ♠❡❞✐❞❛s ♣❛r❛ q✉❡ ♥ã♦ ❡①✐st❛♠ ♣❡r❞❛s s✉♣❡r✐♦r❡s às ♣❡r❞❛s ❡s♣❡r❛❞❛s ❡ ✐♥❡s♣❡r❛❞❛s✳

  ✶✳✸✳✷ ❊①❡♠♣❧♦s ❞❡ ▼♦❞❡❧♦s ♣❛r❛ ❈❛rt❡✐r❛s

❙❡❣✉♥❞♦ ♦ r✐s❝♦ ❞❡ ✉♠ ❞❡✈❡❞♦r ✐♥❞✐✈✐❞✉❛❧ é ♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛

❞❡t❡r♠✐♥❛r ❛ t❛①❛ ❞❡ ❥✉r♦ ❛ ❝♦❜r❛r ♣♦r ✉♠ ❡♠♣rést✐♠♦ ♦✉ ♦❜r✐❣❛çã♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛

♣r✐♦r✐❞❛❞❡ ❞♦s ❜❛♥❝♦s é ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❛ s✉❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❣❧♦❜❛❧✳ ❆ ❡st❛✲

❜✐❧✐❞❛❞❡ ❞♦ ❜❛♥❝♦✱ ❞❡♣❡♥❞❡ ♥ã♦ ❞❡ ❝❛❞❛ ❞❡✈❡❞♦r✱ ♠❛s s✐♠ ❞♦ ❝♦♥❥✉♥t♦ q✉❡ ❢♦r♠❛ ❛

s✉❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✱ ✐st♦ é✱ ❞❡♣❡♥❞❡ ❞♦ ❞❡s❡♠♣❡♥❤♦ ❣❧♦❜❛❧ ❞❛ s✉❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✳

❯♠ ❞♦s ❢❛❝t♦r❡s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♥✉♠❛ ❝❛rt❡✐r❛ é ♦ r✐s❝♦ ❞❡

❝♦♥❝❡♥tr❛çã♦ ✭❝♦♥❝❡♥tr❛t✐♦♥ r✐s❦✮✳ ❙❡ ❛ ❝❛rt❡✐r❛ ❡stá ❝♦♥❝❡♥tr❛❞❛ ♥✉♠❛ ár❡❛ ❣❡♦❣rá✜❝❛

r❡str✐t❛ ♦✉ ♥✉♠❛ só ár❡❛ ✐♥❞✉str✐❛❧✱ ❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ ✜❝❛ ♠✉✐t♦ ♠❛✐s ❡①♣♦st❛ ❛

❝❤♦q✉❡s ❡❝♦♥ó♠✐❝♦s✳ ❖s ❢❛❝t♦r❡s✲❝❤❛✈❡ ♥♦s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❝ré❞✐t♦ ♣❛r❛ ❝❛rt❡✐r❛s sã♦ ❛s ❝♦rr❡❧❛çõ❡s

  ✶✳✸ ▼♦❞❡❧♦s ♣❛r❛ ❈❛rt❡✐r❛s ✶✾

❞❡✈✐❞♦ à ❡s❝❛ss❡③ ❞❡ ❞❛❞♦s✳ P❛r❛ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ❛♣❧✐❝❛r ♦s ♠♦❞❡❧♦s✱ é ♥❡❝❡ssár✐♦

s✐♠♣❧✐✜❝❛r ❛❧❣✉♥s ♣r❡ss✉♣♦st♦s✳ ❖s r❡t♦r♥♦s ❞❡ ❝ré❞✐t♦ ✭r❡t♦r♥♦s ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❡♠♣rést✐♠♦s ♦✉ ♦❜r✐❣❛çõ❡s✮ ♥ã♦

sã♦ ♥♦r♠❛❧♠❡♥t❡ ❞✐str✐❜✉í❞♦s✱ ❛♦ ❝♦♥trár✐♦ ❞♦s r❡t♦r♥♦s ❞❡ ♠❡r❝❛❞♦✱ ♦ q✉❡ t♦r♥❛ ❛✐♥❞❛

♠❛✐s ❝♦♠♣❧✐❝❛❞♦ ❛ ♦❜t❡♥çã♦ ❞❛s ❝♦rr❡❧❛çõ❡s✳ ❆ ❛ss✐♠❡tr✐❛ ❞❛ ❞✐str✐❜✉✐çã♦ é ❝❛✉s❛❞❛

♣❡❧❛s ♣❡r❞❛s ❞❡ ❝ré❞✐t♦✳ ❆ss✐♠✱ ❛ ♠é❞✐❛ ❡ ❛ ✈❛r✐â♥❝✐❛ ♥ã♦ s❡rã♦ ❡st❛tíst✐❝❛s s✉✜❝✐❡♥t❡s

♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❛ ❞✐str✐❜✉✐çã♦ ❞♦s r❡t♦r♥♦s ❞❡ ❝ré❞✐t♦✳ ❆ ❋✐❣✉r❛ ✐❧✉str❛ ✉♠❛

tí♣✐❝❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❘❡t♦r♥♦s ❞❡ ❈ré❞✐t♦ ❡ ❝♦♠♣❛r❛✲❛ ❝♦♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ tí♣✐❝❛ ❞❡

❘❡t♦r♥♦s ❞❡ ▼❡r❝❛❞♦✳

  

❋✐❣✉r❛ ✶✳✸✿ ❉✐str✐❜✉✐çã♦ ❚✐♣í❝❛ ❞❡ ❘❡t♦r♥♦s ❞❡ ❈ré❞✐t♦ ❡ ❘❡t♦r♥♦s ❞❡ ▼❡r❝❛❞♦

◆❡st❛ s✉❜✲s❡❝çã♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣❛r❛ ❝❛r✲

t❡✐r❛s ❞❡ ❝❧✐❡♥t❡s✳ ❚♦❞♦s ♦s ♠♦❞❡❧♦s s❡ ❝♦♥❝❡♥tr❛♠ ♥❛ ❡st✐♠❛t✐✈❛ ❞❛s ♠❡❞✐❞❛s ❞❡ r✐s❝♦

❞❡ ❝ré❞✐t♦ ♣❛r❛ ❝❛rt❡✐r❛s✱ ❡♠❜♦r❛ ✉t✐❧✐③❡♠ ❞✐❢❡r❡♥t❡s té❝♥✐❝❛s✳

  ❙❡❣✉♥❞♦ ❛s r❡❝❡♥t❡s ❛❜♦r❞❛❣❡♥s s♦❜r❡ ❣❡stã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❡stã♦

✐♥t❡r❧✐❣❛❞❛s ❛♦s ♠♦❞❡❧♦s ♣❛r❛ ❝❛rt❡✐r❛s✳ ❖s ♠♦❞❡❧♦s ♣❛r❛ ❝❛rt❡✐r❛s tê♠ ❛ s✉❛ ♦r✐❣❡♠

♥❛ t❡♦r✐❛ ❞❛ ❞✐✈❡rs✐✜❝❛çã♦ ❞❡ ❍❛rr② ▼❛r❦♦✇✐t③ ✭✶✾✺✷✮✳ ❆ ✐❞❡✐❛ ✐♥✐❝✐❛❧ ❞❡ ▼❛r❦♦✇✐t③

❡r❛ q✉❡ ♦ r✐s❝♦ ❞❡ ✉♠ tít✉❧♦ ♣♦❞✐❛ s❡r ❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧❛ s✉❛ ✈❛r✐â♥❝✐❛ ❡ q✉❡ ♦ r❡t♦r♥♦

❞❡ ✉♠❛ ❝❛rt❡✐r❛ ♦✉ ❣r✉♣♦ ❞❡ tít✉❧♦s ❞❡♣❡♥❞✐❛ ❞❛ ✈❛r✐â♥❝✐❛ ❞♦s r❡t♦r♥♦s ✐♥❞✐✈✐❞✉❛✐s ❞♦s

tít✉❧♦s ❡ ❛✐♥❞❛ ❞❛ ❝♦✈❛r✐â♥❝✐❛ ❡♥tr❡ ♦s ♠❡s♠♦s✳

  ❖ ❡st✉❞♦ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❍❛rr② ▼❛r❦♦✇✐t③ ❜❛s❡✐❛✲s❡ ♥✉♠ ♠♦❞❡❧♦ ❞❡ ❞✐✈❡rs✐✜❝❛çã♦

q✉❡ ♣r♦♣õ❡ ❛ r❡❞✉çã♦ ❞♦ r✐s❝♦ ❛té ♣❡rt♦ ❞♦ r✐s❝♦ s✐st❡♠át✐❝♦✳ ❖ r✐s❝♦ s✐st❡♠át✐❝♦ é ♦

r✐s❝♦ ✐♥❡r❡♥t❡ ❛♦ s✐st❡♠❛✱ s❡♥❞♦ ✐♥✢✉❡♥❝✐❛❞♦ ♣♦r ❢❛❝t♦r❡s✴❞❡❝✐sõ❡s ♠❛❝r♦❡❝♦♥ó♠✐❝❛s✳

❆ ❞✐✈❡rs✐✜❝❛çã♦ ❢❛③ ❝♦♠ q✉❡ ♦ r✐s❝♦ t♦t❛❧ ❞❛ ❝❛rt❡✐r❛ s❡❥❛ ♠❡♥♦r q✉❡ ❛ s♦♠❛ ❞♦s r✐s✲

❝♦s ✐♥❞✐✈✐❞✉❛✐s ❞❡ ❝❛❞❛ ❛❝t✐✈♦✳ ❆ss✐♠✱ ♣❛rt❡ ❞♦ r✐s❝♦ t♦t❛❧ ❞♦ ❛❝t✐✈♦ é ❡❧✐♠✐♥❛❞♦ q✉❛♥❞♦

❡st❡ é ✐♥❝❧✉í❞♦ ♥❛ ❝❛rt❡✐r❛✱ s❡♥❞♦ q✉❡ ♦ r✐s❝♦ q✉❡ r❡st❛ r❡♣r❡s❡♥t❛ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡

r✐s❝♦ ❞♦ ❛❝t✐✈♦ ♥❛ ❝❛rt❡✐r❛ t♦t❛❧✳ ❆ ❞✐✈❡rs✐✜❝❛çã♦ ♥ã♦ ❞❡✈❡✱ ♥♦ ❡♥t❛♥t♦✱ s❡r s✉♣ér✢✉❛ ♦✉

❛❧❡❛tór✐❛✳ ❯♠❛ ❞✐✈❡rs✐✜❝❛çã♦ ✏♠á①✐♠❛✑ ♣♦❞❡ r❡❞✉③✐r ❞❡s♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦ r❡t♦r♥♦ ❞❛

❝❛rt❡✐r❛ ❡ ♦s ✐♥✈❡st✐❞♦r❡s ♣r❡❢❡r❡♠ ❝❛rt❡✐r❛s ❝♦♠ r❡t♦r♥♦ ♠❛✐♦r ❡ r✐s❝♦ ♠❛✐♦r ♦✉ r❡t♦r♥♦

♠❡♥♦r ❡ r✐s❝♦ ♠❡♥♦r✳ P❡r❛♥t❡ ✉♠ ♠❡r❝❛❞♦ ❡✜❝✐❡♥t❡✱ ♦ ✐♥✈❡st✐❞♦r ❞❡✈❡r✐❛ ♣r❡♦❝✉♣❛r✲s❡

❝♦♠ ❛ r❡❧❛çã♦ tít✉❧♦✴❝❛rt❡✐r❛✳

  ✷✵ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

❞❛ ❝❛rt❡✐r❛ ❝♦♠♦ ✉♠ t♦❞♦✳ ❆ss✐♠ s❡♥❞♦✱ ♦s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❡ ❝❛rt❡✐r❛s

  

❞✐❢❡r❡♥❝✐❛♠✲s❡ ❞♦s r❡st❛♥t❡s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣♦r t❡r❡♠ ❡♠ ❝♦♥t❛ ♦ r✐s❝♦

❞♦s ❛❝t✐✈♦s✱ ♠❛s t❛♠❜é♠ ♦ r✐s❝♦ ❞❛ ❝❛rt❡✐r❛ ❡♠ ❣❡r❛❧✳ P♦r ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❢♦❝❛❧✐③❛♠ ❛

❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦ ✐♥❞✐✈✐❞✉❛❧ ❡ ❛ ❛♥á❧✐s❡ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ♥❛ ❣❧♦❜❛❧✐❞❛❞❡✳ ❊♠ s✉♠❛✱

♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡❝t✐✈♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❡ ❝❛rt❡✐r❛s é ♦❜t❡r ❛ ❞✐str✐❜✉✐çã♦

❞❡ ♣❡r❞❛s ❞❡ ✉♠❛ ❝❛rt❡✐r❛ r❡s✉❧t❛♥t❡ ❞♦ ✐♥❝✉♠♣r✐♠❡♥t♦ ♦✉ ❞❛ ❞❡s✈❛❧♦r✐③❛çã♦ ❞❛ ♠❡s♠❛

♥✉♠ ❤♦r✐③♦♥t❡ t❡♠♣♦r❛❧ ❞❡t❡r♠✐♥❛❞♦✳ ❖s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❡ ❝❛rt❡✐r❛s ♠❛✐s

❞✐✈✉❧❣❛❞♦s ♥♦ ♠❡r❝❛❞♦ sã♦✿ ❈r❡❞✐t▼❡tr✐❝s✱ ❈r❡❞✐t❘✐s❦✰✱ ❈r❡❞✐tP♦rt❢♦❧✐♦❱✐❡✇ ❡ ❑▼❱✳

◆♦ q✉❡ s❡ s❡❣✉❡✱ s❡rá ❢❡✐t♦ ✉♠❛ ❞❡s❝r✐çã♦ s✉♠ár✐❛ ❞❡st❡s ♠♦❞❡❧♦s✳ ❈r❡❞✐t▼❡tr✐❝s

❊♠ ❆❜r✐❧ ❞❡ ✶✾✾✼✱ ♦ ❇❛♥❝♦ ❏✳P✳ ▼♦r❣❛♥ ❛♥✉♥❝✐♦✉ ♦ ♥♦✈♦ ♠♦❞❡❧♦ ❈r❡❞✐t▼❡tr✐❝s ✭✈❡r

❡st❡ ♠♦❞❡❧♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ❛ ❛❞♠✐♥✐str❛çã♦ ❞♦ r✐s❝♦ t♦t❛❧

❞❛s ❝❛rt❡✐r❛s ❞❡ ❝ré❞✐t♦ ❛tr❛✈és ❞❛ ♠❡t♦❞♦❧♦❣✐❛ ❱❛❘ ✲ ❱❛❧♦r ❡♠ ❘✐s❝♦✳ r❡❢❡r❡ q✉❡ ❡st❡ ♠ét♦❞♦ ❝♦♥s✐st❡ ♥✉♠ ♠ét♦❞♦ ❞❡ ♠✐❣r❛çã♦ ❞❡ ❝ré❞✐t♦

q✉❡ ❡st✐♠❛ ❛s ❛❧t❡r❛çõ❡s ❢✉t✉r❛s ❞♦ ✈❛❧♦r ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❡♠♣rést✐♠♦s ❡ ♦❜r✐❣❛çõ❡s✱

q✉❡ s❡ ❞❡✈❡♠ ❛ ♠✉❞❛♥ç❛s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ❞❡✈❡❞♦r❡s ❞❡ ❝ré❞✐t♦✳ ❖ ♠ét♦❞♦ ❜❛s❡✐❛✲

s❡ ♥❛s ❝❛t❡❣♦r✐❛s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ✉s✉❛✐s ❡ ❛ r❡s♣❡❝t✐✈❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ✭✈❡❥❛✲s❡ ❛

♠❛tr✐③ ❞❡ tr❛♥s✐çã♦

  ❙❡❣✉♥❞♦ ♦ ❱❛❘ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ ♠❡❞✐❞❛ ❞❛ ♠❛✐♦r ♦✉ ♣✐♦r

♣❡r❞❛ ❡s♣❡r❛❞❛ ♥✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❛❝t✐✈♦s✱ ❞❡♥tr♦ ❞❡ ✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ❡ ✐♥t❡r✈❛❧♦ ❞❡

❝♦♥✜❛♥ç❛ ❡st✐♣✉❧❛❞♦✳ ❖ ❱❛❘ é ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♣❡r❞❛ ♠á①✐♠❛ q✉❡ ✉♠❛ ❝❛rt❡✐r❛ ♣♦❞❡

❛♣r❡s❡♥t❛r ❞✉r❛♥t❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦✳ ❇❛s❡✐❛✲s❡ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦

❞♦s ❛❝t✐✈♦s q✉❡ ❛ ❝♦♥st✐t✉❡♠ ❡ é ❝❛❧❝✉❧❛❞❛ ❡♠ ❢✉♥çã♦ ❞❛s ✈♦❧❛t✐❧✐❞❛❞❡s ❡ ❝♦rr❡❧❛çõ❡s

❛♥t❡r✐♦r❡s✳

  ❖ ❱❛❘✱ ❝♦♠♦ ♠❡❞✐❞❛ ❞❡ r✐s❝♦✱ ❝♦♠❡ç♦✉ ❛ s❡r ✉t✐❧✐③❛❞♦ ♥❛ ár❡❛ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦

❛♣ós ❛ ❝r✐❛çã♦ ❞♦ ❈r❡❞✐t▼❡tr✐❝s✱ ❛té ❡♥tã♦ ❡r❛ ❛♣❡♥❛s ✉t✐❧✐③❛❞♦ ♣❛r❛ ♠❡♥s✉r❛r ♦ r✐s❝♦

❞❡ ♠❡r❝❛❞♦✳ ❉❡♥tr♦ ❞❛ ❛♥á❧✐s❡ ❡ ❛✈❛❧✐❛çã♦ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ❡st❛ ♠❡t♦❞♦❧♦❣✐❛ tr❛t❛ ❛s ♠✉❞❛♥✲ ç❛s ❞❡ q✉❛❧✐❞❛❞❡ ❞❡ ❝ré❞✐t♦✱ ❞❡s❡♥✈♦❧✈❡♥❞♦ três ❝♦♠♣♦♥❡♥t❡s ❜ás✐❝❛s✿ ✶✳ ❉❡✜♥✐çã♦ ❞♦ ✈❛❧♦r ❡①♣♦st♦ ❛♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦❀

  ✷✳ ▼❡♥s✉r❛çã♦ ❞❛s ✈♦❧❛t✐❧✐❞❛❞❡s ❞♦ ✈❛❧♦r ❞❡✈✐❞♦ à ♠✉❞❛♥ç❛ ❞❛ q✉❛❧✐❞❛❞❡ ❞♦ ❝ré❞✐t♦❀ ✸✳ ▼❡❞✐çã♦ ❞❛s ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ♦s ❛❝t✐✈♦s ❖ ❈r❡❞✐t▼❡tr✐❝s ♣r♦❝✉r❛ ✐❞❡♥t✐✜❝❛r✱ ❛❧é♠ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✱ ♦ ❱❛❧✉❡✲❛t✲

❘✐s❦ ✭❱❛❘✮ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❡♠♣rést✐♠♦s ♥✉♠ ❤♦r✐③♦♥t❡ q✉❡ ✐♥❝❧✉✐ ✈❛❧♦r✐③❛çõ❡s ❡ ❞❡s✲

  

✈❛❧♦r✐③❛çõ❡s ❞❛ q✉❛❧✐❞❛❞❡ ❞❡ ❝ré❞✐t♦ ❡ ❛s ♣♦ssí✈❡✐s ♠✐❣r❛çõ❡s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ✭r❛t✐♥❣✮✳

❖ ♦❜❥❡❝t✐✈♦ ❞♦ ❈r❡❞✐t▼❡tr✐❝s é ❡st✐♠❛r ❛ ❞✐str✐❜✉✐çã♦ ❞♦ ✈❛❧♦r ❞❡ q✉❛❧q✉❡r ❝❛rt❡✐r❛ ❞❡

  ✶✳✸ ▼♦❞❡❧♦s ♣❛r❛ ❈❛rt❡✐r❛s ✷✶ ❖ ❈r❡❞✐t▼❡tr✐❝s ♦❢❡r❡❝❡✱ ❛ss✐♠✱ ✉♠❛ ❛❜♦r❞❛❣❡♠ ♠❛✐s ♣♦♥❞❡r❛❞❛ ❛♦s ❧✐♠✐t❡s ❞❡

  ❝ré❞✐t♦✳ ❙❡❣✉♥❞♦ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❢♦r❛♠ ❞❡✜♥✐❞♦s ♦s ♣r❡ss✉♣♦st♦s✿ ❼ ❆ ú♥✐❝❛ ❢♦♥t❡ ❞❡ ✐♥❝❡rt❡③❛ é ❛ ♠✐❣r❛çã♦ ❞❡ ❝ré❞✐t♦✱ ♦✉ s❡❥❛✱ ♦s ♠♦✈✐♠❡♥t♦s ♥❛ ❡s❝❛❧❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦✳ ❼ ❉❡♥tr♦ ❞❛ ♠❡s♠❛ ❝❧❛ss❡ t♦❞♦s ♦s ❡♠✐t❡♥t❡s sã♦ ❤♦♠♦❣é♥❡♦s✱ ✐st♦ é✱ t♦❞♦s ❡❧❡s tê♠ ❛s ♠❡s♠❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❡ ❞❡ ❞❡❢❛✉❧t✳ ❼ ❆s t❛①❛s ❞❡ ❥✉r♦ sã♦ ❞❡t❡r♠✐♥✐st❛s ✭❝♦♠♦ s❡ ♥ã♦ ❤♦✉✈❡ss❡ ✐♥❝❡rt❡③❛✮✳ ❆ss✐♠✱ ♦ ✈❛❧♦r ❢✉t✉r♦ ❞♦s tít✉❧♦s é ❝❛❧❝✉❧❛❞♦ ✉t✐❧✐③❛♥❞♦ ❛s t❛①❛s ❞❡ ❤♦❥❡✳ ❼ ❖ ✈❛❧♦r ❞❛s ❛❝çõ❡s ❞❛ ❡♠♣r❡s❛ ✭❊q✉✐t② ✈❛❧✉❡✮ é ✉♠ ❜♦♠ s✉❜st✐t✉t♦ ♣❛r❛ ♦ ✈❛❧♦r ❞♦s ❛❝t✐✈♦s ❞❛ ❡♠♣r❡s❛✳ ❆ss✐♠✱ ❛s ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ♦s ✈❛❧♦r❡s ❞♦s ❛❝t✐✈♦s ♣♦❞❡♠ s❡r ❛♣r♦①✐♠❛❞❛s ♣♦r ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ♦s ♣r❡ç♦s ❞❛s ❛❝çõ❡s✳ ◆❛ ✈❡r❞❛❞❡✱ ❛s ❛❝çõ❡s ✭❡q✉✐t②✮ sã♦ ♠❛✐s ✈♦❧át❡✐s q✉❡ ♦s ❛❝t✐✈♦s ❞❡✈✐❞♦ à ❛❧❛✈❛♥❝❛❣❡♠✳

  ❖ ♠ét♦❞♦ ❞❛ ❈r❡❞✐t▼❡tr✐❝s ✉t✐❧✐③❛ ❞♦✐s t✐♣♦s ♣r✐♥❝✐♣❛✐s ❞❡ ❝♦♥str✉çã♦✿ ❼ ❘✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣❛r❛ ✉♠ ú♥✐❝♦ ✐♥str✉♠❡♥t♦✳ ❼ ❘✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣❛r❛ t♦❞❛ ❛ ❝❛rt❡✐r❛✱ q✉❡ r❡♣r❡s❡♥t❛ ❛s ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ♦s ❡✈❡♥t♦s ❞❡ ❝ré❞✐t♦✳ P❛r❛ ❡st✐♠❛r ❛ ❛❧t❡r❛çã♦ ❞♦ ✈❛❧♦r ❞❡ ✉♠❛ ú♥✐❝❛ ♦❜r✐❣❛çã♦ ❞✉r❛♥t❡ ♦ ❡s♣❛ç♦ ❞❡ ✶

❛♥♦✱ é ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r ♦ ♣r❡ç♦ ❞❛ ♦❜r✐❣❛çã♦ ♣❛r❛ ❝❛❞❛ ♠✐❣r❛çã♦ ❞❡ ❝ré❞✐t♦ ♣♦ssí✈❡❧✳

  

P❛r❛ ❝❛❧❝✉❧❛r ❛s ♠❡❞✐❞❛s ❞❡ r✐s❝♦ ❡ ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♦❜r✐❣❛çã♦

s❡❣✉❡✲s❡ ♦s ♣❛ss♦s s❡❣✉✐♥t❡s✿ ✶✳ ❊st✐♠❛r✴♦❜t❡r ❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦✳ P♦❞❡ ✈❡r✲s❡✱ ♥❛ ❚❛❜❡❧❛ ❡♠ ❛♥❡①♦✱ ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦✳ ✷✳ ❊st✐♠❛r ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ♣❛r❛ ❛s ❛♥t✐❣❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ♦❜r✐❣❛çã♦✳ ✸✳ ❯t✐❧✐③❛r ❛s t❛①❛s ❞❡ ❥✉r♦ ♣❛r❛ ❝❛❞❛ ❝❛t❡❣♦r✐❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♣❛r❛ ❛✈❛❧✐❛r ❛ ♦❜r✐✲

  ❣❛çã♦ ❡♠ ❝❛❞❛ ❡st❛❞♦✳ ❱ é ♦ ✈❛❧♦r ❞❛ ♦❜r✐❣❛çã♦✳ ✹✳ ❈á❧❝✉❧♦ ❞❡ ♠❡❞✐❞❛s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ❖ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❛ ♦❜r✐❣❛çã♦ é ♦ ✈❛❧♦r ♠é❞✐♦ ❞❡ t♦❞♦s ♦s ❡st❛❞♦s✿

  

X

µ ≡ p V ,

  

V s s

s

s

  ♦♥❞❡ p é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♣❡rt❡♥❝❡r ❛♦ ❡st❛❞♦ s✳ ❉❡✈❡ t❡r✲s❡ ❡♠ ❝♦♥t❛ q✉❡ ♦ ❈r❡❞✐t▼❡tr✐❝s ❝♦♥s✐❞❡r❛ ❛s ♣❡r❞❛s ♦r✐❣✐♥❛❞❛s ♣♦r ❞❡❢❛✉❧t

  ✷✷ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ❯♠❛ ♠❡❞✐❞❛ ❛❧t❡r♥❛t✐✈❛ ❞❡ r✐s❝♦ t❛♠❜é♠ ❝❛❧❝✉❧❛❞♦ ♣❡❧♦ ❈r❡❞✐t▼❡tr✐❝s é ♦ ❞❡s✈✐♦

  ♣❛❞rã♦ ❞♦ ✈❛❧♦r ❞❛ ♦❜r✐❣❛çã♦✱ ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡✿

  X X

  2

  2

  2

σ ≡ p (V − µ ) = p (∆V ) .

s s V s S

V s s

  ❈r❡❞✐t❘✐s❦✰

❖ ❈r❡❞✐t❘✐s❦✰ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❡❧❛ ❈r❡❞✐t ❙✉✐ss❡ ❋✐♥❛♥❝✐❛❧ Pr♦❞✉❝ts ✭❈❙❋P✮✱ ❛tr❛✈és

❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✐❞❡✐❛s ❥á ✉t✐❧✐③❛❞❛s ♥❛ ár❡❛ ❞♦s s❡❣✉r♦s ♣❛r❛ ♠♦❞❡❧❛r ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳

❖ ❈r❡❞✐t❘✐s❦✰ é ✉♠ ♠♦❞❡❧♦ ❡st❛tíst✐❝♦ ♣❛r❛ r✐s❝♦ ❞❡ ❞❡❢❛✉❧t q✉❡ ❝♦♥s✐❞❡r❛ ❛ t❛①❛

❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ❝♦♠♦ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♥tí♥✉❛✳ ◆❡st❡ ♠♦❞❡❧♦ ♥ã♦ sã♦ ❢❡✐t❛s

s✉♣♦s✐çõ❡s s♦❜r❡ ❛s ❝❛✉s❛s ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦✱ ❝♦♥s✐❞❡r❛♠✲s❡ ❛♣❡♥❛s ❝❡♥ár✐♦s ❞❡ ✐♥❝✉♠✲

♣r✐♠❡♥t♦✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ p✱ ❡ ❞❡ ❝✉♠♣r✐♠❡♥t♦ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ 1 − p✳ ❚❡♠♦s

❡♥tã♦ q✉❡✿

  ❼ P❛r❛ ✉♠ ❡♠♣rést✐♠♦ ♦✉ tít✉❧♦✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♥✉♠ ❞❛❞♦ ♣❡rí♦❞♦✱ ✉♠ ❛♥♦✱ ♣♦r ❡①❡♠♣❧♦✱ é ✐❣✉❛❧ ❡♠ q✉❛❧q✉❡r ♦✉tr♦ ❛♥♦❀ ❼ P❛r❛ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❡♠♣rést✐♠♦s ♦✉ tít✉❧♦s✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❞❡ ✉♠ tít✉❧♦ ♦✉ ❡♠♣rést✐♠♦ ❡♠ ♣❛rt✐❝✉❧❛r é ♣❡q✉❡♥❛✱ ❡ ♦ ♥ú♠❡r♦ ❞❡ ❞❡❢❛✉❧ts q✉❡

  ♦❝♦rr❡♠ ♥✉♠ ❞❛❞♦ ♣❡rí♦❞♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ♥ú♠❡r♦ ❞❡ ❞❡❢❛✉❧ts q✉❡ ♦❝♦rr❡♠ ❡♠ q✉❛❧q✉❡r ♦✉tr♦ ♣❡rí♦❞♦✳ P❛rt✐♥❞♦ ❞❡st❡s ♣r❡ss✉♣♦st♦s✱ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ ♥ú♠❡r♦ ❞❡ ❞❡❢❛✉❧ts✱

◆✱ ♥✉♠❛ ❝❛rt❡✐r❛ ❡ ♥✉♠ ❞❛❞♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦✱ ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞❛ ♣♦r ✉♠ Pr♦❝❡ss♦

  ❞❡ P♦✐ss♦♥ ❍♦♠♦❣é♥❡♦✿

  n −µ

  µ e P [

◆❂♥] = ✱ ♣❛r❛ n ∈ N

n!

  

♦♥❞❡ µ é ♦ ✈❛❧♦r ♠é❞✐♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❞❡❢❛✉❧ts ♣♦r ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ✭♥♦r♠❛❧♠❡♥t❡ ✶

❛♥♦✮✳ ❙❡❣✉♥❞♦ ♦ ❈r❡❞✐t❘✐s❦✰ ✉t✐❧✐③❛ ❡st❡ ♠♦❞❡❧♦ ❜ás✐❝♦ ❞❡ ❡st❛tíst✐❝❛✱

♣❛r❛ ❞❡r✐✈❛r ❡①♣r❡ssõ❡s ❞❡ ❢♦r♠❛ ❢❡❝❤❛❞❛ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ♣❡r❞❛ ❞❡ ✉♠❛ ❝❛rt❡✐r❛✳

❖ ❈r❡❞✐t❘✐s❦✰ ❛♣❡♥❛s ✐♥❝✐❞❡ s♦❜r❡ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✱ ♦ q✉❡ t❡♠ ✈❛♥t❛❣❡♥s ❡

❞❡s✈❛♥t❛❣❡♥s✳ ❆ ♣r✐♥❝✐♣❛❧ ✈❛♥t❛❣❡♠ é q✉❡ é ❢á❝✐❧ ❞❡ ✐♠♣❧❡♠❡♥t❛r ❡ é ❝♦♠♣✉t❛❝✐♦♥❛❧✲

  

♠❡♥t❡ ❛tr❛❝t✐✈♦✳ ❆ ♣r✐♥❝✐♣❛❧ ❞❡s✈❛♥t❛❣❡♠ é q✉❡ ✐❣♥♦r❛ ♦ r✐s❝♦ ❞❡ ♠✐❣r❛çã♦ ✭❞❡s❝❧❛ss✐✲

✜❝❛çã♦ ✲ ❞♦✇♥❣r❛❞✐♥❣✮✳ ❈r❡❞✐tP♦rt❢♦❧✐♦❱✐❡✇

❖ ♠♦❞❡❧♦ ❈r❡❞✐t▼❡tr✐❝s ❝♦♥s✐❞❡r❛✱ ❝♦♠♦ s❡ ✈✐✉ ❛♥t❡r✐♦r♠❡♥t❡✱ ✉♠❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦

❝♦♥st❛♥t❡✳ ❆ ♠❛tr✐③ é ✐♥❝♦♥❞✐❝✐♦♥❛❧✱ ♥♦ s❡♥t✐❞♦ ❡♠ q✉❡ é ✉♠❛ ♠é❞✐❛ ❤✐stór✐❝❛ ❜❛s❡❛❞❛

  ✶✳✸ ▼♦❞❡❧♦s ♣❛r❛ ❈❛rt❡✐r❛s ✷✸

q✉❡ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t ❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ♠✐❣r❛çã♦ ❞❡ ❝ré❞✐t♦ ❡stã♦ ❧✐❣❛❞❛s ❛♦

  

❡st❛❞♦ ❞❛ ❡❝♦♥♦♠✐❛✱ ❝♦♠ ♦s ❝✐❝❧♦s ❞❡ ❝ré❞✐t♦ ❛ s❡❣✉✐r✲s❡ ♣❡rt♦ ❞♦s ❝✐❝❧♦s ❡❝♦♥ó♠✐❝♦s✳

❊♠ ♣❡rí♦❞♦s ❞❡ r❡❝❡ssã♦✱ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ❡ ❞♦✇♥❣r❛❞❡ ❛✉♠❡♥t❛♠❀ ❡♠

♣❡rí♦❞♦s ❞❡ ❡①♣❛♥sã♦ ✈❡r✐✜❝❛✲s❡ ♦ ♦♣♦st♦✳ ❲✐❧s♦♥ ✭✶✾✾✼❛✱ ❜✮ ♣r♦♣õ❡ ✉♠ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞♦ ♠ét♦❞♦ ❞❡ ♠✐❣r❛çã♦ ❞❡ ❝ré❞✐t♦

❞❡♥♦♠✐♥❛❞❛ ❈r❡❞✐tP♦rt❢♦❧✐♦❱✐❡✇✱ q✉❡ ❡stá ❛❣♦r❛ ❝♦♠❡r❝✐❛❧♠❡♥t❡ ✐♠♣❧❡♠❡♥t❛❞♦ ♣❡❧❛

▼❝❑✐♥s❡②✳

  ◆❡st❡ ♠ét♦❞♦✱ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ✈❛r✐❛♠ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❡❧❛s sã♦ ❝♦♥❞✐❝✐♦♥❛✐s ❛♦ ❡st❛❞♦ ❞❛ ❡❝♦♥♦♠✐❛✳ ❖ ♠♦❞❡❧♦ ❈r❡❞✐tP♦rt❢♦❧✐♦❱✐❡✇ ✉t✐❧✐③❛ ✉♠ í♥❞✐❝❡ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ❡st❛❞♦ ❞❛ ❡❝♦♥♦✲ ♠✐❛✳ ❖ í♥❞✐❝❡ é ❢✉♥çã♦ ❞❡ ✈❛r✐á✈❡✐s ♠❛❝r♦✲❡❝♦♥ó♠✐❝❛s✱ t❛✐s ❝♦♠♦✿ ❼ ❚❛①❛ ❞❡ ❞❡s❡♠♣r❡❣♦❀ ❼ ❚❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞♦ P■❇❀ ❼ ❚❛①❛s ❞❡ ❥✉r♦❀ ❼ ❚❛①❛s ❞❡ ❝â♠❜✐♦❀ ❼ ●❛st♦s ❞♦ ❣♦✈❡r♥♦❀ ❼ ❚❛①❛ ❞❡ ♣♦✉♣❛♥ç❛ ❛❣r❡❣❛❞❛✳ ❆s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❝♦♥❞✐❝✐♦♥❛❞❛s ❞❡ tr❛♥s✐çã♦ sã♦ ❡st✐♠❛❞❛s ❝♦♠♦ ❢✉♥çõ❡s ❞❡st❡ ♠❛❝r♦✲í♥❞✐❝❡✳ ❆ ❧✐♠✐t❛çã♦ ❞❡st❛ ❛❜♦r❞❛❣❡♠ é q✉❡✱ s❡♥❞♦ ❝♦♥❞✐❝✐♦♥❛❧✱ ❡①✐❣❡ ✉♠❛ ♠❛✐♦r q✉❛♥t✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧ts ❡ ♠✐❣r❛çã♦ ❞❡ ❞❛❞♦s ♣❛r❛ ❝❛❧✐❜r❛r ♦ ♠♦❞❡❧♦✳ ❖ ♠♦❞❡❧♦ ❈r❡❞✐tP♦rt❢♦❧✐♦❱✐❡✇ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ♠♦❞❡❧♦ ♠✉❧t✐✲❢❛❝t♦r ✉t✐❧✐③❛❞♦ ❡♠

s✐♠✉❧❛çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ❝♦♥❞✐❝✐♦♥❛✐s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♠✐❣r❛çã♦ ❞❡ r❛t✐♥❣✱ ♦✉

  

❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦✱ ♣❛r❛ ❞✐❢❡r❡♥t❡s ❝♦♥❥✉♥t♦s ❞❡ ❡♠♣r❡s❛s ♦✉ ♣❛ís❡s✱ q✉❡ ❡stã♦ ❝♦♥✲

❞✐❝✐♦♥❛❞❛s ❛ ❢❛❝t♦r❡s ♠❛❝r♦❡❝♦♥ó♠✐❝♦s✱ ❝♦♠♦ ❛ t❛①❛ ❞❡ ❞❡s❡♠♣r❡❣♦✱ t❛①❛s ❞❡ ❥✉r♦ ❡

❝â♠❜✐♦✱ ❡t❝✳ ❆ss✐♠✱ ❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡st❡ ♠♦❞❡❧♦ ✐♥t❡r❧✐❣❛ ♦s ❢❛❝t♦r❡s ♠❛❝r♦❡❝♦♥ó♠✐❝♦s

à ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡ ♠✐❣r❛çã♦ ❞❛ q✉❛❧✐❞❛❞❡ ❞❡ ❝ré❞✐t♦✳

  ◆❡st❡ ♠♦❞❡❧♦✱ ♦s ❞❡✈❡❞♦r❡s sã♦ ❛❣r✉♣❛❞♦s ❡♠ s❡❝t♦r❡s ❞❡✜♥✐❞♦s ♣♦r ❝❧❛ss❡s ❡❝♦♥ó✲ ♠✐❝❛s✱ ♣❛ís❡s ❡ r❛t✐♥❣s✳ ❊st❛ ❛❜♦r❞❛❣❡♠ ❝♦♥stró✐ ✉♠ ♠♦❞❡❧♦ ❡❝♦♥♦♠étr✐❝♦ q✉❡ ♣r♦❝✉r❛ ❡①♣❧✐❝❛r ❡ ❞❡t❡r✲

♠✐♥❛r ❛ ♣r♦❛❜✐❧✐❞❛❞❡ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ♣❛r❛ ♦s ❞✐❢❡r❡♥t❡s s❡❝t♦r❡s✱ ❜❛s❡❛♥❞♦✲s❡ ❡♠

  

✈❛r✐á✈❡✐s ♠❛❝r♦❡❝♦♥ó♠✐❝❛s ❤✐stór✐❝❛s ❡ ❡♠ sér✐❡s t❡♠♣♦r❛✐s ❞❡ t❛①❛s ♠é❞✐❛s ❞❡ ✐♥❝✉♠✲

♣r✐♠❡♥t♦✳ ❑▼❱

❖ ♠♦❞❡❧♦ ❑▼❱ ❢♦✐ ❝r✐❛❞♦ ♣❡❧❛ ❑▼❱ ❈♦r♣♦r❛t✐♦♥ ♣❛r❛ ❡st✐♠❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡

❞❡❢❛✉❧t ❞❡ ❡♠♣r❡s❛s ❡ ❜❛s❡✐❛✲s❡ ♥❛ ❛❜♦r❞❛❣❡♠ ❞❡ ▼❡rt♦♥ ❞❡ ✶✾✼✹ ✭♣♦❞❡ ✈❡r✲s❡ ❡♠

♥❛ q✉❛❧ ✉♠❛ ❡♠♣r❡s❛ é ❝♦♥s✐❞❡r❛❞❛ ❡♠ ✐♥s♦❧✈ê♥❝✐❛ q✉❛♥❞♦

  ✷✹ ❈❛♣ít✉❧♦ ✶✳ ❆♥á❧✐s❡ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

❞❡ ❞❡❢❛✉❧t é ❞❛❞❛ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ✈❛❧♦r ❞❡ ♠❡r❝❛❞♦ ❞♦ ❛❝t✐✈♦ ❡ ♦ ✈❛❧♦r ♦✜❝✐❛❧ ❞♦s

  ♣❛ss✐✈♦s✱ ✈❡r P❛r❛ ❛♣❧✐❝❛çã♦ ❞❡st❡ ♠♦❞❡❧♦ é ♥❡❝❡ssár✐♦ ❝♦♠❡ç❛r ♣♦r ❡st✐♠❛r ♦ ✈❛❧♦r ❞❡ ♠❡r❝❛❞♦

❞♦ ❛❝t✐✈♦ ❞❛ ❡♠♣r❡s❛✱ ♣♦✐s ❡st❡ ♥ã♦ é ♦❜s❡r✈á✈❡❧✳ ❈♦♥s✐❞❡r❛✲s❡ ♦ ❝❛♣✐t❛❧ ♣ró♣r✐♦ ❞❛

❡♠♣r❡s❛ ❝♦♠♦ ✉♠❛ ♦♣çã♦ ❞❡ ❝♦♠♣r❛ s♦❜r❡ ♦s s❡✉s ❛❝t✐✈♦s ❡ ❡st❛❜❡❧❡❝❡✲s❡ ✉♠❛ r❡❧❛çã♦

❡♥tr❡ ♦s ❞❛❞♦s ❞❡ ♠❡r❝❛❞♦✱ ♦❜s❡r✈á✈❡✐s ❡♠ r❡❧❛çã♦ às ❛❝çõ❡s ❞❛ ❡♠♣r❡s❛✱ ❝♦♠ ♦s ✈❛❧♦r❡s

♥ã♦ ♦❜s❡r✈á✈❡✐s ❞♦s ❛❝t✐✈♦s✳

  ❖ ♦❜❥❡❝t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞♦ ❑▼❱ é ❛♣r❡ç❛r ♦ ❝ré❞✐t♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❡♠♣r❡s❛ ❝♦♠♦ ✉♠❛ ♦♣çã♦ ❞❡ ❝♦♠♣r❛✳ ❚♦❞♦s ♦s ♠♦❞❡❧♦s ❞❡ ❝ré❞✐t♦ ♣❛r❛ ❝❛rt❡✐r❛s ❞✐s❝✉t✐❞♦s ♥❡st❛ s✉❜s❡❝çã♦ ♣❛r❡❝❡♠ s❡r

❞✐❢❡r❡♥t❡s✱ ♥♦ ❡♥t❛♥t♦✱ ✈ár✐♦s ❡st✉❞♦s ❝♦♠♣❛r❛t✐✈♦s tê♠ ♠♦str❛❞♦ q✉❡✱ q✉❛♥❞♦ ❡st❡s

  

♠♦❞❡❧♦s sã♦ ❡①❡❝✉t❛❞♦s ✉t✐❧✐③❛♥❞♦ ♣❛râ♠❡tr♦s ❝♦♥s✐st❡♥t❡s✱ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s sã♦

❜❛st❛♥t❡ s❡♠❡❧❤❛♥t❡s✳ ◆❡st❛ s❡❝çã♦✱ ❢♦✐ r❡❛❧✐③❛❞♦ ✉♠ r❡s✉♠♦ t❡ór✐❝♦ ❞♦s ♣r✐♥❝✐♣❛✐s ♠♦❞❡❧♦s ❞❡ ❣❡stã♦ ❞❡

r✐s❝♦ ❞❡ ❝ré❞✐t♦ q✉❡ sã♦ ✉t✐❧✐③❛❞♦s ❛❝t✉❛❧♠❡♥t❡ ❡♠ ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s ❝♦♠ ♦ ✐♥t✉✐t♦

  

❞❡ ❢❛③❡r ✉♠ ❡♥q✉❛❞r❛♠❡♥t♦ ❣❡r❛❧ ❞♦s ♠♦❞❡❧♦s ❡ té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ♣❛r❛ ♠♦❞❡❧❛çã♦ ❞❡

r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ❘❡ss❛❧t❡✲s❡ q✉❡✱ ♥ã♦ t❡♥❞♦ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ❡s❣♦t❛r ♦ ❛ss✉♥t♦✱ ❢♦r❛♠

❛❜♦r❞❛❞♦s ❛♣❡♥❛s ❛❧❣✉♥s ❛s♣❡❝t♦s ♠❛✐s r❡❧❡✈❛♥t❡s ❞❡ss❡s ♠♦❞❡❧♦s✳

  ✷✺ ❈❛♣ít✉❧♦ ✷ ❘✐s❝♦ ❡ ❙♣r❡❛❞

❙♣r❡❛❞ é ♣♦r ❞❡✜♥✐çã♦ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♣r❡ç♦ ❞❡ ❝♦♠♣r❛ ❡ ♦ ♣r❡ç♦ ❞❡ ✈❡♥❞❛ ❞❡ ✉♠❛

♠❡s♠❛ ❛❝çã♦✱ tít✉❧♦ ♦✉ ❣❛r❛♥t✐❛✳ ❖ s♣r❡❛❞ ❜❛♥❝ár✐♦✱ é ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ t❛①❛ ❞❡

❥✉r♦ q✉❡ ❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s ♣❛❣❛♠ ♥❛ ❛q✉✐s✐çã♦ ❞♦ ❞✐♥❤❡✐r♦ ❡ ❛ q✉❡ ❝♦❜r❛♠ ❞♦s

❝❧✐❡♥t❡s✳ ➱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✏t❛①❛ ❞❡ r✐s❝♦✑✳

  ❖ s♣r❡❛❞ ❞❡✈❡rá r❡♣r❡s❡♥t❛r✱ ♣❛r❛ ❛❧é♠ ❞♦ ❧✉❝r♦ ❞❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s✱ ✉♠❛

♠❡❞✐❞❛ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ q✉❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝❧✐❡♥t❡ r❡♣r❡s❡♥t❛ ♣❛r❛ ❛ ✐♥st✐t✉✐çã♦

✜♥❛♥❝❡✐r❛ ♣❡r♠✐t✐♥❞♦ ❛ ❝♦♠♣❡♥s❛çã♦ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❛ss✉♠✐❞♦ ♣❡❧❛ ✐♥st✐t✉✐çã♦✳ ❚❡♥❞♦ ♣♦r ❜❛s❡ ❡st❛ ✐❞❡✐❛ ♣r❡t❡♥❞❡✲s❡✱ ♥❡st❡ ❝❛♣ít✉❧♦✱ ♦❜t❡r ❡①♣r❡ssõ❡s s✐♠♣❧❡s q✉❡

❞❡✜♥❛♠ ♦ s♣r❡❛❞ ❛ ❛♣❧✐❝❛r ❛ ❝❛❞❛ ❝❧✐❡♥t❡✱ ❡♠ ❢✉♥çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦

✭❞❡❢❛✉❧t✮ ❡ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ✭r❡❝♦✈❡r② r❛t❡✮✳

  ❚♦❞❛✈✐❛✱ ❞❡t❡r♠✐♥❛r ♦ s♣r❡❛❞ ♣❛r❛ ❝❛❞❛ ❝❧✐❡♥t❡ ❞❡ ✉♠❛ ❞❛❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦

♥ã♦ é s✐♠♣❧❡s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣rát✐❝♦✳ ❆❞♠✐t❡✲s❡ q✉❡✱ ♣❛r❛ ❡ss❛ ❝❛rt❡✐r❛✱ ❤á ❞❛❞♦s

r❡❧❛t✐✈♦s ❛ ❢❛❝t♦r❡s ❞❡ r✐s❝♦ ❜❡♠ ❝♦♠♦ ❞❛❞♦s s♦❜r❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ❡ ❞❛s

t❛①❛s ❞❡ r❡❝✉♣❡r❛çã♦✳

  ❖ ♦❜❥❡❝t✐✈♦ ♥❛s s❡❝çõ❡s s❡❣✉✐♥t❡s é ❡st❛❜❡❧❡❝❡r✱ ❡♠ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✱ ✉♠❛ ❢ór♠✉❧❛

q✉❡ ♣❡r♠✐t❛ ❞❡t❡r♠✐♥❛r ♦ s♣r❡❛❞ ❡♠ ❢✉♥çã♦ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❡ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡

❞❡ ❞❡❢❛✉❧t✳ ❖❜s❡r✈❛♠♦s q✉❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣rát✐❝❛ ❤❛❜✐t✉❛❧ ✭✈❡r

❡①✐st❡♠ ❞✉❛s ♠❡t♦❞♦❧♦❣✐❛s ❞❡ ❛♣r❡ç❛♠❡♥t♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✿ ❆ ♠❡t♦❞♦❧♦❣✐❛ ❛❝t✉❛✲

r✐❛❧ ❡ ❛ ♠❡t♦❞♦❧♦❣✐❛ ✜♥❛♥❝❡✐r❛ ♦✉ ❞❡ ❛♣r❡ç❛♠❡♥t♦ ❛tr❛✈és ❞❡ ✉♠❛ ♠❡❞✐❞❛ ♥❡✉tr❛ ❢❛❝❡

❛♦ r✐s❝♦✳ ❊st❛s ♠❡t♦❞♦❧♦❣✐❛s sã♦ ❞✐❢❡r❡♥❝✐❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❡❧♦ ✉s♦ ❞❛ ♠❡❞✐❞❛

❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❛t✉r❛❧ ❡ ❞❛ ♠❡❞✐❞❛ ♠❛rt✐♥❣❛❧❛ ❡q✉✐✈❛❧❡♥t❡✳ ❈♦♠ ❜❛s❡ ♥♦ ♠♦❞❡❧♦ ❞✐s❝r❡t♦ ❛ ✉♠ ♣❡rí♦❞♦ ❞❡s❡♥✈♦❧✈❡✲s❡✱ ❛tr❛✈és ❞❛s ♠❡t♦❞♦❧♦❣✐❛s

❛❝t✉❛r✐❛❧ ❡ ✜♥❛♥❝❡✐r❛✱ ✉♠❛ ❡①♣r❡ssã♦ q✉❡ ❞❡✜♥❡ ♦ s♣r❡❛❞ ♠í♥✐♠♦ ❛ ❛♣❧✐❝❛r ❛ ✉♠ ❞❛❞♦

❝❧✐❡♥t❡ ❡♠ ❢✉♥çã♦ ❞♦ r✐s❝♦ q✉❡ ❡st❡ r❡♣r❡s❡♥t❛✳

  P♦st❡r✐♦r♠❡♥t❡ ❞❡s❡♥✈♦❧✈❡✲s❡ ❛ ♠❡s♠❛ ❡①♣r❡ssã♦ ❝♦♠ ❜❛s❡ ♥♦ ♠♦❞❡❧♦ ❝♦♥tí♥✉♦✱ ❛tr❛✈és ❞❛ ♠❡t♦❞♦❧♦❣✐❛ ❛❝t✉❛r✐❛❧✳ ❆ ❡①♣r❡ssã♦ ♦❜t✐❞❛ é t❛♠❜é♠ ❡st❛❜❡❧❡❝✐❞❛✱ ❡♠ ❡st✉❞♦s s♦❜r❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♣♦r

❞✐✈❡rs♦s ❛✉t♦r❡s✱ ♥♦♠❡❛❞❛♠❡♥t❡✱ ❆♠❜♦s

  ✷✻ ❈❛♣ít✉❧♦ ✷✳ ❘✐s❝♦ ❡ ❙♣r❡❛❞

✐♠♣♦rtâ♥❝✐❛ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❡ ❛✐♥❞❛ ❞❡ ❢♦r♠❛ ❛ ♣♦ss✐❜✐❧✐t❛r ❛❧❣✉♠❛s ❛♥❛❧♦❣✐❛s✳

  ❆❧❣✉♥s ❞♦s ❝♦♥❝❡✐t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡st❡ ❝❛♣ít✉❧♦ ❡stã♦ ❞❡✜♥✐❞♦s ♥♦ ●❧♦ssár✐♦✳ ✷✳✶ ❱❛r✐á✈❡✐s ❡ P❛râ♠❡tr♦s

❊♠ q✉❛❧q✉❡r ❛❜♦r❞❛❣❡♠ s♦❜r❡ ♠♦❞❡❧❛çã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ é ♥❡❝❡ssár✐♦ ❝♦♥s✐❞❡r❛r

❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s✱ ✈❛r✐á✈❡✐s ❡ ♣❛râ♠❡tr♦s ❜ás✐❝♦s✳

  ❈♦♥s✐❞❡r❡✲s❡ q✉❡✿ ❯♠❛ ♦❜r✐❣❛çã♦ ❞❡ ❝✉♣ã♦ ③❡r♦ ✭❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞✮ é ✉♠ ❝♦♥tr❛t♦ ❞❡ ❞í✈✐❞❛ q✉❡✱ ♥❛

♠❛t✉r✐❞❛❞❡ T ✱ ♣❛❣❛ ✉♠❛ q✉❛♥t✐❛ ✭♥♦♠✐♥❛❧✮ ♥♦ ✈❛❧♦r ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ♠♦♥❡tár✐❛✱ ❡ ♥ã♦

♣❛❣❛ ❝✉♣õ❡s✳

  ❈♦♥s✐❞❡r❡♠✲s❡✿ ) t t≥0

  ❼ (X ✲ ❱❛r✐á✈❡❧ q✉❡ ❞❡s❝r❡✈❡ ❛ ❡✈♦❧✉çã♦ ❞♦ ♣r❡ç♦ ❞❡ ✉♠ ❛❝t✐✈♦ ❡ q✉❡ s❡ ❝♦♥s✐❞❡r❛ ✶ s❡r ✉♠ Pr♦❝❡ss♦ ❞❡ ▼❛r❦♦✈ ❀ ❼ τ ✲ ▼♦♠❡♥t♦ ❞❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t q✉❡ ❞❡✈❡ s❡r ✉♠ t❡♠♣♦ ❞❡ ♣❛r❛❣❡♠✳ ❆s✲ s♦❝✐❛❞❛ ❛ ❡st❛ ✈❛r✐á✈❡❧ t❡r✲s❡✲á ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✱ q✉❡ ❞❡✈❡rá s❡r ✉♠ ♣❛râ♠❡tr♦ ❞♦ ♠♦❞❡❧♦❀

  ❼ R(T ) ✲ ❚❛①❛ ❞❡ ❘❡❝✉♣❡r❛çã♦ ❡ q✉❡ t❛♠❜é♠ s❡ ❝♦♥s✐❞❡r❛ s❡r ✉♠ ♣r♦❝❡ss♦ ❞❡ ▼❛r❦♦✈✳ ◆♦t❡✲s❡ q✉❡ ❛ ✐♥❢♦r♠❛çã♦ ❞✐s♣♦♥í✈❡❧ s♦❜r❡ ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ♥❡♠ s❡♠♣r❡ é ✜á✈❡❧✱ ♦ q✉❡ s✉❣❡r❡ q✉❡ ❡st❡ ♣r♦❝❡ss♦ ❞❡✈❛ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ ♣❛râ♠❡tr♦ ❝♦♥st❛♥t❡ ✭♦s ✈❛❧♦r❡s ❞❡ R r❡❣✐st❛❞♦s ♣♦r ✉♠❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ r❡❢❡r❡♠✲s❡ ❛♣❡♥❛s ❛♦ ✈❛❧♦r r❡❝✉♣❡r❛❞♦ ♣❡❧❛ ♠❡s♠❛✱ ♥♦ ❡♥t❛♥t♦✱ ❛s ❡♠♣r❡s❛s ❡♥❝❛rr❡❣✉❡s ❞❡ r❡❝✉♣❡r❛r ♦ ♣❛tr✐♠ó♥✐♦ ♣♦❞❡♠ r❡❝✉♣❡r❛r k > R ♦✉ k < R ♠❛s ❞❡✈♦❧✈❡♠ ❛♣❡♥❛s à ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛ ❛ ♣❛rt❡ ❛❝♦r❞❛❞❛ ❝♦♠ ❛ ♠❡s♠❛✮✳

  ❼ B(t, T ) ✲ Pr❡ç♦✱ ♥♦ ✐♥st❛♥t❡ t✱ ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦ ❞❡ ❝✉♣ã♦ ③❡r♦ ❧✐✈r❡ ❞❡ ✐♥❝✉♠✲ ♣r✐♠❡♥t♦ ✭❞❡❢❛✉❧t✲❢r❡❡ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞✮ ❝♦♠ ✈❡♥❝✐♠❡♥t♦ ❡♠ T ❀ B(t, T ) ❼ ˜ ✲ Pr❡ç♦✱ ♥♦ ✐♥st❛♥t❡ t✱ ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦ ❞❡ ❝✉♣ã♦ ③❡r♦ ❝♦♠ ♣♦ss✐❜✐❧✐❞❛❞❡

❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ✭❞❡❢❛✉❧t❛❜❧❡ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞✮ ❝♦♠ ✈❡♥❝✐♠❡♥t♦ ❡♠ T ❀

❼ s(t, T ) ✲ ❙♣r❡❛❞ ❞❡ ❝ré❞✐t♦ ♥♦ ✐♥st❛♥t❡ t✱ ❡ T ❛ ❞❛t❛ ❞❡ ✈❡♥❝✐♠❡♥t♦ ❞♦ ❝♦♥tr❛t♦ ❞❡ ❝ré❞✐t♦✳ ❈♦♥s✐❞❡r❛✲s❡ q✉❡ ♦ s♣r❡❛❞ t❛♠❜é♠ ❞❡✈❡rá s❡r ✉♠ ♣r♦❝❡ss♦ ❞❡ ▼❛r❦♦✈✳

  ✶ ●❡♥❡r✐❝❛♠❡♥t❡✱ ✉♠ ♣r♦❝❡ss♦ é ❞❡ ▼❛r❦♦✈ s❡ ♦s ❡st❛❞♦s ❛♥t❡r✐♦r❡s sã♦ ✐rr❡❧❡✈❛♥t❡s ♣❛r❛ ❛ ♣r❡❞✐çã♦

❞♦s ❡st❛❞♦s s❡❣✉✐♥t❡s✱ ❞❡s❞❡ q✉❡ ♦ ❡st❛❞♦ ❛❝t✉❛❧ s❡❥❛ ❝♦♥❤❡❝✐❞♦✳ ❆s ❞❡✜♥✐çõ❡s ❢♦r♠❛✐s ♣♦❞❡♠ s❡r ❧✐❞❛s

  ❡♠

  ✷✳✷ ❆♣r❡ç❛♠❡♥t♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✷✼ ❼ ❆❞✐❝✐♦♥❛❧♠❡♥t❡ ❞❡✈❡ ❝♦♥s✐❞❡r❛r✲s❡ t❛♠❜é♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛❧❣✉♠❛ ❡str✉t✉r❛ t❡♠✲

  ♣♦r❛❧ ✭t❡r♠ str✉❝t✉r❡✮✱ ✐st♦ é✱ ❝♦♥s✐❞❡r❛r✲s❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ r❡❧❛çõ❡s ❡♥tr❡ ❛s t❛①❛s ❞❡ ❥✉r♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❡ ❞✉❛s ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞s ❝♦♠ ❞✐❢❡r❡♥t❡s ♠❛t✉r✐❞❛❞❡s✳ ❙❡❥❛ B(0, T )✱ ♦ ♣r❡ç♦✱ ❤♦❥❡✱ ❞❡ ✉♠❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❧✐✈r❡ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ✭r✐s❦

❢r❡❡✮ ❡ ❝♦♠ ♠❛t✉r✐❞❛❞❡ T ✳ ■♥❡r❡♥t❡ ❛ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦ ✜①❛✱ r✱ ❡ t❡♥❞♦ ❡♠ ❝♦♥t❛ ❛

  

✐♥❡①✐stê♥❝✐❛ ❞❡ ❛r❜✐tr❛❣❡♠✱ ♣♦❞❡✲s❡ r❡♣r❡s❡♥t❛r ♦ ♣r❡ç♦ ❞❡st❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝♦♠♦

❡♠ (rT )

  B(0, T ) × e = 1 (−rT )

  ⇔ B(0, T ) = e ✭✷✳✶✮ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♦ ♣r❡ç♦✱ ❡♠ t > 0✱ ❞❡ ✉♠❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❧✐✈r❡ ❞❡ r✐s❝♦ ✭r✐s❦✲❢r❡❡✮ ❡ ❝♦♠ ♠❛t✉r✐❞❛❞❡ T ❝♦♠ t❛①❛ ❞❡ ❥✉r♦ ✜①❛ r✱ s❡rá✿ r(T −t)

  B(t, T ) × e = 1 −r(T −t)

  ⇔ B(t, T ) = e ✭✷✳✷✮ ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡❣✉♥❞♦ t❡♥❞♦ ❡♠ ❝♦♥t❛ q✉❡ ♦ ♣r❡ç♦ ❞❡ ✉♠❛

❞❡❢❛✉❧t❛❜❧❡ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ t❡♠ ❞❡ s❡r ♠❡♥♦r q✉❡ ♦ ♣r❡ç♦ ❞❡ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❧✐✈r❡

  

❞❡ r✐s❝♦ ✭♣❛r❛ ❝♦♠♣❡♥s❛r ♦ r❡s♣❡❝t✐✈♦ r✐s❝♦✮✱ ♦ ♣r❡ç♦ ❞❡ ✉♠❛ ❞❡❢❛✉❧t❛❜❧❡ ③❡r♦ ❝♦✉♣♦♥

❜♦♥❞ ❝♦♠ s♣r❡❛❞ ❞❡ ❝ré❞✐t♦✱ s(t, T )✱ ♥♦ t❡♠♣♦ t ❡ ❝♦♠ ✈❡♥❝✐♠❡♥t♦ ❡♠ T ✱ é t❛❧ q✉❡✿ (T −t)×s(t,T )

  ˜

B(t, T )e = B(t, T ).

❆ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❞❡ ❢♦r♠❛ ❛ ♦❜t❡r✲s❡ ♦ ♣r♦❝❡ss♦ q✉❡ ❞❡s❝r❡✈❡ ♦

s♣r❡❛❞✿

  ! ˜

  1 B(t, T )

s(t, T ) = − ln .

  ✭✷✳✸✮ T − t B(t, T ) ✷✳✷ ❆♣r❡ç❛♠❡♥t♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦

◆❡st❛ s❡❝çã♦✱ ❞✐s❝✉t❡♠✲s❡ ❡ ❝♦♠♣❛r❛♠✲s❡ ❛s ♠❡t♦❞♦❧♦❣✐❛s ❛❝t✉❛r✐❛❧ ❡ ✜♥❛♥❝❡✐r❛✱ ❝♦♠

✈✐st❛ ❛♦ ❛♣r❡ç❛♠❡♥t♦ ❞❡ tít✉❧♦s ❞❡ ❝ré❞✐t♦ ❝♦♠ r✐s❝♦✳ ❆♣r❡s❡♥t❛✲s❡ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ♦

♠✉♥❞♦ r❡❛❧ ♦✉ ♠❡❞✐❞❛ ❢ís✐❝❛✱ q✉❡ ♠♦❞❡❧❛ ❛ r❡❛❧ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✱ ❡ ✉♠❛ ♠❡❞✐❞❛

❡q✉✐✈❛❧❡♥t❡ ♠❛rt✐♥❣❛❧❛ ♦✉ ♠❡❞✐❞❛ ♥❡✉tr❛ ❢❛❝❡ ❛♦ r✐s❝♦ ✳ ✷✳✷✳✶ ▼♦❞❡❧♦ ❉✐s❝r❡t♦ ❛ ❯♠ P❡rí♦❞♦ ❆ ▼❡t♦❞♦❧♦❣✐❛ ❆❝t✉❛r✐❛❧

◆❛ ❛❜♦r❞❛❣❡♠ ❛❝t✉❛r✐❛❧✱ s❡❣✉♥❞♦ ♦s ♣r❡ç♦s sã♦ ❝❛❧❝✉❧❛❞♦s ❝♦♠♦

  ✷✽ ❈❛♣ít✉❧♦ ✷✳ ❘✐s❝♦ ❡ ❙♣r❡❛❞

  ✶

❞♦ ♣ré♠✐♦ ❞❡ r✐s❝♦ é ♠✉✐t❛s ✈❡③❡s r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ ♥♦çã♦ ❞❡ ❝❛♣✐t❛❧ ❡❝♦♥ó♠✐❝♦ ✳

❊♠ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ❛ ❛❜♦r❞❛❣❡♠ ❛❝t✉❛r✐❛❧ é ❛♣❧✐❝❛❞❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ❛♣r❡ç❛♠❡♥t♦

❞❡ ❡♠♣rést✐♠♦s ♥ã♦ ♥❡❣♦❝✐❛❞♦s ✭♥♦♥✲tr❛❞❡❞✮ ♦✉ ❞❡ ♣r♦❞✉t♦s ❡str✉t✉r❛❞♦s r❡❧❛❝✐♦♥❛❞♦s

❝♦♠ tít✉❧♦s s❡♠ ❧✐q✉✐❞❡③✳ ❙❡❥❛ (Ω, F, P)✱ ♦ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡♠ q✉❡ P é ❛ ♠❡❞✐❞❛

♥❛t✉r❛❧ ♦❜s❡r✈á✈❡❧✳ ■❧✉str❛çã♦✿

❈♦♥s✐❞❡r❡✲s❡ ✉♠❛ ♦❜r✐❣❛çã♦ ♦✉ ✉♠ ❡♠♣rést✐♠♦ ❞❡ 100u.m.✱ s✉❥❡✐t♦s ❛ ✉♠❛ t❛①❛ ❞❡

❥✉r♦ ❝♦♥st❛♥t❡ r✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ✐❣✉❛❧ ❛ p✱ ✉♠❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ R

❡ ✉♠ s♣r❡❛❞ s✳ ◆♦ ❝❛s♦ ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦ ❧✐✈r❡ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ✭❞❡❢❛✉❧t✲❢r❡❡✮ ♦✉ ❞❡ ✉♠ ❡♠♣rés✲

t✐♠♦✱ ❛s 100u.m.✱ ♥❛ ♠❛t✉r✐❞❛❞❡✱ ❝♦♥✈❡rt❡r✲s❡✲ã♦ ❡♠ 100(1 + r)u.m.✳ ❆ ❛✉sê♥❝✐❛ ❞❡

r✐s❝♦ ❞❡ ❝ré❞✐t♦ ✐♠♣❧✐❝❛ q✉❡ ♦ s♣r❡❛❞ ❡ ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ♥ã♦ s❡❥❛♠ ♥❡❝❡ssár✐♦s✳

◆♦ ❝❛s♦ ❞❡ ✉♠ ❡♠♣rést✐♠♦ ♦✉ ♦❜r✐❣❛çã♦ ❞❡❢❛✉❧t❛❜❧❡ ✭s✉❥❡✐t♦s ❛♦ r✐s❝♦ ❞❡ ❞❡❢❛✉❧t✮ ❤á

q✉❡ t❡r ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✱ ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❡ ♦ s♣r❡❛❞✳

❆ss✐♠✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ (1 − p) ✭♥ã♦ ♦❝♦rr❡r ❞❡❢❛✉❧t✮ ❛s 100u.m. ❝♦♥✈❡rt❡r✲s❡✲ã♦ ❡♠

100(1 + r)(1 + s)u.m.

  ❡ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ p ❝♦♥✈❡rt❡r✲s❡✲ã♦ ❡♠ 100R(1+r)(1+s)u.m.✳ ❆ ❚❛❜❡❧❛ ✐❧✉str❛ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞♦s ❝❛s❤✲✢♦✇s ❞♦ ❡♠♣rést✐♠♦ ♦✉ ♦❜r✐❣❛çã♦ ♥♦s ❞♦✐s ❝❡♥ár✐♦s ❞✐st✐♥t♦s✳ ❚✐♠❡ ❉❡❢❛✉❧t✲❢r❡❡ ❉❡❢❛✉❧t❛❜❧❡ t = 0 100 ✶✵✵

t = 1 100(1 + r) [100(1 − p) + R100p] (1 + r)(1 + s)

❚❛❜❡❧❛ ✷✳✶✿ ❈❛s❤✲❋❧♦✇s ❛ ❯♠ P❡rí♦❞♦

  ◆♦t❛ ✷✳✶✳ ◆♦t❡✲s❡ q✉❡✱ ❝♦♠♦ R < 1✱ t❡♠♦s q✉❡

100(1 − p) + R100p < 100

♣❡❧♦ q✉❡ ♦ s♣r❡❛❞ r❡♣r❡s❡♥t❛ ✉♠❛ t❛①❛ ❞❡ ♣ré♠✐♦ ❞❡ r✐s❝♦ q✉❡ r❡❧❛❝✐♦♥❛ ♦ ✐♥✈❡st✐♠❡♥t♦

  ❞❡ r✐s❝♦ ❝♦♠ ♦ ✐♥✈❡st✐♠❡♥t♦ s❡♠ r✐s❝♦✳ ❊♠ t❡r♠♦s ❞❡ ❡①♣❡❝t❛t✐✈❛s ❞❡ ♠❡r❝❛❞♦✱ ❞❡✈❡rá t❡r✲s❡ q✉❡✿ E [

  ❝❛s❤✲✢♦✇ ❞❡ ✉♠❛ ❉❡❢❛✉❧t✲❢r❡❡] = E[ ❝❛s❤✲✢♦✇ ❞❡ ✉♠❛ ❉❡❢❛✉❧t❛❜❧❡], ✭✷✳✹✮ ✶

  ❈❛♣✐t❛❧ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦ ❜❛♥❝♦ ♣❡r♠❛♥❡❝❡r s♦❧✈❡♥t❡ ❡♠ ❝❛s♦ ❞❡ ♣❡r❞❛ ❡①tr❡♠❛✳

  ✷✳✷ ❆♣r❡ç❛♠❡♥t♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✷✾ ♦✉ s❡❥❛✱

h i

100(1 + r) = 100(1 − p) + R100p (1 + r)(1 + s),

  ✭✷✳✺✮ ❝❛s♦ ❝♦♥trár✐♦✱ ♥✐♥❣✉é♠ ✐♥✈❡st✐r✐❛ ♥❛ ♦❜r✐❣❛çã♦ ♠❡♥♦s r❡♥tá✈❡❧✳ ❈♦♥s✐❞❡r❡✲s❡ ❡♥tã♦ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿ ❚❡♦r❡♠❛ ✷✳✶✳

  

◆♦ ♠♦❞❡❧♦ ❞✐s❝r❡t♦ ❛ ✉♠ ♣❡rí♦❞♦✱ ♣❛r❛ ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♦ s♣r❡❛❞ é ❢✉♥çã♦ ❞❛ t❛①❛

❞❡ r❡❝✉♣❡r❛çã♦ ❡ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♥❛ ♠❡❞✐❞❛ ♥❛t✉r❛❧ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ P ❡ é

❞❛❞♦ ♣♦r✿ (1 − R)p

s = .

  ✭✷✳✻✮ 1 − (1 − R)p ❉❡♠♦♥str❛çã♦✳ ❇❛st❛ r❡s♦❧✈❡r ❛ ❊q✉❛çã♦ ❡♠ ♦r❞❡♠ ❛ s✳ Pr♦♣♦s✐çã♦ ✷✳✶✳ P❛r❛ (1 − R)p << 1 t❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ❝❧áss✐❝♦✿

s ≈ (1 − R)p.

  ✭✷✳✼✮ ❉❡♠♦♥str❛çã♦✳ ❉❡❞✉③✲s❡ ❛ ♣❛rt✐r ❞❛ ❊q✉❛çã♦ ❋❛③❡♥❞♦ ❛ ❛♥á❧✐s❡✱ ❛tr❛✈és ❞❛ ▼❡t♦❞♦❧♦❣✐❛ ❋✐♥❛♥❝❡✐r❛✱ ♦ ♦❜❥❡❝t✐✈♦ é ♦❜t❡r ✉♠ r❡✲ s✉❧t❛❞♦ ❝♦♠♣❛rá✈❡❧✳

  ❆ ▼❡t♦❞♦❧♦❣✐❛ ❋✐♥❛♥❝❡✐r❛

❙♦❜ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ ❛♣r❡ç❛♠❡♥t♦✱ r❡❢❡r❡ q✉❡✱ ❝♦♠ ❜❛s❡ ♥❛ ♣r♦✲

❜❛❜✐❧✐❞❛❞❡ ♥❡✉tr❛ ❢❛❝❡ ❛♦ r✐s❝♦✱ ♦s ♣r❡ç♦s sã♦ ❝❛❧❝✉❧❛❞♦s ❝♦♥❢♦r♠❡ ♦s ✈❛❧♦r❡s ❡s♣❡r❛❞♦s

  ✷

❞❡s❝♦♥t❛❞♦s s♦❜ ❛❧❣✉♠❛ ♠❡❞✐❞❛ ♠❛rt✐♥❣❛❧❛ ❡q✉✐✈❛❧❡♥t❡✳ ❊st❛ ❛❜♦r❞❛❣❡♠ é ❜❛s❡❛❞❛

♥❛s ♥♦çõ❡s ❞❡ ❛✉sê♥❝✐❛ ❞❡ ❛r❜✐tr❛❣❡♠ ❡ ❞❡ ❤❡❞❣✐♥❣ ❞✐♥â♠✐❝♦✳

  ❍♦❥❡ ❡♠ ❞✐❛✱ ❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ❛♣r❡ç❛♠❡♥t♦✱ ❝♦♠ ❜❛s❡ ♥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❡✉tr❛

❢❛❝❡ ❛♦ r✐s❝♦✱ é st❛♥❞❛r❞ ♣❛r❛ ❛♣r❡ç❛r tít✉❧♦s ♥♦♥✲❞❡❢❛✉❧t❛❜❧❡s✳ ❊♠ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ é

✉s❛❞♦ ♣❛r❛ ❛♣r❡ç❛r tít✉❧♦s ♥❡❣♦❝✐❛❞♦s✱ ♥♦♠❡❛❞❛♠❡♥t❡✱ ❝♦r♣♦r❛t❡ ❜♦♥❞s✱ ❞❡❢❛✉❧t s✇❛♣s

❞❡ ❝ré❞✐t♦ ❡ tít✉❧♦s ❞❡r✐✈❛❞♦s s❡♠❡❧❤❛♥t❡s ❛ ❡st❡s ♣r♦❞✉t♦s✳

  ❯♠❛ ♠❡❞✐❞❛ ♠❛rt✐♥❣❛❧❛ ❡q✉✐✈❛❧❡♥t❡ ♦✉ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❡✉tr❛ ❢❛❝❡ ❛♦

r✐s❝♦✱ é ✉♠❛ ♠❡❞✐❞❛ ❛rt✐✜❝✐❛❧ Q ❡q✉✐✈❛❧❡♥t❡ à ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❢ís✐❝❛ P✱ ❞❡ t❛❧

❢♦r♠❛ q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ♣r❡ç♦s ❞❡s❝♦♥t❛❞♦s ❞♦ ❛❝t✐✈♦ r❡❧❡✈❛♥t❡ é ✉♠❛ Q−martingala✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛ ✭♦ ❞❡s✐❣♥❛❞♦✱ ❋✐rst ❋✉♥❞❛♠❡♥t❛❧

❚❤❡♦r❡♠ ♦❢ ❆ss❡t Pr✐❝✐♥❣✮✱ ✉♠ ♠♦❞❡❧♦ ♣❛r❛ ♦s ♣r❡ç♦s ❞♦s tít✉❧♦s é ❧✐✈r❡ ❞❡ ❛r❜✐tr❛❣❡♠

  ✶ ●❡♥❡r✐❝❛♠❡♥t❡✱ ✉♠❛ ♠❛rt✐♥❣❛❧❛ é ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ s❡♠ t❡♥❞ê♥❝✐❛✱ q✉❡ ❞❡s❝r❡✈❡ ✉♠ ❥♦❣♦ ❡q✉✐❧✐❜r❛❞♦✳ ❆ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ♣♦❞❡ s❡r ❝♦♥s✉❧t❛❞❛ ❡♠

  ✸✵ ❈❛♣ít✉❧♦ ✷✳ ❘✐s❝♦ ❡ ❙♣r❡❛❞

s❡ ❡ só s❡ ❛❞♠✐t❡ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ♠❛rt✐♥❣❛❧❛ ❡q✉✐✈❛❧❡♥t❡ Q✳ ◆♦ ♠♦❞❡❧♦ ❞✐s❝r❡t♦✱ Q é

  ❞❛❞❛ s✐♠♣❧❡s♠❡♥t❡ ♣❡❧❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❛rt✐✜❝✐❛❧ q✳ ❈♦♥s✐❞❡r❡✲s❡ ♥♦✈❛♠❡♥t❡ ✉♠ ❡♠♣rést✐♠♦ ♦✉ ✉♠❛ ♦❜r✐❣❛çã♦ ❞❡ 100u.m.✱ s✉❥❡✐t♦s ❛

✉♠❛ t❛①❛ ❞❡ ❥✉r♦ ❝♦♥st❛♥t❡ r✱ ❝♦♠ ✉♠❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ R ❡ ✉♠ s♣r❡❛❞ s✳ ◆♦ ❝❛s♦

❞❡ ✉♠ ❡♠♣rést✐♠♦ ♦✉ ♦❜r✐❣❛çã♦ ❞❡❢❛✉❧t✲❢r❡❡✱ ❛s 100u.m.✱ ♥❛ ♠❛t✉r✐❞❛❞❡✱ ❝♦♥✈❡rt❡r✲s❡✲

ã♦ ❡♠ 100(1 + r)u.m.✳ P♦❞❡ ❞✐③❡r✲s❡ q✉❡ é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ r❡❧❛çã♦ s❡♠❡❧❤❛♥t❡ à ❞♦ ❚❡♦r❡♠❛ ♠❛s

❝♦♠ ✉♠❛ ♠❡❞✐❞❛ ♠❛rt✐♥❣❛❧❛ ✭♣♦❞❡ ✈❡r✲s❡ ✉♠ ❡①❡♠♣❧♦ ❡♠ ➱

s✉✜❝✐❡♥t❡ ❞❡✜♥✐r ❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ Q = (q, 1 − q)✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡✿

  100 B(0, 1) = 1 + r

  1 ˜

B(0, 1) = (1 − q)100 + R100q

✭✷✳✽✮

  1 + r

❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❡①♣r❡ssõ❡s ✈❡r✐✜❝❛✲s❡ q✉❡✱ ♦ s♣r❡❛❞ é ❞❛❞♦ ♣♦r✿

s(0, 1) = − ln ˜ B(0, 1) − lnB(0, 1) 100 100

  

= −ln (1 − q) + Rq + ln

1 + r 1 + r

 

  100 (1 − q) + Rq

  1+r

 

= −ln

  

100

1+r

= −ln (1 − (1 − R)q)) ✭✷✳✾✮

  Pr♦♣♦s✐çã♦ ✷✳✷✳ P❛r❛ (1 − R)q << 1✱ t❡♠✲s❡ q✉❡✿

s(0, 1) ≈ (1 − R)q.

❊st❛ r❡❧❛çã♦ ❡♥tr❡ ♦ s♣r❡❛❞✱ ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t é ✉♠❛

r❡❧❛çã♦ s❡♠❡❧❤❛♥t❡ à ♦❜t✐❞❛ ♥❛ Pr♦♣♦s✐çã♦ ♠❛s ❛❣♦r❛ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡

  ❞❡ ❞❡❢❛✉❧t ❞❛❞❛ ♥❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❡✉tr❛ ❢❛❝❡ ❛♦ r✐s❝♦✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛s sér✐❡s ❣❡♦♠étr✐❝❛s✱

  n

  R P ∞ P ∞ n 1 x

  1 x = = dx = −ln(1 − x) ❡ ✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ♦ ❡rr♦ ❝♦♠❡t✐❞♦ n=0 n=1 1−x n 1−x

  ❛♦ ❞❡s♣r❡③❛r ❛s ♣♦tê♥❝✐❛s ❞❡ x ❞❡ ❣r❛✉ s✉♣❡r✐♦r ❛ ✷ é ♣❡q✉❡♥♦ s❡ x << 1✳ P♦rt❛♥t♦✱ ♣♦❞❡ ❡s❝r❡✈❡r✲s❡ −ln(1 − x) ≈ x✱ ❡ ❛♥❛❧♦❣❛♠❡♥t❡✱ −ln (1 − (1 − R)q)) ≈ (1 − R)q.

  ✷✳✷ ❆♣r❡ç❛♠❡♥t♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✸✶

  ✷✳✷✳✷ ▼♦❞❡❧♦ ❡♠ ❚❡♠♣♦ ❈♦♥tí♥✉♦ ❆ ▼❡t♦❞♦❧♦❣✐❛ ❆❝t✉❛r✐❛❧

❈♦♥s✐❞❡r❡✲s❡ ♥♦✈❛♠❡♥t❡ ✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ (Ω, F, P) ❡ ❛✐♥❞❛ ✉♠ ♣r♦❝❡ss♦

  ) t t≥0

  

❇r♦✇♥✐❛♥♦ (B s♦❜r❡ ❡st❡ ❡s♣❛ç♦✳ ❙✉♣♦♥❤❛✲s❡ t❛♠❜é♠ q✉❡ ❡①✐st❡ ✉♠ ❤♦r✐③♦♥t❡ ❞❡

θ

t❡♠♣♦ T ❡ q✉❡ ♦ ♣r❡ç♦ ❞❛ ♦❜r✐❣❛çã♦ ❝♦♠ r✐s❝♦ é ❞❛❞♦ ♣♦r ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ X ✱

t m

  

❡♠ ❢✉♥çã♦ ❞❡ ✉♠ ♣❛râ♠❡tr♦ θ ∈ Θ ⊂ R t❛❧ q✉❡✱ ♥♦ ❝❛s♦ ❞❛ ♥ã♦ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✱

s❡ t❡♠✿ (

  θ θ θ

dX = µ dt + σ dB , t ∈]0, T ]

t t t t

  ✭✷✳✶✵✮ θ

  X = X(0) θ

  ) P❡❧♦ q✉❡✱ (X t≥0 ✱ ♦ ♣r❡ç♦ ❞❛ ♦❜r✐❣❛çã♦ s❡♠ ✐♥❝✉♠♣r✐♠❡♥t♦ ✭❞❡❢❛✉❧t✮✱ é ❞❛❞♦ ♣❡❧❛ t

  θ

s♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❊st♦❝ást✐❝❛ ✭❊❉❊✮ ❡♠ q✉❡ µ t é ♦ ❝♦❡✜❝✐❡♥t❡

  θ

❞❡ t❡♥❞ê♥❝✐❛ ✭❞r✐❢t✮ ❡ σ é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞✐❢✉sã♦ ✭✈♦❧❛t✐❧✐t②✮✳ ❖s ❝♦❡✜❝✐❡♥t❡s ❞❡

t m

t❡♥❞ê♥❝✐❛ ❡ ❞✐❢✉sã♦ ♣♦❞❡♠ ❞❡♣❡♥❞❡r ❞❡ ✉♠ ♣❛râ♠❡tr♦ ✈❡❝t♦r✐❛❧ θ ∈ Θ ⊂ R ✳ P♦r

❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❱❛s✐❝❡❦ ♣❛r❛ ❛ ❞❡s❝r✐çã♦ ❞❛ t❛①❛ ❞❡ ❥✉r♦ r ✭✈❡❥❛✲s❡

♣❛❣✳✷✽✽✮✱ t❡♠✲s❡ q✉❡✿

dr = a(b − r )dt + σdB ,

t t t

  

3

× R × R ⊆ R )

❧♦❣♦ θ = (a, b, σ) ∈ (R ✳ ◆♦t❡✲s❡ q✉❡✱ ♥❡st❡ ♠♦❞❡❧♦✱ ♦ ♣r❡ç♦ ❞❛

  

), t > 0

t

  ♦❜r✐❣❛çã♦ ❞❡ ❝✉♣ã♦ ③❡r♦ é ❢✉♥çã♦ ❞❡ (r ✳ ❙✉♣♦♥❤❛✲s❡ ❛❣♦r❛ q✉❡ ♦❝♦rr❡ ❞❡❢❛✉❧t ♥♦ ✐♥st❛♥t❡ τ ∈ [0, T ]✳ ◆❡ss❡ ❝❛s♦✱ t❡r❡♠♦s ❛ θ

  ) t∈[0,T ] ❞✐♥â♠✐❝❛ ❞♦ ♣r❡ç♦ ❞❛ ♦❜r✐❣❛çã♦ ❞❛❞❛ ♣♦r (Y t ✱ ♦♥❞❡

  ′

  θ θ θ

Y = X [0,τ [ (t) + X [τ,T ] (t).

  ✶■ ✶■ ✭✷✳✶✶✮ t t t

  

♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ ❛♥t❡s ❞❡ ♦❝♦rr❡r ❞❡❢❛✉❧t ♦ ♣r♦❝❡ss♦ ❡✈♦❧✉✐ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❊❉❊

♣❛r❛ ♦ ✈❛❧♦r ❞♦ ♣❛râ♠❡tr♦ θ ❡✱ ❛♣ós ♦❝♦rr❡r ❞❡❢❛✉❧t✱ r❡s♣❡✐t❛ ❛ ♠❡s♠❛ ❊❉❊✱ ♠❛s

  

❝♦♠ ♦✉tr♦ ♣❛râ♠❡tr♦ ❞❡ ✈❛❧♦r θ ✳ ◆♦ q✉❡ s❡ s❡❣✉❡✱ ❛ss✉♠❡✲s❡ ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦ ❞❡

❤✐♣ót❡s❡s✿

  ′

  θ θ = RX

❼ ❍✶✿ P❛r❛ t♦❞♦ ♦ t ∈ [0, T ]✱ t❡♠✲s❡ X t t ✱ ❝♦♠ R ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✳

  ❼ ❍✷✿ ❖s ♣❛râ♠❡tr♦s r ❡ R sã♦ ❝♦♥st❛♥t❡s ♥♦ t❡♠♣♦❀ ♦ s♣r❡❛❞ ✐♥st❛♥tâ♥❡♦✱ s(t) ♣❛r❛ t ≥ 0✱ é ❞❡t❡r♠✐♥íst✐❝♦✳ ❼ ❍✸✿ ❖ ♠♦♠❡♥t♦ ❞❡ ❞❡❢❛✉❧t τ✱ é ✉♠ t❡♠♣♦ ❞❡ ♣❛r❛❣❡♠ ❡♠ r❡❧❛çã♦ à ✜❧tr❛çã♦ ) t t≥0

  ❇r♦✇♥✐❛♥❛✱ ♠❛s τ ❡ (B sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳ θ

  ] 6= 0 ❼ ❍✹✿ ∀t ≥ 0✱ E[X t ✳ ❘❡❝♦rr❡♥❞♦ ❛♦ ♣r✐♥❝í♣✐♦ ❞❡ q✉❡ ♦ ♣r❡ç♦ ❞❡ ✉♠ tít✉❧♦ ❝♦♠ r✐s❝♦ é ♠❡♥♦r q✉❡ ♦ ♣r❡ç♦

  θ θ ≤ Y

❞♦ ♠❡s♠♦ tít✉❧♦ s❡♠ r✐s❝♦✱ ♦❜s❡r✈❛♠♦s q✉❡✱ ❝♦♠ R < 1 ❡ X T T ✱ ♦ s♣r❡❛❞✱ ❛tr❛✈és

  ✶ P❛r❛ ♦s r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s s♦❜r❡ ❡st❡ ❛ss✉♥t♦ ♣♦❞❡ ❝♦♥s✉❧t❛r✲s❡

  ✸✷ ❈❛♣ít✉❧♦ ✷✳ ❘✐s❝♦ ❡ ❙♣r❡❛❞ R t s(u)du

❞♦ ❢❛❝t♦r ❛❝✉♠✉❧❛❞♦ exp ✱ ❝♦rr✐❣❡ ❛ ❞✐s❝r❡♣â♥❝✐❛ ❡♥tr❡ ♦s tít✉❧♦s ❝♦♠ r✐s❝♦ ❡

s❡♠ r✐s❝♦ ❡✱ ❛ss✐♠✱ ❡♠ t❡r♠♦s ❞❡ ❡①♣❡❝t❛t✐✈❛s ❞❡ ♠❡r❝❛❞♦✱ ❝♦♠ ♦s ♣r❡ç♦s ❞❡s❝♦♥t❛❞♦s

à ❞❛t❛ t = 0✱ ❞❡✈❡ t❡r✲s❡✿

  " # θ R t θ

  Y exp s(u)du

  X P P t t E = E .

  ✭✷✳✶✷✮ exp(rt) exp(rt) ◆♦t❡✲s❡ q✉❡ ❛ ❡①♣r❡ssã♦ ✐♠♣❧✐❝❛ q✉❡✿ (

  1 s❡ t < τ (t) + c (t) =

  [0,τ [ [τ,T ] ✶■ ✶■

  ✭✷✳✶✸✮ c s❡ t ≥ τ , ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❡q✉❛çõ❡s t❡♠✲s❡ q✉❡✿

  θ θ Z t X indep.

  X P t P t P

exp − s(u)du · E = E · E (t) + R (t) ⇔

[0,τ [ [τ,T ]

  ✶■ ✶■ exp(rt) exp(rt) Z t

  P

E

⇔ exp − s(u)du = [0,τ [ (t) + R [τ,T ] (t) ✶■ ✶■

  

P

= [τ > t] + RP [τ ≤ t] = 1 − P [τ ≤ t] + RP [τ ≤ t] = 1 + (R − 1)P [τ ≤ t] .

  ✭✷✳✶✹✮ ❆ ❡①♣r❡ssã♦ q✉❡ r❡s✉❧t❛ ❞♦ ❚❡♦r❡♠❛ s❡❣✉✐♥t❡✱ r❡♣r❡s❡♥t❛ ♦ s♣r❡❛❞ ❡♠ ❢✉♥çã♦ ❞❛

t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❡ ❞❡ p := P [τ ≤ t]✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♥❛ ♠❡❞✐❞❛ ♥❛t✉r❛❧✳

❚❡♦r❡♠❛ ✷✳✷✳ ◆♦ ♠♦❞❡❧♦ ❛ t❡♠♣♦ ❝♦♥tí♥✉♦ ♣❛r❛ ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♦ s♣r❡❛❞ r❡❧❛❝✐♦♥❛✲

s❡ ❝♦♠ ❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❛tr❛✈és ❞❡✿ Z t

  1 s(u)du = − ln 1 + (R − 1)P [τ ≤ t] .

  ✭✷✳✶✺✮ t ❉❡♠♦♥str❛çã♦✳ ❉❡❞✉çã♦ ✐♠❡❞✐❛t❛ ❛ ♣❛rt✐r ❞❡

Pr♦♣♦s✐çã♦ ✷✳✸✳ ❊s❝♦❧❤❡♥❞♦ ❛s ✉♥✐❞❛❞❡s ❞❡ ❢♦r♠❛ ❛ q✉❡ ♥❛ ♠❛t✉r✐❞❛❞❡ s❡ t❡♥❤❛ t = 1

❡ ♣❛r❛ (1−R)p << 1 ♥❛ ♠❡❞✐❞❛ ♥❛t✉r❛❧✱ t❡♠✲s❡ q✉❡ ♦ ✈❛❧♦r ♠é❞✐♦ ❞♦ s♣r❡❛❞✱ s✱ ✈❡r✐✜❝❛✿

Z

  1

s := s(u)du ≈ (1 − R)p.

  ❉❡♠♦♥str❛çã♦✳ ❆♥á❧♦❣❛ à ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦

  ✷✳✸ ❈á❧❝✉❧♦ ❆♣r♦①✐♠❛❞♦ ❞♦ ❙♣r❡❛❞ ✸✸ ❖ r❡s✉❧t❛❞♦ ❞❛ Pr♦♣♦s✐çã♦

  ❆ ❡s❝♦❧❤❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ✭♥❛ ♠❡❞✐❞❛ ♥❛t✉r❛❧ ♦✉ ♥❛ ♠❡❞✐❞❛ ♠❛r✲

t✐♥❣❛❧❛✮ ❛ ✉s❛r✱ ❞❡✈❡ s❡r ❝♦❡r❡♥t❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ s❡ ✉♠❛ ♥❡❝❡ss✐t❛ ❞❛ t❡r♠ str✉❝t✉r❡

❛ss♦❝✐❛❞❛ ❛♦ s♣r❡❛❞✳ P❛r❡❝❡ ❝❧❛r♦ ✉t✐❧✐③❛r ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ♥❛t✉r❛✐s ✭♠❡t♦❞♦❧♦❣✐❛ ❛❝✲

t✉❛r✐❛❧✮✱ s❡ ❛ ✐♥t❡♥çã♦ ❢♦r ❡st✐♠❛r ♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦✳ ❆s ❝♦♥❡①õ❡s ❝♦♠ ♦ ♠♦❞❡❧♦

❞❡ ▼❡rt♦♥ ✭✈❡r ✮♣❛r❛ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❡✈❡♠ s❡r ❛♥❛❧✐s❛❞❛s

♣♦st❡r✐♦r♠❡♥t❡✳

  P❛r❛ r❡❢♦rç❛r ♦s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s✱ ❛♣r❡s❡♥t❛♠✲s❡ s❡❣✉✐❞❛♠❡♥t❡ ❞✉❛s ❛❜♦r❞❛❣❡♥s

❞❡ ❞✐❢❡r❡♥t❡s ❢♦♥t❡s ♠❛s q✉❡ ❝♦♥✈❡r❣❡♠ ♣❛r❛ ✉♠ r❡s✉❧t❛❞♦ ✐❞ê♥t✐❝♦ ❛♦ ♦❜t✐❞♦ ♥❡st❛

s❡❝çã♦✳ ✷✳✸ ❈á❧❝✉❧♦ ❆♣r♦①✐♠❛❞♦ ❞♦ ❙♣r❡❛❞

◆❡st❛ s❡❝çã♦ ❢❛③✲s❡ ✉♠❛ ♣❡q✉❡♥❛ ✐❧✉str❛çã♦ ❞♦ ♠ét♦❞♦ ❛♣r♦①✐♠❛❞♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦

s♣r❡❛❞ ❜❛s❡❛❞♦ ♥✉♠❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ q✉❛❧q✉❡r✱ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ ❞❛❞♦s ❤✐stór✐❝♦s

❞❛ ❝❛rt❡✐r❛✳ ❱❡❥❛✲s❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ❞❛ ▼♦♦❞②✬s ♥❛ ❚❛❜❡❧❛

❡♠ ❛♥❡①♦✳

  ❝♦♥s✐❞❡r❛♠ q✉❡ ♦ ♣r❡ç♦✱ ❤♦❥❡✱ ❞❡ ✉♠❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞

❧✐✈r❡ ❞❡ r✐s❝♦ ❡ ❝♦♠ ♠❛t✉r✐❞❛❞❡ T é B(0, T )✳ ■♥❡r❡♥t❡ ❛ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦ ❝♦♠♣♦st❛

❝♦♥t✐♥✉❛♠❡♥t❡✱ r✱ ❡ ♣❛r❛ q✉❡ ♥ã♦ ❤❛❥❛ ❛r❜✐tr❛❣❡♠✱ ♣♦❞❡✲s❡ r❡♣r❡s❡♥t❛r ♦ ♣r❡ç♦ ❞❡st❛

③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝♦♠♦ ❡♠

  B(0, T ) × exp(rT ) = 1

⇔ B(0, T ) = exp(−rT ).

  ✭✷✳✶✻✮ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ss✉♠❡✲s❡ q✉❡ ♦ r❡t♦r♥♦ ❞❡ ✉♠❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝♦♠ r✐s❝♦ ❡ ❝♦♠

❛ ♠❡s♠❛ ♠❛t✉r✐❞❛❞❡ s❡rá r + s✱ ♦♥❞❡ s r❡♣r❡s❡♥t❛ ✉♠ s♣r❡❛❞ ❞❡ ❝ré❞✐t♦✱ q✉❡ é ✉♠❛

❝♦♠♣❡♥s❛çã♦ ♣❛r❛ ♦ r✐s❝♦ ❞❡ ❞❡❢❛✉❧t✳ ◆❡st❡ ❝❛s♦✱ ♦ ♣r❡ç♦ ❞❡st❛ ♦❜r✐❣❛çã♦✱ à ❞❛t❛ ③❡r♦✱

❞❡✈❡ s❡r ♠❡♥♦r q✉❡ ♦ ❞❡ ✉♠❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝♦♠ ❛s ♠❡s♠❛s ❝❛r❛❝t❡ríst✐❝❛s ♠❛s s❡♠

r✐s❝♦✱ ❡ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✿

  ˜

B(0, T ) = k exp(−rT ),

✭✷✳✶✼✮

❝♦♠ 0 < k < 1 ❡✱ ♣♦rt❛♥t♦✱ ♣♦❞❡ ❞✐③❡r✲s❡ q✉❡ ∃s > 0 t❛❧ q✉❡ k = exp(−sT )✳ ▲♦❣♦ ❛

  ♣❛rt✐r ❞❡ ♣♦❞❡ ♦❜t❡r✲s❡✿ ˜

B(0, T ) = exp [−(r + s)T ] .

  ✭✷✳✶✽✮ ❈♦♥s✐❞❡r❛♥❞♦ r ✜①♦ ❡♠ R t❡♠♦s q✉❡✱ ♦ ♣r❡ç♦ ❞❡ ✉♠❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝♦♠ r✐s❝♦✱

  • ˜ B(0, T )

  ✸✹ ❈❛♣ít✉❧♦ ✷✳ ❘✐s❝♦ ❡ ❙♣r❡❛❞ ❞❛ ❢✉♥çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✿

  R(T ) 1u.m. P =

  ✭✷✳✶✾✮ p(0, T ) (1 − p(0, T )) ♥♦ ❝❛s♦ ❞❡ ♦❝♦rrê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❞❡❢❛✉❧t✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦rt❛♥t♦✱ é ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ♦ ♣r❡ç♦ ❥✉st♦ ❞❡ ✉♠❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝♦♠ r✐s❝♦✱ ❡♠

  B(0, T )

t❡r♠♦s ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦ ❧✐✈r❡ ❞❡ r✐s❝♦✱ ˜ ✱ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✱ R(T )✱ ❡ ❞❛

♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✱ p(0, T )✱ ♥♦ ✐♥t❡r✈❛❧♦ [0, T ] ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ˜

  B(0, T ) = E[P ] ⇔

⇔ ˜ B(0, T ) = B(0, T ) [1 − p(0, T )] + R(T )B(0, T )p(0, T )

= B(0, T ) [1 − (1 − R(T ))p(0, T )] .

  ✭✷✳✷✵✮ ❊♥tã♦✱ ❛ ♣❛rt✐r ❞❛s ❡q✉❛çõ❡s ❛♥t❡r✐♦r❡s✱ ♣♦❞❡♠♦s ♦❜t❡r ♦ s♣r❡❛❞ ❞❡ ❝ré❞✐t♦ ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦ ❞❡ r✐s❝♦✱ ❡♠ t❡r♠♦s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❤✐stór✐❝❛✱ p(0, T )✳ ˜

  B(0, T ) exp [−(r + s)T ] = B(0, T ) exp [−rT ] = exp [−sT ]

  = [1 − (1 − R(T ))p(0, T )] .

  ✭✷✳✷✶✮ ❖ ✈❛❧♦r p(0, T ) ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❡❧❡✈❛♥❞♦ ❛ ♠❛tr✐③ ❞❡ tr❛♥s✐çã♦ ❛ ✉♠ ❛♥♦ à

♣♦tê♥❝✐❛ ❛♣r♦♣r✐❛❞❛✳ ◆♦t❡✲s❡ q✉❡ é ✉♠❛ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛ ❞❡ ❡①tr❛✐r ♦ s♣r❡❛❞ ❜❛s❡❛❞♦

❡♠ ❞❛❞♦s ❤✐stór✐❝♦s✱ ♠❛s ♥ã♦ é✱ ♥♦ ❡♥t❛♥t♦✱ ❛ ❢♦r♠❛ ❛♣r♦♣r✐❛❞❛ ♣❛r❛ ❛♣r❡ç❛r ✉♠❛

♦❜r✐❣❛çã♦ ❞❡ r✐s❝♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥ã♦ é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥❡✉tr❛ ❢❛❝❡ ❛♦

r✐s❝♦✳ Pr♦♣♦s✐çã♦ ✷✳✹✳ ❆ ❡q✉❛çã♦ ♣♦❞❡ s❡r s✐♠♣❧✐✜❝❛❞❛ r❡s✉❧t❛♥❞♦✿

s ≈ (1 − R)p.

  ✭✷✳✷✷✮ x

  

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❚❛②❧♦r ❞❛ ❢✉♥çã♦ f(x) = e ❝♦♠

r❡st♦ ❞❡ ❣r❛✉ ✷ ♣♦❞❡ ❡s❝r❡✈❡r✲s❡✿ ∞ n

  X x

exp(x) = ≈ 1 + x; x << 1

n

  

✷✳✹ Pr♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❉❡❢❛✉❧t ■♠♣❧í❝✐t❛s ❞❡ ▼❡r❝❛❞♦ ✸✺

❊♥tã♦ ❢❛③❡♥❞♦✿

exp [−sT ] = [1 − (1 − R(T ))p(0, T )]

1 − sT ≈ [1 − (1 − R(T ))p(0, T )]

  (1 − R(T ))p(0, T ) s ≈ , T ♣❛r❛ T = 1✱ r❡s✉❧t❛ s ≈ (1 − R)p✳

  ❖ r❡s✉❧t❛❞♦ ❞❛ Pr♦♣♦s✐çã♦ ❝♦✐♥❝✐❞❡✱ ♠❛✐s ✉♠❛ ✈❡③✱ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r✲ ♠❡♥t❡ ♦❜t✐❞♦s✳

✷✳✹ Pr♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❉❡❢❛✉❧t ■♠♣❧í❝✐t❛s ❞❡ ▼❡r✲

❝❛❞♦

❯♠❛ ♦✉tr❛ ❛❜♦r❞❛❣❡♠ ♣♦ssí✈❡❧ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ s♣r❡❛❞ ♣♦❞❡ s❡r ❢❡✐t❛ ❛ ♣❛rt✐r ❞❡

❝♦r♣♦r❛t❡ ❜♦♥❞s✳ ❊st✐♠❛çã♦ ❛ ♣❛rt✐r ❞❡ ❈♦r♣♦r❛t❡ ❇♦♥❞s

❙❡❣✉♥❞♦ t❡♥❞♦ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ♦s ♣r❡ç♦s ❞❡ ♠❡r❝❛❞♦ ❞❡ ❝♦r♣♦r❛t❡

❜♦♥❞s é ♣♦ssí✈❡❧ ❢❛③❡r ✏r❡✈❡rs❡ ❡♥❣✐♥❡❡r✐♥❣✑ ♣❛r❛ ❝❛❧❝✉❧❛r ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s q✉❡ ❞❡✈❡♠

s❡r ✉t✐❧✐③❛❞❛s ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦s ♣r❡ç♦s ❞❛s ♠❡s♠❛s ❝♦r♣♦r❛t❡ ❜♦♥❞s✳ ❆ ❡st❛s ♣r♦❜❛✲

❜✐❧✐❞❛❞❡s ❞á✲s❡ ♦ ♥♦♠❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ✐♠♣❧í❝✐t❛s ❞❡ ♠❡r❝❛❞♦✳ ❖ ♣r❡ç♦ ❛ ✶✲♣❡rí♦❞♦ ❞❡ ✉♠❛ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝♦♠ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♣♦❞❡ s❡r ❞❡✲

t❡r♠✐♥❛❞♦ ❛tr❛✈és ❞❡ ❞♦✐s ♠ét♦❞♦s ❡q✉✐✈❛❧❡♥t❡s✳ ❈♦♥s✐❞❡r❡♠✲s❡ ❛s ♥♦t❛çõ❡s ❞❡✜♥✐❞❛s

❛♥t❡r✐♦r♠❡♥t❡✿

  B(0, T ) ✶✳ ❙♦❜ ♦ r✐s❝♦ ♥❡✉tr♦ ✭r✐s❦✲♥❡✉tr❛❧✮ ❞❡ ❛♣r❡ç❛♠❡♥t♦✱ ♦ ✈❛❧♦r ❛❝t✉❛❧ ˜ ❞❡✈❡ s❡r ♦ ♣❛②♦✛ ❡s♣❡r❛❞♦ ✭❝♦♠ ♥♦♠✐♥❛❧✱ B(0, T )✮ ❞❡s❝♦♥t❛❞♦ à t❛①❛ s❡♠ r✐s❝♦ ✭r✐s❦✲❢r❡❡ r❛t❡✮✱ r✿

  Q E [P ] (1 − q(0, T ))B(0, T ) + q(0, T )R(T )B(0, T ) ˜

B(0, T ) = =

  , ✭✷✳✷✸✮ 1 + r 1 + r ♦♥❞❡ q(0, T ) é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ✳

  P♦❞❡✲s❡ ❞❡s✐❣♥❛r ♣♦r ❛♣r❡ç❛♠❡♥t♦ ❝♦♠ r✐s❝♦ ♥❡✉tr♦ ♣♦✐s ♦ ❢❛❝t♦r ✉s✉❛❧ ❞❡ ❝♦r✲ r❡❝çã♦ ❞❡ r✐s❝♦✱ ♦ s♣r❡❛❞✱ ♥ã♦ ❝♦♥st❛ ♥♦ ❞❡♥♦♠✐♥❛❞♦r✳ ❊♠ ✈❡③ ❞✐ss♦ ❛ ❛✈❡rsã♦ ❛♦ r✐s❝♦ é ❞❡s❝r✐t❛ ♣❡❧❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✳ ❉❡st❡ ♠♦❞♦✱ q é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❛♣r❡ç❛♠❡♥t♦ ♦✉ ❞❡ ♠❡❞✐❞❛ ♠❛rt✐♥❣❛❧❛✳ ❊st❛

  ✸✻ ❈❛♣ít✉❧♦ ✷✳ ❘✐s❝♦ ❡ ❙♣r❡❛❞ ♠✐♦ ❞♦ r✐s❝♦ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ (q > p)✳

  ✷✳ ❆❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣♦❞❡✲s❡ ❛❝r❡s❝❡♥t❛r ♦ s♣r❡❛❞✱ s✱ ♥♦ ❞❡♥♦♠✐♥❛❞♦r✿ ˜ B(0, T ) = B(0, T ))

  1 + r + s .

  ✭✷✳✷✹✮ ❯t✐❧✐③❛♥❞♦ ❛s ❡①♣r❡ssõ❡s ✈❡♠ q✉❡✿ (1 − q) + qR 1 + r

  =

  1 1 + r + s q✉❡ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛ ❞❡ ❢♦r♠❛ ❛ ♦❜t❡r♠♦s q✱ q = s (1 + r + s)(1 − R)

  ≈

s

1 − R

.

  ✭✷✳✷✺✮ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ú❧t✐♠❛ ❛♣r♦①✐♠❛çã♦ ❞✐③ q✉❡ ♦ s♣r❡❛❞ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♣❡❧❛ ❧♦ss ❣✐✈❡♥ ❞❡❢❛✉❧t✱ ♦✉ s❡❥❛✱ s ≈ q(1 − R) = qL.

  ✭✷✳✷✻✮ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ♦ s♣r❡❛❞ é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ à ♣❡r❞❛ ❡s♣❡r❛❞❛ ✭❝♦♠ r✐s❝♦ ♥❡✉tr♦✮✳ ❆s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ✐♠♣❧í❝✐t❛s ❞❡ ♠❡r❝❛❞♦ sã♦ ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛♣r❡ç❛♠❡♥t♦

  

❞❡ ❞❡r✐✈❛❞♦s ❞❡ ❝ré❞✐t♦✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ sã♦ ✉t✐❧✐③❛❞❛s ♥❛ ❣❡stã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦

✭♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛s r❡s❡r✈❛s ❞❡ ❝❛♣✐t❛❧✮✱ ♣♦✐s ❡stã♦ ❞✐st♦r❝✐❞❛s ♣❡❧♦

♣ré♠✐♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♦✉ s❡❥❛✱ ♥ã♦ sã♦ ✐❣✉❛✐s às ❢r❡q✉ê♥❝✐❛s ❡s♣❡r❛❞❛s ❞❡ ♣❡r❞❛✳

❋♦r❛♠ ❛♣r❡s❡♥t❛❞❛s ♥❡st❛ s❡❝çã♦ ❞✐✈❡rs❛s ♠❡t♦❞♦❧♦❣✐❛s ♣❛r❛ ♦❜t❡r ✉♠❛ ❡①♣r❡ssã♦

q✉❡ ❞❡✜♥✐ss❡ ♦ s♣r❡❛❞ ❡♠ ❢✉♥çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✳

❊♠❜♦r❛ ❝♦♠ ♠❡t♦❞♦❧♦❣✐❛s ❞✐st✐♥t❛s✱ ♦ r❡s✉❧t❛❞♦ ❛ q✉❡ s❡ ❝❤❡❣♦✉ ❡♠ ❝❛❞❛ ✉♠❛ ❞❡❧❛s

❢♦✐ ✐❞ê♥t✐❝♦✱ ❝♦♠♦ t❛❧✱ ❝♦♥s✐❞❡r❛✲s❡✱ ♥❡st❡ tr❛❜❛❧❤♦✱ q✉❡ ♦ s♣r❡❛❞ q✉❡ s❡ ❞❡✈❡ ❛♣❧✐❝❛r ❛

✉♠ ❞❛❞♦ ❝❧✐❡♥t❡ s♦❧✐❝✐t❛♥t❡ ❞❡ ❝ré❞✐t♦ ♣♦❞❡ s❡r ❞❛❞♦ ♣❡❧❛ ❡①♣r❡ssã♦✿

s ≈ p(1 − R),

  

✷✳✹ Pr♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❉❡❢❛✉❧t ■♠♣❧í❝✐t❛s ❞❡ ▼❡r❝❛❞♦ ✸✼

❡♠ q✉❡ p é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♥❛ s✉❛ ♠❡❞✐❞❛ ♥❛t✉r❛❧ ❡ R ❛ t❛①❛ ❞❡ r❡❝✉✲ ♣❡r❛çã♦✳ ◆♦ ✜♥❛❧ ❞♦ ♣ró①✐♠♦ ❝❛♣✐t✉❧♦✱ ❞❛❞❛ ✉♠❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ R✱ ♣♦❞❡r✲s❡✲á ✉t✐❧✐③❛r

❛ ❡q✉❛çã♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ s♣r❡❛❞ q✉❡ s❡ ❞❡✈❡rá ❛♣❧✐❝❛r ❛ ❛❧❣✉♥s ❝❧✐❡♥t❡s

s♦❧✐❝✐t❛♥t❡s ❞❡ ❝ré❞✐t♦ à ❍❛❜✐t❛çã♦✱ ❝♦♠ ❜❛s❡ ♥❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t q✉❡ s❡rã♦

❡st✐♠❛❞❛s ❡ ♥✉♠❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❝♦♥st❛♥t❡✳

  ✸✾ ❈❛♣ít✉❧♦ ✸ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦ ✸✳✶ ❆ ❈❛rt❡✐r❛ ❞❡ ❈ré❞✐t♦ ✸✳✶✳✶ ❉❡✜♥✐çã♦ ❞❡ ❈❧✐❡♥t❡ ✐♥❝✉♠♣r✐❞♦r

◆♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ tr❛❜❛❧❤♦ s♦❜r❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ♥♦ q✉❛❧ s❡ ❡st✉❞❛♠ ♠♦❞❡❧♦s

❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✱ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦✱ ✐st♦ é✱ ♦

❝♦♥❝❡✐t♦ ❞❡ ❝❧✐❡♥t❡ ❝✉♠♣r✐❞♦r ❡ ❝❧✐❡♥t❡ ✐♥❝✉♠♣r✐❞♦r ✉t✐❧✐③❛❞♦ ♥❡st❡ tr❛❜❛❧❤♦✱ ✈✐st♦ q✉❡

❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ♥♦ ♥♦ss♦ ♠♦❞❡❧♦ ❞❡ ❛✈❛❧✐❛çã♦ ❞❡ r✐s❝♦ é ❛ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦✳

  ❉❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ ❝♦♥s✐❞❡r❛♠✲s❡ ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s ❛q✉❡❧❡s ❝❧✐❡♥t❡s ❝♦♠ ✉♠

❛tr❛s♦ ♥♦ ♣❛❣❛♠❡♥t♦ s✉♣❡r✐♦r ❛ ✸✵ ❞✐❛s ❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ♣r❡st❛çã♦ ❞♦ ❡♠♣rést✐♠♦✳

❖s ❝❧✐❡♥t❡s ❝✉♠♣r✐❞♦r❡s sã♦ ❛q✉❡❧❡s q✉❡ ♥ã♦ ♣♦ss✉❡♠ ❛tr❛s♦s ♦✉ q✉❡ ♣♦ss✉❡♠ ❛tr❛s♦s

♥♦ ♠á①✐♠♦ ❞❡ ✸✵ ❞✐❛s ❡♠ ❛❧❣✉♠❛ ♣❛r❝❡❧❛ ❞♦ ❡♠♣rést✐♠♦✳ P♦r ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ ♦

♥ú♠❡r♦ ❞❡ ♣r❡st❛çõ❡s ❥á ♣❛❣❛s ❢♦r ✐♥❢❡r✐♦r ❛♦ ♥ú♠❡r♦ ❞❡ ♣r❡st❛çõ❡s ❞❡❝♦rr✐❞❛s ❡♥tã♦ ♦

❝❧✐❡♥t❡ ❡stá ❡♠ ✐♥❝✉♠♣r✐♠❡♥t♦✳

  P♦❞❡♠♦s r❡s✉♠✐r ❛s ❝♦♥s✐❞❡r❛çõ❡s ❞❛ ✈❛r✐á✈❡❧ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦✱ ♥♦ ♣r❡s❡♥t❡ ❡st✉❞♦✱ ❝♦♠ ❛ t❛❜❡❧❛

❆tr❛s♦ ♥♦ P❛❣❛♠❡♥t♦ ❙✐t✉❛çã♦

✭❉✐❛s✮

  ❉❡ ✵ ❛ ✸✵ ❞✐❛s ❈✉♠♣r✐❞♦r ▼❛✐s ❞❡ ✸✵ ❞✐❛s ■♥❝✉♠♣r✐❞♦r

P❛❣❛s ❁ ❉❡❝♦rr✐❞❛s ■♥❝✉♠♣r✐❞♦r

  

❚❛❜❡❧❛ ✸✳✶✿ ❉❡✜♥✐çã♦ ❞❡ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦

  

✹✵ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

✸✳✶✳✷ ❈♦♠♣♦s✐çã♦ ❞❛ ❈❛rt❡✐r❛ ❞❡ ❈ré❞✐t♦

❉❡ ❢♦r♠❛ ❛ ♠♦❞❡❧❛r ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝❧✐❡♥t❡s✱ r❡❝♦rr❡✉✲s❡ ❛ ❞❛❞♦s

❤✐stór✐❝♦s ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❡♠♣rést✐♠♦s ❜❛♥❝ár✐♦s ♣❛r❛ ❝ré❞✐t♦s à ❍❛❜✐t❛çã♦✳

  ❆ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ✉t✐❧✐③❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ♥✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦

à ❍❛❜✐t❛çã♦ ❡ é ❝♦♠♣♦st❛ ♣♦r ✶✷✳✵✵✵ ❝❧✐❡♥t❡s ❞♦s q✉❛✐s ❝❡r❝❛ ❞❡ ✸% ❡♥tr❛r❛♠ ❡♠

✐♥❝✉♠♣r✐♠❡♥t♦✳ ❉❡✈✐❞♦ à ❢❛❧t❛ ❞❡ ❞❛❞♦s r❡❛✐s t♦r♥♦✉✲s❡ ♥❡❝❡ssár✐♦ s✐♠✉❧❛r ♦s ✈❛❧♦r❡s ❞❡ t♦❞❛s ❛s

✈❛r✐á✈❡✐s q✉❡ ❝♦♥s✐❞❡rá♠♦s ❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ ♦ ❡st✉❞♦✳ ❆ s✐♠✉❧❛çã♦ ❞❛ ❝❛rt❡✐r❛ ❞❡

❝ré❞✐t♦ ❢❡③ ✉s♦ ❞❡ ♠ét♦❞♦s ❞❡ ❛❧❡❛t♦r✐❡❞❛❞❡✱ ❡ ❢♦✐ ♦❜t✐❞❛ r❡❝♦rr❡♥❞♦ à ❛♣❧✐❝❛çã♦ ▼✐✲

❝r♦s♦❢t ❊①❝❡❧✳ ◆♦ ♣r❡s❡♥t❡ ❡st✉❞♦✱ ❛ ✈❛r✐á✈❡❧ ❞❡ ✐♥t❡r❡ss❡ é ❛ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦ ❞♦s

❝❧✐❡♥t❡s✱ q✉❡ ❝♦♥s✐st❡ ♥✉♠❛ ✈❛r✐á✈❡❧ ♥♦♠✐♥❛❧✱ ❡♠ q✉❡ sã♦ ❛tr✐❜✉í❞♦s ♦s ✈❛❧♦r❡s 0 ♣❛r❛

❝❧✐❡♥t❡s ❝✉♠♣r✐❞♦r❡s ❡ 1 ♣❛r❛ ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s✳ ❆♣ós ❛ s✐♠✉❧❛çã♦ ❞❡ t♦❞❛s ❛s

✈❛r✐á✈❡✐s ♣❛r❛ ♦s ✶✷✵✵✵ ❝❧✐❡♥t❡s✱ ❛❣r✉♣❛r❛♠✲s❡ ♦s ❞❛❞♦s ❞❡ ❢♦r♠❛ ❛ t❡r♠♦s ✉♠❛ ❜❛s❡ ❞❡

❞❛❞♦s ♣r♦♥t❛ ♣❛r❛ ✉t✐❧✐③❛r ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✳ P♦❞❡✲s❡

✈✐s✉❛❧✐③❛r ♥❛ ❚❛❜❡❧❛ ❡♠ ❛♥❡①♦✱ ✉♠❛ ♣❛rt❡ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ s✐♠✉❧❛❞❛✳

  P❛r❛ ♦ tr❛t❛♠❡♥t♦ ❞♦s ❞❛❞♦s ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❞✐✈✐❞✐✉✲s❡ ❛ ♠❡s♠❛ ❡♠ ❞✉❛s

❛♠♦str❛s✱ ❛ ❛♠♦str❛ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❛ ❛♠♦str❛ ❞❡ ✈❛❧✐❞❛çã♦✳ ❆ ❛♠♦str❛ ❞❡

❞❡s❡♥✈♦❧✈✐♠❡♥t♦ é ❛ss✐♠ ❝♦♥st✐t✉í❞❛ ♣♦r ✶✵✵✵✵ ❞♦s ❝❧✐❡♥t❡s ❞♦ t♦t❛❧ ❞❛ ❝❛rt❡✐r❛ ❞❡

❝ré❞✐t♦ ❡ ❛ ❛♠♦str❛ ❞❡ t❡st❡ é ❝♦♥st✐t✉í❞❛ ♣❡❧♦s r❡st❛♥t❡s ✷✵✵✵ ❝❧✐❡♥t❡s✳ ❆ ❛♠♦str❛ ❞❡

❞❡s❡♥✈♦❧✈✐♠❡♥t♦ é ❛ ❛♠♦str❛ q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ❞❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡ ♣❛r❛ ❛ ❡st✐♠❛çã♦ ❞♦

♠♦❞❡❧♦ q✉❡ ♠❡❧❤♦r ❡①♣❧✐❝❛ ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❡♠ ❢✉♥çã♦ ❞❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✳

❆ ❛♠♦str❛ ❞❡ t❡st❡ ❛♣❡♥❛s s❡rá ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛✈❛❧✐❛r ❛ ❝❛♣❛❝✐❞❛❞❡ ♣r❡❞✐t✐✈❛ ❞♦ ♠♦❞❡❧♦

❛❥✉st❛❞♦ ❝♦♠ ❛ ❛♠♦str❛ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✳ ❚r❛t❛♠❡♥t♦ ❞❡ ❞❛❞♦s

❖ tr❛t❛♠❡♥t♦ ❞❡ ❞❛❞♦s ♥✉♠❛ ❛♠♦str❛ ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ r❡❛❧ ❝♦♥s✐st✐r✐❛✱ ❝♦♠

❜❛s❡ ♥♦s ❞❛❞♦s r❡❝♦❧❤✐❞♦s✱ ♥❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛s ✈❛r✐á✈❡✐s ❡st❛t✐st✐❝❛♠❡♥t❡ s✐❣♥✐✜❝❛t✐✈❛s

❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✳

  ◆♦ ❡♥t❛♥t♦✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ❢♦r❛♠ ❡s❝♦❧❤✐❞❛s✱ ❝♦♠ ❜❛s❡ ❡♠ ❡st✉❞♦s ❛♥t❡r✐♦r❡s✱ ❛s

✈❛r✐á✈❡✐s q✉❡ s❡ ❝♦♥s✐❞❡r❛r❛♠ ✐♠♣♦rt❛♥t❡s r❡❝♦❧❤❡r ❛q✉❛♥❞♦ ✉♠ ♥♦✈♦ ♣❡❞✐❞♦ ❞❡ ❝ré✲

❞✐t♦✳ P♦st❡r✐♦r♠❡♥t❡✱ ❢♦r❛♠ s✐♠✉❧❛❞♦s ♦s ❞❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ t♦❞❛s ❛s ✈❛r✐á✈❡✐s

❡ ♦❜t❡♥❞♦✲s❡✱ ❞❡st❛ ❢♦r♠❛✱ ❛ ❜❛s❡ ❞❡ ❞❛❞♦s q✉❡ s❡r✈✐rá ❞❡ ❜❛s❡ ❛ ❡st❡ tr❛❜❛❧❤♦✳ ❋✐✲

♥❛❧♠❡♥t❡✱ ❝♦♠ ❛ ❜❛s❡ ❞❡ ❞❛❞♦s ❝♦♥str✉í❞❛✱ ❛♣❧✐❝♦✉✲s❡ ✉♠ ❞♦s ♠ét♦❞♦s ❡st❛tíst✐❝♦s

✉t✐❧✐③❛❞♦s ❡♠ ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✱ ❛ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛✱ ❡ ❢♦r❛♠ ❡❢❡❝t✉❛❞♦s

❡st✉❞♦s ❡st❛tíst✐❝♦s ♣❛r❛ ✐❞❡♥t✐✜❝❛r ❛s ✈❛r✐á✈❡✐s ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛s q✉❡ ♣❡r♠✐t❡♠ ✏❡①✲

♣❧✐❝❛r✑ ❛ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦ ❞❡ ✉♠ ❝❧✐❡♥t❡✳ ✸✳✶✳✸ ❆s ❱❛r✐á✈❡✐s ❯t✐❧✐③❛❞❛s

◆♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✱ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ✭❞❡♣❡♥❞❡♥t❡✮ é ❛ ◗✉❛❧✐❞❛❞❡ ❞❡

❈ré❞✐t♦ ✭❞❡❢❛✉❧t ♦✉ ♥♦♥✲❞❡❢❛✉❧t✮✳ P❛r❛ ❝❧❛ss✐✜❝❛r ❛s ♦❜s❡r✈❛çõ❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ◗✉❛✲

  ✸✳✶ ❆ ❈❛rt❡✐r❛ ❞❡ ❈ré❞✐t♦ ✹✶ ♣✉❞❡ss❡♠ ✐♥✢✉❡♥❝✐❛r ❛ s✐t✉❛çã♦ ❞♦s ❝❧✐❡♥t❡s ♥❛s s✉❛s ♦❜r✐❣❛çõ❡s ❡♥q✉❛♥t♦ ❝r❡❞♦r❡s✳

  P❛r❛ ♠❡❧❤♦r ♦r✐❡♥t❛çã♦ ♥❛ ❞❡✜♥✐çã♦ ❡ ❡s❝♦❧❤❛ ❞❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❢♦r❛♠ t❛♠✲

❜é♠ ❝♦♥s✉❧t❛❞♦s ♦✉tr♦s tr❛❜❛❧❤♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✱ ❡♥tr❡ ❡❧❡s✱ ✉♠ tr❛❜❛❧❤♦ ❡❧❛❜♦r❛❞♦

♣❡❧❛ ❈r❡❞✐❘✐s❦ ♣❛r❛ ✉♠ ❜❛♥❝♦ ♣♦rt✉❣✉ês✱ ♦ tr❛❜❛❧❤♦ ❞❡ ❞✐ss❡rt❛çã♦ ❞❡ ✳

  ■♥✐❝✐❛❧♠❡♥t❡✱ ❢♦✐ ❝♦♥str✉í❞❛ ✉♠❛ ❜❛s❡ ❞❡ ❞❛❞♦s q✉❡ ❛❣r❡❣❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦ssí✈❡✐s

✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ♣ré✲s❡❧❡❝❝✐♦♥❛❞❛s ♣❛r❛ ✉t✐❧✐③❛çã♦ ♥❛ ❝♦♥str✉çã♦ ❞♦s ♠♦❞❡❧♦s✳ ❆

♣❛rt✐r ❞❡st❛ ❜❛s❡ ✐♥✐❝✐❛❧✱ ❢♦r❛♠ s❡❧❡❝❝✐♦♥❛❞❛s✱ ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s ❡st❛tís✲

t✐❝♦s✱ ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ♠❛✐s r❡❧❡✈❛♥t❡s✱ ❛s q✉❛✐s ❡❢❡❝t✐✈❛♠❡♥t❡ ❝♦♠♣✉s❡r❛♠ ♦s

♠♦❞❡❧♦s✳ ❆ ❚❛❜❡❧❛ ✐❧✉str❛ ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❛❞♦♣t❛❞❛s ❜❡♠ ❝♦♠♦ ❛❧❣✉♠❛s ❝❛r❛❝✲ t❡ríst✐❝❛s ❞❛s ♠❡s♠❛s✳

  ❱❛r✐á✈❡✐s ❈ó❞✐❣♦ ❞❛ ◆❛t✉r❡③❛ ❞❛ ❚✐♣♦ ❙✐♠✉❧❛çã♦ ❱❛r✐á✈❡❧ ❱❛r✐á✈❡❧

✶ ■❞❛❞❡ ■❞❛❞❡ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✷ ◆➸ ❞❡ ❉❡♣❡♥❞❡♥t❡s ❉❡♣ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✸ ❚✐♣♦ ❍❛❜✐t❛çã♦ ❍❛❜ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✹ ■❞❛❞❡ ❝❛s❛ ■❞❝❛s❛ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✺ ❊st❛❞♦ ❈✐✈✐❧ ❈✐✈✐❧ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✻ ❈ó❞✐❣♦ P♦st❛❧ P♦st❛❧ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✼ ❚✐♣♦ Pr♦✜ssã♦ Pr♦❢ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✽ ❆♥t✐❣✉✐❞❛❞❡ Pr♦✜ssã♦ ❆♥t❣ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✾ ❘❡♥❞✐♠❡♥t♦ ▲íq✉✐❞♦ ❘❡♥❞ ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✶✵ ❈ré❞✐t♦ ❈r❡❞ ◗✉❛♥t✐t❛t✐✈❛ ◆✉♠ér✐❝❛ ❉✐str✐❜✉✐çã♦ ●❛♠♠❛✭α❀β✮

  

✶✶ % ❞❡ ❊♥tr❛❞❛ ❊♥tr ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ❉✐str✐❜✉✐çã♦ ❇❡t❛✭α❀β✮

✶✷ ◆➸❚♦t❛❧ Pr❡st❛çõ❡s Pt♦t❛❧ ◗✉❛♥t✐t❛t✐✈❛ ◆✉♠ér✐❝❛ ❉✐str✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✭❛✱❜✮

✶✸ ◆➸Pr❡st❛çõ❡s P❛❣❛s P❛❣❛s ◗✉❛♥t✐t❛t✐✈❛ ◆✉♠ér✐❝❛ ❆❧❡❛tór✐♦ [Min, P total]

✶✹ P❛❣❛s✴❚♦t❛❧ (%) P❛❣♣❡r❝❡♥t ◗✉❛❧✐t❛t✐✈❛ ❈❛t❡❣ór✐❝❛ ✲

  

❚❛❜❡❧❛ ✸✳✷✿ ▲✐st❛ ■♥✐❝✐❛❧ ❞❡ ❱❛r✐á✈❡✐s ❊①♣❧✐❝❛t✐✈❛s

❆ ❚❛❜❡❧❛ ❡♠ ❛♥❡①♦✱ ✐❧✉str❛ ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❡ ❛s r❡s♣❡❝t✐✈❛s ❝❛t❡❣♦r✐❛s ✉t✐❧✐③❛❞❛s✳ ❆s ✈❛r✐á✈❡✐s ❝❛t❡❣ór✐❝❛s t❛♠❜é♠ sã♦ ❞❡s✐❣♥❛❞❛s ♣♦r ✈❛r✐á✈❡✐s ❞✉♠♠②✱ ❞✐❝♦tó♠✐❝❛s

♦✉ ❜✐♥ár✐❛s ❡ sã♦ ✈❛r✐á✈❡✐s ❝♦♥str✉í❞❛s ❛rt✐✜❝✐❛❧♠❡♥t❡ ♣❛r❛ ♠❡❞✐r ❛ ♣r❡s❡♥ç❛ ♦✉ ❛✉sê♥❝✐❛

❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❛tr✐❜✉t♦✳

  ❆ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✱ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦✱ t❛♠❜é♠ ❢♦✐ ✐♥s❡r✐❞❛ ♥❛ ❜❛s❡ ❞❡ ❞❛❞♦s

❛tr❛✈és ❞♦ ✉s♦ ❞❡ ✈❛r✐á✈❡✐s ❞✉♠♠②✳ ❆ss✐♠ ❢♦✐ ❝r✐❛❞❛ ✉♠❛ ❞✉♠♠②✱ ❞❡♥♦♠✐♥❛❞❛ ✏❉❡✲

❢❛✉❧t✑✱ q✉❡ ❛ss✉♠❡ ♦ ✈❛❧♦r 0 ♣❛r❛ ❝❧✐❡♥t❡s ❝✉♠♣r✐❞♦r❡s ❡ 1 ♣❛r❛ ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s✱

✈✐s❛♥❞♦ ❞✐❢❡r❡♥❝✐❛r ♦s ❝❧✐❡♥t❡s ❡♠ r❡❧❛çã♦ à ♣r❡s❡♥ç❛ ♦✉ ❛✉sê♥❝✐❛ ❞❛ ❝♦♥❞✐çã♦ ❞❡ ❞❡❢❛✉❧t✳

✸✳✶✳✹ ❙✐♠✉❧❛çã♦ ❞❛ ❈❛rt❡✐r❛ ❞❡ ❈ré❞✐t♦

◆❡st❛ s❡❝çã♦ s❡rá ❞❡s❝r✐t❛ ❛ ♠❡t♦❞♦❧♦❣✐❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❣❡r❛r ❝❛❞❛ ✉♠❛ ❞❛s ✈❛r✐á✈❡✐s

  

✹✷ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❛❧❣✉♥s ♣r❡ss✉♣♦st♦s s♦❜r❡ ♦ ♣❡s♦ r❡❧❛t✐✈♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝❛t❡❣♦r✐❛s ❞❛ ✈❛r✐á✈❡❧✱ ♦✉

s❡❥❛✱ ❛ ♣❡r❝❡♥t❛❣❡♠ ❞❡ ❝❧✐❡♥t❡s ❝♦♠ ❡ss❛ ❝❛r❛❝t❡ríst✐❝❛✱ ❞❡ ❢♦r♠❛ ❛ q✉❡ ❛ s✐♠✉❧❛çã♦

r❡s✉❧t❛ss❡ ♥✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝❧✐❡♥t❡s ♠✐♥✐♠❛♠❡♥t❡ r❡❛❧✐st❛✳ ❱❛r✐á✈❡✐s ■♥❞❡♣❡♥❞❡♥t❡s ❆s ❝♦✈❛r✐á✈❡✐s s✐♠✉❧❛❞❛s sã♦✿

  ■❞❛❞❡ ✭■❞❛❞❡✮✿ P❛r❛ ❛ ✈❛r✐á✈❡❧ ■❞❛❞❡ ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞❛s três ❝❛t❡❣♦r✐❛s✱ ❝♦♥❢♦r♠❡ ❛ ❚❛❜❡❧❛ ❡♠ ❛♥❡①♦✳ P❛r❛ ❝❛❞❛ ❝❧✐❡♥t❡ ❢♦✐ ❣❡r❛❞♦ ✉♠ ♥ú♠❡r♦ ❛❧❡❛tór✐♦ ❡♥tr❡ 0 ❡ 1 ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐s✲

tr✐❜✉✐çã♦ ❯♥✐❢♦r♠❡✳ ❖ ❝❧✐❡♥t❡ é ❛❢❡❝t♦ à ❝❛t❡❣♦r✐❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ♥ú♠❡r♦ ❛❧❡❛tór✐♦

❣❡r❛❞♦✳ P♦r ❡①❡♠♣❧♦ s❡ ♦ ♥ú♠❡r♦ ❛❧❡❛tór✐♦ ♦❜t✐❞♦ s❡ ❡♥❝♦♥tr❛ ❡♥tr❡ 0 ❡ 0.13 ♦ ❝❧✐❡♥t❡

é ✐♥s❡r✐❞♦ ♥❛ ❝❛t❡❣♦r✐❛ ✏❏♦✈❡♥s✑✳ ❆ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞❛s ✈❛r✐á✈❡✐s ❞❛ ❚❛❜❡❧❛ ❢♦r❛♠ ❝♦♥str✉í❞❛s ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ✈❛r✐á✈❡❧ ■❞❛❞❡✳ ◆➸ ❞❡ ❉❡♣❡♥❞❡♥t❡s ✭❉❡♣✮✿

  ❆ ✈❛r✐á✈❡❧ ◆➸❞❡ ❉❡♣❡♥❞❡♥t❡s ❢♦✐ s✐♠✉❧❛❞❛ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ✈❛r✐á✈❡❧ ■❞❛❞❡

t❡♥❞♦ ❡♠ ❝♦♥t❛ ❛s ♣❡r❝❡♥t❛❣❡♥s ❞❛s r❡s♣❡❝t✐✈❛s ❝❛t❡❣♦r✐❛s✳ ◆♦ ❡♥t❛♥t♦✱ ♥❡st❛ ✈❛r✐á✲

✈❡❧✱ ❝♦♥s✐❞❡r❛✲s❡ q✉❡ ♦s ❝❧✐❡♥t❡s ❝♦♠ ♠❡♥♦s ❞❡ ✷✺ ❛♥♦s ♥ã♦ tê♠ q✉❛tr♦ ♦✉ ♠❛✐s ✜❧❤♦s✱

♦✉ s❡❥❛✱ ❢❡③✲s❡ ❞❡♣❡♥❞❡r ❡st❛ ✈❛r✐á✈❡❧ ❞❛ ✈❛r✐á✈❡❧ ■❞❛❞❡✱ ♠❛s ❛♣❡♥❛s ♣❛r❛ ❛ ❝❛t❡❣♦r✐❛

✏▼✉✐t♦s✑✳ ❚✐♣♦ ❍❛❜✐t❛çã♦ ✭❍❛❜✮✿

  ❆ ✈❛r✐á✈❡❧ ❚✐♣♦ ❍❛❜✐t❛çã♦ ❢♦✐ t❛♠❜é♠ s✐♠✉❧❛❞❛ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ✈❛r✐á✈❡❧ ■❞❛❞❡✱ t❡♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛s ♣❡r❝❡♥t❛❣❡♥s ❞❛s r❡s♣❡❝t✐✈❛s ❝❛t❡❣♦r✐❛s✳ ■❞❛❞❡ ❈❛s❛ ✭■❞❝❛s❛✮✿ ❆ ✈❛r✐á✈❡❧ ■❞❛❞❡ ❈❛s❛ ❢♦✐ s✐♠✉❧❛❞❛ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ t❡♥❞♦ ❡♠ ❝♦♥t❛ ❛s ♣❡r❝❡♥t❛✲

  ❣❡♥s ❞❛s r❡s♣❡❝t✐✈❛s ❝❛t❡❣♦r✐❛s✳ ❊st❛❞♦ ❈✐✈✐❧ ✭❈✐✈✐❧✮ ❆ ✈❛r✐á✈❡❧ ❊st❛❞♦ ❈✐✈✐❧ ❢♦✐ s✐♠✉❧❛❞❛ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ✈❛r✐á✈❡❧ ■❞❛❞❡✱ ♠❛s

❝♦♠ ❛s ♣❡r❝❡♥t❛❣❡♥s r❡s♣❡❝t✐✈❛s✳ ◆❡st❡ ❝❛s♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❛s ♣❡r❝❡♥t❛❣❡♥s ❢♦r❛♠

r❡t✐r❛❞❛s ❞♦s ❝❡♥s♦s ❢❡✐t♦s ❡♠ P♦rt✉❣❛❧ ♥♦ ❛♥♦ ❞❡ ✷✵✵✶ ✭✈❡❥❛✲s❡ ❈ó❞✐❣♦ P♦st❛❧ ✭P♦st❛❧✮✿

  ❆ ✈❛r✐á✈❡❧ ❈ó❞✐❣♦ P♦st❛❧ ❢♦✐ s✐♠✉❧❛❞❛ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ✈❛r✐á✈❡❧ ■❞❛❞❡✱ t❡♥❞♦

❡♠ ❝♦♥t❛ ❛s r❡s♣❡❝t✐✈❛s ♣❡r❝❡♥t❛❣❡♥s✳ ◆♦ ❡♥t❛♥t♦✱ ♦s ❣r✉♣♦s ❢♦r♠❛❞♦s t✐✈❡r❛♠ ❝♦♠♦

  ✸✳✶ ❆ ❈❛rt❡✐r❛ ❞❡ ❈ré❞✐t♦ ✹✸

❝♦♥s✐❞❡r❛♠♦s ❛s ♠❡s♠❛s ❝❛t❡❣♦r✐❛s✱ ✈✐st♦ ♥ã♦ t❡r♠♦s ❞❛❞♦s r❡❛✐s ♣❛r❛ ❢❛③❡r ✉♠ ❡st✉❞♦

  ❡st❛tíst✐❝♦ s♦❜r❡ ❛s ③♦♥❛s q✉❡ s❡ ❞❡✈❡r✐❛♠ ❛❣r✉♣❛r✳ ❖s ✈❛❧♦r❡s q✉❡ ❝♦♥st❛♠ ♥❛ ❞❡s❝r✐çã♦ ❞❛ ❝♦✈❛r✐á✈❡❧ ❈ó❞✐❣♦ P♦st❛❧ ♥❛ ❚❛❜❡❧❛ ❝♦rr❡s♣♦♥❞❡♠ ❛s ❞♦✐s ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞♦s ❝ó❞✐❣♦s ♣♦st❛✐s ♥❛❝✐♦♥❛✐s✳ ❚✐♣♦ Pr♦✜ssã♦ ✭Pr♦❢✮✿

  ❊st❛ ✈❛r✐á✈❡❧ ❢♦✐ s✐♠✉❧❛❞❛ ❡①❛❝t❛♠❡♥t❡ ♥❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❞❛ ✈❛r✐á✈❡❧ ❈ó❞✐❣♦ P♦st❛❧✱ ✉t✐❧✐③❛♥❞♦ t❛♠❜é♠ ❛ ✐♥❢♦r♠❛çã♦ ❞♦ ❡st✉❞♦ ❞❛ ❈r❡❞✐❘✐s❦✳ ❆♥t✐❣✉✐❞❛❞❡ Pr♦✜ssã♦ ✭❆♥t❣✮✿ P❛r❛ ❛ ✈❛r✐á✈❡❧ ❆♥t✐❣✉✐❞❛❞❡ Pr♦✜ssã♦ ❝♦♥s✐❞❡r❛r❛♠✲s❡ ❛♣❡♥❛s ❞♦✐s ❣r✉♣♦s ❡ ❛ s✐✲

  ♠✉❧❛çã♦ ❢♦✐ ❢❡✐t❛ ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣❛r❛ ❛ ✈❛r✐á✈❡❧ ■❞❛❞❡✳ ❘❡♥❞✐♠❡♥t♦ ▲íq✉✐❞♦ ✭❘❡♥❞✮✿ ▼❛✐s ✉♠❛ ✈❡③✱ ❡st❛ ✈❛r✐á✈❡❧ ❢♦✐ s✐♠✉❧❛❞❛ t❡♥❞♦ ❡♠ ❝♦♥t❛ ❛s ♣❡r❝❡♥t❛❣❡♥s q✉❡ ❢♦r❛♠

❛tr✐❜✉í❞❛s ❛ ❝❛❞❛ ❝❛t❡❣♦r✐❛✳ ◆ã♦ s❡ ❝♦♥s✐❞❡r❛♠ ♦s ❝❧✐❡♥t❡s ❝♦♠ ♦r❞❡♥❛❞♦s ✐♥❢❡r✐♦r❡s ❛

e 450

  ❡ s✉♣❡r✐♦r❡s ❛ e20000✱ ♦✉ s❡❥❛✱ ♥❛ s✐♠✉❧❛çã♦ ❞♦s ✈❛❧♦r❡s ❞♦s ♦r❞❡♥❛❞♦s ♣r♦♣r✐❛✲ ♠❡♥t❡ ❞✐t♦s✱ ❛♣❡♥❛s s❡ ❣❡r♦✉ ✉♠ ♥ú♠❡r♦ ❛❧❡❛tór✐♦ ❡♥tr❡ 450 ❡ 20000✳ ❈ré❞✐t♦ ✭❈r❡❞✮✿ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ❛❞♦♣t❛❞♦ ♥❛ s✐♠✉❧❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❈ré❞✐t♦ é ❜❛st❛♥t❡ ❞✐st✐♥t♦ ❞❛s

r❡st❛♥t❡s ✈❛r✐á✈❡✐s✳ P♦r ✉♠ ❧❛❞♦✱ ♥ã♦ é ✉♠❛ ✈❛r✐á✈❡❧ ❝❛t❡❣ór✐❝❛ ❡ ♣♦r ♦✉tr♦✱ t❡♠

❝❛r❛❝t❡ríst✐❝❛s ♠✉✐t♦ ♣ró♣r✐❛s✳

  ❈♦♥s✐❞❡r♦✉✲s❡ q✉❡ ❛ ✈❛r✐á✈❡❧ ❈ré❞✐t♦ ♣♦❞❡r✐❛ s❡r ♠♦❞❡❧❛❞❛ ♣♦r ✉♠❛ ❞✐str✐❜✉✐çã♦ Y = aX + b ✱ ♦♥❞❡ X ∼ Gama(α = 1.5; β = 3)✱ a = 20000 ❡ b = 75000✳ ❖s ♣❛râ♠❡tr♦s

❢♦r❛♠ ❡s❝♦❧❤✐❞♦s ❞❡ ❢♦r♠❛ ❛ ♦❜t❡r ♠♦♥t❛♥t❡s ❞❡ ❝ré❞✐t♦ ❝♦♠♣❛tí✈❡✐s ❝♦♠ ✉♠❛ ❝❛rt❡✐r❛

  

❞❡ ❝ré❞✐t♦ à ❤❛❜✐t❛çã♦ r❡❛❧✐st❛✳ ◆♦t❡✲s❡ q✉❡ ♦ ♠♦♥t❛♥t❡ ♠é❞✐♦ ❞❡ ❝ré❞✐t♦ ❝♦♥❝❡❞✐❞♦ é

❛♣r♦①✐♠❛❞❛♠❡♥t❡ e✶✻✺✳✵✵✵ ✳ P❡r❝❡♥t❛❣❡♠ ❞❡ ❊♥tr❛❞❛ ✭❊♥tr✮✿ P❛r❛ s✐♠✉❧❛r ❡st❛ ✈❛r✐á✈❡❧ ❝♦♥s✐❞❡r♦✉✲s❡ q✉❡ ❛ P❡r❝❡♥t❛❣❡♠ ❞❡ ❡♥tr❛❞❛ ♣♦❞❡r✐❛ s❡r

  

♠♦❞❡❧❛❞❛ ❛tr❛✈és ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ Beta(α; β)✳ ❖s ♣❛râ♠❡tr♦s ❢♦r❛♠ ❡s❝♦❧❤✐❞♦s

❛ss✉♠✐♥❞♦ ✉♠ ✈❛❧♦r ♠í♥✐♠♦ ❞❡ ♣❡r❝❡♥t❛❣❡♠ ❞❡ ❡♥tr❛❞❛ ✐❣✉❛❧ ❛ 0✱ ✉♠ ✈❛❧♦r ♠á①✐♠♦

✐❣✉❛❧ ❛ 40% ❡ ❝♦♥s✐❞❡r❛♥t♦ ♦ ✈❛❧♦r 5% ❝♦♠♦ ♦ ✈❛❧♦r ♠❛✐s ♣r♦✈á✈❡❧✳ ❆ss✐♠ ❝♦♥s✐❞❡r♦✉✲s❡

q✉❡ P❡r❝❡♥t❛❣❡♠ ❞❡ ❊♥tr❛❞❛ ∼ Beta(α = 1.4375; β = 4.3125)

  ◆➸ ❚♦t❛❧ Pr❡st❛çõ❡s ✭Pt♦t❛❧✮✿ ◆❛ s✐♠✉❧❛çã♦ ❞❡st❛ ✈❛r✐á✈❡❧✱ ♥ã♦ t❡♥❞♦ ❞✐s♣♦♥í✈❡✐s q✉❛✐sq✉❡r ❡st✉❞♦s s❡♠❡❧❤❛♥t❡s

♣❛r❛ ❡st✐♠❛çã♦ ❞❛ ♣❡r❝❡♥t❛❣❡♠ ❞❡ ❝❧✐❡♥t❡s ❡♠ ❝❛❞❛ ❝❛t❡❣♦r✐❛✱ ❝♦♠❡ç♦✉✲s❡ ♣♦r s✐♠✉❧❛r

❛ ♣❡r❝❡♥t❛❣❡♠ ❞❡ ❝❧✐❡♥t❡s✱ ❡♠ ❝❛❞❛ ✉♠❛ ❞❛s ❞❡③ ❝❛t❡❣♦r✐❛s ❞❡✜♥✐❞❛s✱ ✉t✐❧✐③❛♥❞♦ ♣❛r❛

t❛❧ ✉♠❛ ❞✐str✐❜✉✐çã♦ Binomial(n = 10, p = 0.6)✳ ❆ ❡s❝♦❧❤❛ ❞♦ ♣❛râ♠❡tr♦ p r❡✢❡❝t❡✱

♥♦✈❛♠❡♥t❡✱ ❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ s✐♠✉❧❛r ✉♠❛ ❝❛rt❡✐r❛ r❡❛❧✐st❛✳ ❯♠❛ ✈❡③ ❡st✐♠❛❞❛s ❛s ♣❡r✲

  

✹✹ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❛ ✈❛r✐á✈❡❧ ■❞❛❞❡✳ ◆➸ Pr❡st❛çõ❡s P❛❣❛s ✭P❛❣❛s✮✿ ❆ s✐♠✉❧❛çã♦ ❞❡st❛ ✈❛r✐á✈❡❧ ❢♦✐ r❡❛❧✐③❛❞❛ ❣❡r❛♥❞♦ ✉♠ ♥ú♠❡r♦ ❛❧❡❛tór✐♦✱ ✉♥✐❢♦r♠❡✲

  

♠❡♥t❡ ❞✐str✐❜✉í❞♦✱ ❡♥tr❡ ✻✵ ♣r❡st❛çõ❡s ✭❛ss✉♠✐✉✲s❡ q✉❡ ♦s ✐♥❝✉♠♣r✐♠❡♥t♦s ♦❝♦rr❡♠

❛♣❡♥❛s ❛♣ós ✺ ❛♥♦s ❞❡ ❝ré❞✐t♦✮ ❡ ♦ ◆➸ ❚♦t❛❧ ❞❡ Pr❡st❛çõ❡s ❞♦ r❡s♣❡❝t✐✈♦ ❝❧✐❡♥t❡✳ ◆➸ Pr❡st❛çõ❡s P❛❣❛s✴❚♦t❛❧ ✭✪✮✭P❛❣♣❡r❝❡♥t✮✿ ❊st❛ ✈❛r✐á✈❡❧ ♥ã♦ é ♠❛✐s q✉❡ ♦ q✉♦❝✐❡♥t❡ ❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ ♣r❡st❛çõ❡s ♣❛❣❛s ❡ ♦ t♦t❛❧ ❞❡ ♣r❡st❛çõ❡s ♣❛r❛ ♦ ♠❡s♠♦ ❝❧✐❡♥t❡✳ ❱❛r✐á✈❡❧ ❉❡♣❡♥❞❡♥t❡ ❉❡❢❛✉❧t✿ ❆ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✱ ♥❡st❡ ♠♦❞❡❧♦✱ s❡rá✱ ❝♦♠♦ r❡❢❡r✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❛ ◗✉❛✲

❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦ ❞❡ ❝❛❞❛ ❝❧✐❡♥t❡✱ ♠❡❞✐❞❛ ❝♦♠♦ ❛ ♦❝♦rrê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❉❡❢❛✉❧t ♦✉

✐♥❝✉♠♣r✐♠❡♥t♦✳

  ❈♦♠♦ r❡❢❡r✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❛ ♠❡t♦❞♦❧♦❣✐❛ ✉t✐❧✐③❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦ ♣r❡ss✉♣õ❡ ❛

❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❤✐stór✐❝♦s ❞❛ ❝❛rt❡✐r❛✱ ❞❡ ❢♦r♠❛ ❛ ♣♦❞❡r ✐♥❢❡r✐r s♦❜r❡ ❛s ✈❛r✐á✈❡✐s

❡①♣❧✐❝❛t✐✈❛s ❞❡ ✐♥t❡r❡ss❡ ♥♦ ❢❡♥ó♠❡♥♦ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦✳ ❉❡st❛ ❢♦r♠❛✱ t♦r♥♦✉✲s❡ t❛♠❜é♠ ♥❡❝❡ssár✐♦ s✐♠✉❧❛r ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✱ ♣❛r❛ t♦❞♦s ♦s ❝❧✐❡♥t❡s q✉❡ ❝♦♠♣õ❡♠ ❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❝♦♥s✐❞❡r❛❞❛✳ ◆❛ s✐♠✉❧❛çã♦ ❞❡st❛ ✈❛r✐á✈❡❧ ♦♣t♦✉✲s❡ ♣♦r ❢❛③❡r ✉♠❛ ✏r❡✈❡rs❡ ❡♥❣✐♥❡❡r✐♥❣✑✱ ♦✉ s❡❥❛✱

❢❛③❡r ❛❧❣✉♥s ♣r❡ss✉♣♦st♦s s♦❜r❡ q✉❛✐s ❛s ❝♦✈❛r✐á✈❡✐s q✉❡✱ r❡❣r❛ ❣❡r❛❧✱ t❡rã♦ ♠❛✐s ✐♥✢✉ê♥✲

❝✐❛ ♥❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✳ ❊st❛ ♦♣çã♦ ♣r❡♥❞❡✉✲s❡✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ❝♦♠ ♦ ♦❜❥❡❝t✐✈♦

❞❡ ✐♥tr♦❞✉③✐r ❛❧❣✉♠❛ ♦r❞❡♠ ♥❛ ❛❧❡❛t♦r✐❡❞❛❞❡ ❞❛s ❝♦✈❛r✐á✈❡✐s✱ ✉♠❛ ✈❡③ q✉❡ ❢♦r❛♠ ♦❜✲

t✐❞❛s ♣♦r s✐♠✉❧❛çã♦ ❡ ♥ã♦ r❡✢❡❝t❡♠ ❛s ✐♥t❡r❛❝çõ❡s q✉❡✱ ❡♠ s✐t✉❛çõ❡s ♥♦r♠❛✐s✱ q✉❛s❡

❝❡rt❛♠❡♥t❡ s❡ ✈❡r✐✜❝❛rã♦✳ ❆ ❡s❝♦❧❤❛ ❞❛s ❝♦✈❛r✐á✈❡✐s ❝♦♥s✐❞❡r❛❞❛s ❝♦♠ ♠❛✐s r❡❧❡✈â♥❝✐❛ ♥❛ ❡①♣❧✐❝❛çã♦ ❞❛ ♦❝♦r✲

rê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t ❜❛s❡♦✉✲s❡ ♥♦s ❡st✉❞♦s ❛♥t❡r✐♦r❡s ❥á r❡❢❡r✐❞♦s✱ ❡ ♥❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❛ ❡st❡

❛ss✉♥t♦ r❡♣♦rt❛❞❛ ♣♦r ❡s♣❡❝✐❛❧✐st❛s ❞❛ ár❡❛ ❞❡ ❝ré❞✐t♦✳

  ❆ ❝❛❞❛ ❝♦✈❛r✐á✈❡❧ ❢♦✐ ❛tr✐❜✉í❞♦ ✉♠ ♣❡s♦✱ ❝♦♥❢♦r♠❡ ❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ♦❝♦rrê♥❝✐❛

❞❡ ❞❡❢❛✉❧t✳ ◆❛s ✈❛r✐á✈❡✐s ❝❛t❡❣ór✐❝❛s✱ ❛tr✐❜✉✐✉✲s❡ t❛♠❜é♠ ✉♠ ♣❡s♦ ❛ ❝❛❞❛ ❝❛t❡❣♦r✐❛✱

❞❡✜♥✐♥❞♦ ❛ss✐♠✱ q✉❛✐s ❛s ❝❛t❡❣♦r✐❛s ♠❛✐s ♣r♦♣❡♥s❛s à ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✳ ❖s ♣r❡ss✉♣♦st♦s ❡st❛❜❡❧❡❝✐❞♦s ♥♦ ♣❡s♦ ❞❡ ❝❛❞❛ ❝♦✈❛r✐á✈❡❧ ❡ ❡♠ ❝❛❞❛ ✉♠❛ ❞❛s ❝❛t❡✲ ❣♦r✐❛s ❡♥❝♦♥tr❛♠✲s❡ ✐❧✉str❛❞♦s ♥❛ ❚❛❜❡❧❛ ❆ s✐♠✉❧❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❉❡❢❛✉❧t ❞❡♣❡♥❞❡ ❞❛ ❝♦♠❜✐♥❛çã♦ ❞❛s ♣♦♥❞❡r❛çõ❡s ❞❛s ❝♦✈❛✲

r✐á✈❡✐s ❞❡ ❝❛❞❛ ❝❧✐❡♥t❡✳ ❆ ♣♦♥❞❡r❛çã♦✱ ♣❛r❛ ❝❛❞❛ ❝❧✐❡♥t❡✱ ♦❜té♠✲s❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

n

  X pv i × pc i , ✭✸✳✶✮

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✹✺

◆♦♠❡ P❡s♦ P❡s♦ ❈❛t❡❣♦r✐❛s

❱❛r✐á✈❡❧ ❱❛r✐á✈❡❧ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵

■❞❛❞❡ ✵✱✶✵✵✵ ✶ ✵✱✻✻ ✵✱✸✸ ◆➸ ❞❡ ❉❡♣❡♥❞❡♥t❡s ✵✱✵✺✵✵ ✶ ✵✱✸✸ ✵✱✻✻ ❚✐♣♦ ❍❛❜✐t❛çã♦ ✵✱✵✵✺✵ ✶ ✵✱✸✸ ✵✱✻✻ ■❞❛❞❡ ❝❛s❛ ✵✱✵✵✺✵ ✶ ✵✱✻✻ ✵✱✸✸ ❊st❛❞♦ ❈✐✈✐❧ ✵✱✵✹✵✵ ✶ ✵✱✻✻ ✵✱✸✸ ❈ó❞✐❣♦ P♦st❛❧ ✵✱✶✵✵✵ ✵✱✷✺ ✵✱✺ ✵✱✼✺ ✶ ❚✐♣♦ Pr♦✜ssã♦ ✵✱✵✵✾✺ ✶ ✵✱✼✺ ✵✱✺ ✵✱✷✺ ❆♥t✐❣✉✐❞❛❞❡ Pr♦✜ssã♦ ✵✱✵✵✵✺ ✶ ✵✱✺ ❘❡♥❞✐♠❡♥t♦ ▲íq✉✐❞♦ ✵✱✷✹✵✵ ✶ ✵✱✼✺ ✵✱✺ ✵✱✷✺ ❈ré❞✐t♦ ✵✱✵✺✵✵ ✶ ✵✱✼✺ ✵✱✺ ✵✱✷✺ % ❞❡ ❊♥tr❛❞❛ ✵✱✶✵✵✵ ✶ ✵✱✺

◆➸❚♦t❛❧ Pr❡st❛çõ❡s ✵✱✶✵✵✵ ✵✱✶ ✵✱✷ ✵✱✸ ✵✱✹ ✵✱✺ ✵✱✻ ✵✱✼ ✵✱✽ ✵✱✾ ✶

◆➸Pr❡st❛çõ❡s P❛❣❛s ✵✱✷✵✵✵ ✶ ✵✱✼✺ ✵✱✺ ✵✱✷✺

  ✶✱✵✵✵✵

❚❛❜❡❧❛ ✸✳✸✿ ❚❛❜❡❧❛ ❞❡ P❡s♦s ❞♦s ❋❛❝t♦r❡s ❈❤❛✈❡

♦♥❞❡ i = 1, ..., n sã♦ ❛s ❝♦✈❛r✐á✈❡✐s✱ pv i

  é ♦ ♣❡s♦ ❞❛ ❝♦✈❛r✐á✈❡❧ i ❡ pc i

  é ♦ ♣❡s♦ ❞❛ ❝❛t❡❣♦r✐❛ ❞❛ ❝♦✈❛r✐á✈❡❧ i ♦♥❞❡ ♦ ❝❧✐❡♥t❡ s❡ ✐♥s❡r❡✳ ❆ ♣♦♥❞❡r❛çã♦ s❡rá ✉♠ ✈❛❧♦r ❡♥tr❡ 0 ❡ 1✱ ✉♠❛ ✈❡③ q✉❡ t♦❞♦s ♦s ♣❡s♦s s❡ ❡♥❝♦♥tr❛♠

♥❡ss❡ ✐♥t❡r✈❛❧♦✳ ❆♣ós ♦ ❝á❧❝✉❧♦ ❞❛ ♣♦♥❞❡r❛çã♦ ♣❛r❛ ❝❛❞❛ ❝❧✐❡♥t❡ ❞❡✜♥❡✲s❡ s❡ ♦ ❝❧✐❡♥t❡

  

❡♥tr♦✉ ♦✉ ♥ã♦ ❡♠ ✐♥❝✉♠♣r✐♠❡♥t♦✳ P❛r❛ t❛❧✱ ❞❡✜♥✐✉✲s❡ ✉♠ ❧✐♠✐t❡ ❛ ♣❛rt✐r ❞♦ q✉❛❧ s❡

❝♦♥s✐❞❡r❛r✐❛ q✉❡ ♦ ❝❧✐❡♥t❡ t❡r✐❛ ♦r✐❣✐♥❛❞♦ ❞❡❢❛✉❧t✳ ❖ ✈❛❧♦r ♠í♥✐♠♦ ❞❡ ♣♦♥❞❡r❛çã♦ ♣❛r❛

❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t ❢♦✐ ❡s❝♦❧❤✐❞♦ ❞❡ ❢♦r♠❛ ❛ ❣❛r❛♥t✐r q✉❡ ❛ ♣❡r❝❡♥t❛❣❡♠ ❞❡ ❝❧✐❡♥t❡s

✐♥❝✉♠♣r✐❞♦r❡s ❢♦ss❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 3% ❞♦s ❝❧✐❡♥t❡s ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✳ P❛r❛ ❛

❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ s✐♠✉❧❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦✱ ♦ ❧✐♠✐t❡ ✉t✐❧✐③❛❞♦ ❢♦✐ ❞❡ 0.85✳ ✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s

❆ ❝♦♥str✉çã♦ ❞❡ ♠♦❞❡❧♦s q✉❛♥t✐t❛t✐✈♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣✱ ❛ ✜♠ ❞❡ ❛✉①✐❧✐❛r ♦ ♣r♦❝❡ss♦

❞❡ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦ à ❍❛❜✐t❛çã♦ ❡ ❛✈❛❧✐❛çã♦ ❞❡ r✐s❝♦ ♥✉♠❛ ✐♥st✐t✉✐çã♦ ❜❛♥❝ár✐❛

P♦rt✉❣✉❡s❛✱ é ❛ ✐❞❡✐❛ ❢✉❧❝r❛❧ ❞❡st❡ tr❛❜❛❧❤♦✳ ◆♦s ♠♦❞❡❧♦s ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ♦ ♦❜❥❡❝t✐✈♦ ♣r✐♥❝✐♣❛❧ é ✐❞❡♥t✐✜❝❛r ♦s ❢❛❝t♦r❡s q✉❡

❞❡t❡r♠✐♥❛♠ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ✭❞❡❢❛✉❧t✮ ❞♦s ❝❧✐❡♥t❡s✳ ❯♠ ❞♦s ♣r❡s✲

s✉♣♦st♦s ♣r✐♥❝✐♣❛✐s é ❛❞♠✐t✐r✲s❡ q✉❡ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ❝❧✐❡♥t❡s q✉❡ ❡♥tr❛rã♦ ❡♠

✐♥❝✉♠♣r✐♠❡♥t♦ ♥♦ ❢✉t✉r♦ sã♦ s❡♠❡❧❤❛♥t❡s às ❝❛r❛❝t❡ríst✐❝❛s ❞❛q✉❡❧❡s q✉❡ ❡♥tr❛r❛♠ ❡♠

✐♥❝✉♠♣r✐♠❡♥t♦ ♥♦ ♣❛ss❛❞♦✳

  ◆♦s ♠♦❞❡❧♦s ❞❡ ❛♣r♦✈❛çã♦ ❞❡ ❝ré❞✐t♦ ✉t✐❧✐③❛♠✲s❡ ✐♥❢♦r♠❛çõ❡s ❡①✐st❡♥t❡s s♦❜r❡ ♦

❝❧✐❡♥t❡✳ ❊st❡s ♠♦❞❡❧♦s sã♦ ❡ss❡♥❝✐❛✐s ❝♦♠♦ ❝♦♠♣❧❡♠❡♥t♦ ♥❛ ❛✈❛❧✐❛çã♦ ❡ ♥❛ t♦♠❛❞❛ ❞❡

  

✹✻ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

✸✳✷✳✶ ❆♥á❧✐s❡ ❊st❛tíst✐❝❛ ❞❛s ❱❛r✐á✈❡✐s ❯t✐❧✐③❛♥❞♦ ♦ s♦❢t✇❛r❡ ❘

  ✶ ♣♦❞❡♠♦s tr❛t❛r ❡st❛t✐st✐❝❛♠❡♥t❡ ❛ ❜❛s❡ ❞❡ ❞❛❞♦s ❞❛ ❝❛rt❡✐r❛

❞❡ ❝ré❞✐t♦✳ ❈♦♠❡ç❛♥❞♦ ♣♦r ✉♠❛ ❛♥á❧✐s❡ ♣r❡❧✐♠✐♥❛r ❞❛s ❡st❛tíst✐❝❛s ❞❡s❝r✐t✐✈❛s ❡ ❞❡♣♦✐s

  

♣♦r ✉♠❛ ❛♥á❧✐s❡ ❣rá✜❝❛ é ♣♦ssí✈❡❧ t❡r ✉♠❛ ✈✐sã♦ ♠❛✐s ❣❡r❛❧ ❞❛s ✈❛r✐á✈❡✐s q✉❡ ❝♦♥st✐t✉❡♠

❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❡ ❞❛s r❡❧❛çõ❡s ❡♥tr❡ ❛s ♠❡s♠❛s✳ ■♥✐❝✐❛❧♠❡♥t❡ ❝❛rr❡❣❛✲s❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❡ r❡❛❧✐③❛✲s❡ ✉♠❛ ❛♥á❧✐s❡ ❞❛s ❡st❛tíst✐❝❛s ❞❡s❝r✐t✐✈❛s ❞❛ q✉❛❧ r❡s✉❧t❛ ❛ ❚❛❜❡❧❛

  

■❞❛❞❡ ❉❡♣ ❍❛❜

❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪

❆❞✉❧t♦s ✽✺✻✻ ✽✺✳✻✻✪ ❆❧❣✉♥s ✶✺✹✵ ✶✺✳✹✪ ❈❛s❛ ✽✽✷✷ ✽✽✳✷✷✪

■❞♦s♦s ✷✵✼ ✷✳✵✼✪ ▼✉✐t♦s ✹✻✵ ✹✳✻✪ ❈♦♠❡r❝✐♦ ✽✸✵ ✽✳✸✪

❏♦✈❡♥s ✶✷✷✼ ✶✷✳✷✼✪ P♦✉❝♦s ✽✵✵✵ ✽✵✱✵✵✪ ❊♠♣r❡s❛ ✸✹✽ ✸✳✹✽✪

  

■❞❝❛s❛ ❈✐✈✐❧ P♦st❛❧

❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪

◆♦✈❛ ✼✹✼✹ ✼✹✳✼✹✪ ❈❛s ✹✾✸✺ ✹✾✳✸✺✪ ❩♦♥❛✶ ✹✺✸ ✹✳✺✸✪

❯s❛❞❛ ✷✵✻✵ ✷✵✳✻✪ ❉✐✈ ✷✻✷ ✷✳✻✷✪ ❩♦♥❛✷ ✶✵✶✹ ✶✵✳✶✹✪

❱❡❧❤❛ ✹✻✻ ✹✳✻✻✪ ❙♦❧ ✹✽✵✸ ✹✽✳✵✸✪ ❩♦♥❛✸ ✷✵✻✹ ✷✵✳✻✹✪

❩♦♥❛✹ ✻✹✻✾ ✻✹✳✻✾✪

  

Pr♦❢ ❆♥t❣ ❘❡♥❞

❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪

❚✐♣♦✶ ✶✾✺✻ ✶✾✳✺✻✪ ❆♥t✐❣♦ ✾✷✾✺ ✾✷✳✾✺✪ ❆❧t♦ ✸✵✷ ✸✳✵✷✪

❚✐♣♦✷ ✷✾✻✽ ✷✾✳✻✽✪ ❘❡❝❡♥t❡ ✼✵✺ ✼✳✵✺✪ ❇❛✐①♦ ✸✻✾✾ ✸✻✳✾✾✪

❚✐♣♦✸ ✸✻✶✻ ✸✻✳✶✻✪ ▼❡❞✐♦ ✶✾✾✻ ✶✾✳✾✻✪

❚✐♣♦✹ ✶✹✻✵ ✶✹✳✻✪ ▼✐♥✐♠♦ ✹✵✵✸ ✹✵✳✵✸✪

  

❈r❡❞ ❊♥tr Pt♦t❛❧

▼❡❞✐❞❛ ▲♦❝❛❧✐③❛çã♦ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪ ▼❡❞✐❞❛ ▲♦❝❛❧✐③❛çã♦

  ▼✐♥✳ ✼✺✶✻✸ ▼❛✐s✶✵ ✹✸✾✸ ✹✸✳✾✸✪ ▼✐♥✳ ✻✵✳✵ ✶st ◗✉✳ ✶✶✶✹✹✵ ▼❡♥♦s✶✵ ✺✻✵✼ ✺✻✳✵✼✪ ✶st ◗✉✳ ✸✵✵✳✵ ▼❡❞✐❛♥ ✶✹✻✾✵✸ ▼❡❞✐❛♥ ✸✻✵✳✵ ▼❡❛♥ ✶✻✺✺✷✶ ▼❡❛♥ ✸✻✶✳✾ ✸r❞ ◗✉✳ ✶✾✾✵✻✷ ✸r❞ ◗✉✳ ✹✷✵✳✵ ▼❛①✳ ✻✻✶✽✼✹ ▼❛①✳ ✻✵✵✳✵

  

P❛❣❛s P❛❣♣❡r❝♥t ❉❡❢❛✉❧t

▼❡❞✐❞❛ ▲♦❝❛❧✐③❛çã♦ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪ ❈❛t❡❣♦r✐❛ ❋r❡q✳ ✪

  

▼✐♥✳ ✻✵✳✵ ❊♥tr❡✷✺❡✺✵ ✷✽✽✼ ✷✽✳✽✼✪ ✵ ✾✻✼✸ ✾✻✳✼✸✪

✶st ◗✉✳ ✶✷✹✳✵ ❊♥tr❡✺✵❡✼✺ ✸✵✾✷ ✸✵✳✾✷✪ ✶ ✸✷✼ ✸✳✷✼✪

▼❡❞✐❛♥ ✶✾✹✳✵ ▼❛✐s✼✺ ✸✵✺✺ ✸✵✳✺✺✪ ▼❡❛♥ ✷✵✽✳✻ ▼❡♥♦s✷✺ ✾✻✻ ✾✳✻✻✪ ✸r❞ ◗✉✳ ✷✼✾✳✵ ▼❛①✳ ✺✼✽✳✵

  

❚❛❜❡❧❛ ✸✳✹✿ ❆♥á❧✐s❡ Pr❡❧✐♠✐♥❛r ❞❛s ❱❛r✐á✈❡✐s

❯♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❛♥❛❧✐s❛r ♦s ❞❛❞♦s ❞❛ ❚❛❜❡❧❛ é ♦ r❡❝✉rs♦ ❛ ❤✐st♦❣r❛♠❛s ❡ ❛ ❞✐❛❣r❛♠❛s ✏❝❛✐①❛✲❡✲❜✐❣♦❞❡s✑✳

  ✶ ❚❤❡ ❘ Pr♦❥❡❝t ❢♦r ❙t❛t✐st✐❝❛❧ ❈♦♠♣✉t✐♥❣ ✲ ❤tt♣✿✴✴✇✇✇✳r✲♣r♦❥❡❝t✳♦r❣✴

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✹✼

P❛r❛ ✐❧✉str❛r ♦s ❞❛❞♦s ❞❛s ✈❛r✐á✈❡✐s q✉❛❧✐t❛t✐✈❛s ❞❛ ❚❛❜❡❧❛ r❡❛❧✐③❛r❛♠✲s❡ ♦s ❤✐st♦❣r❛♠❛s ❞❛ ❋✐❣✉r❛

Idade Nº Dependentes Tipo de Habitação

  90

  70

  70

  70

  50

  50

  50 (%) (%) (%)

  30

  30

  30

  10

  10

  10 poucos Alguns Jovens Adultos Idosos Muitos Casa Empresa Comércio Idade Dep Hab

Idade da Casa Estado Cívil Código Postal

  50

  70

  50

  50

  30

  30 (%) (%) (%)

  30

  10

  10

  10 Nova Usada Velha Div Sol Cas Zona1 Zona2 Zona3 Zona4 Idcasa Civil Postal

Tipo de Profissão Antiguidade na Profissão Rendimento Líquido

  90

  30

  70

  30

  50 (%) (%) (%)

  30

  10

  10

  10 Tipo1 Tipo2 Tipo3 Tipo4 Antigo Recente Mínimo Baixo Médio Alto

Prof Antg Rend

  

❋✐❣✉r❛ ✸✳✶✿ ❍✐st♦❣r❛♠❛s ❱❛r✐á✈❡✐s ◗✉❛❧✐t❛t✐✈❛s

❘❡❛❧✐③❛r❛♠✲s❡ t❛♠❜é♠✱ ❛❧❣✉♥s ❣rá✜❝♦s ❝♦♠ ♦s ❞❛❞♦s ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✱ ♣❛r❛

✐❧✉str❛r ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ❝♦✈❛r✐á✈❡✐s ❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛✱ ✏❉❡❢❛✉❧t✑✳ ❈♦♠♦ ❛s ✈❛r✐á✈❡✐s

❡①♣❧✐❝❛t✐✈❛s sã♦ ✈❛r✐á✈❡✐s ❝❛t❡❣ór✐❝❛s ♥ã♦ é ♣♦ssí✈❡❧ ❢❛③❡r ✉♠❛ ❛♥á❧✐s❡ ❛tr❛✈és ❞❡ ❣rá✜❝♦s

❞❡ ❞✐s♣❡rsã♦✱ ❡♥tã♦ ✉t✐❧✐③❛♠✲s❡ ♦s ❣rá✜❝♦s ✏❝❛✐①❛✲❡✲❜✐❣♦❞❡s✑✳ ❙❡❣✉♥❞♦ ♦ r❡❝tâ♥❣✉❧♦ ✭✏❛ ❝❛✐①❛✑✮ é ❞❡s❡♥❤❛❞♦ ❞❡ t❛❧ ♠♦❞♦✱

  

✹✽ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

q✉❡ ❞✐✈✐❞❡ ❛ ❝❛✐①❛ ❡♠ ❞✉❛s ♣❛rt❡s ❝♦rr❡s♣♦♥❞❡ à ♠❡❞✐❛♥❛ ❡ é ♣♦ssí✈❡❧ ✐❞❡♥t✐✜❝❛r ❛s

❛ss✐♠❡tr✐❛s ♥♦s ❞❛❞♦s✱ ❝❛s♦ ❛s ♣❛rt❡s✱ ❡♠ q✉❡ ❛ ❝❛✐①❛ ❡stá ❞✐✈✐❞✐❞❛✱ s❡❥❛♠ ♠✉✐t♦ ❞✐❢❡✲

r❡♥t❡s✳ ❆ ❝❛✐①❛ ♥♦s s❡✉s ❧✐♠✐t❡s ❤♦r✐③♦♥t❛✐s✱ ✐st♦ é✱ ❡♥tr❡ ♦ ♣r✐♠❡✐r♦ ❡ ♦ t❡r❝❡✐r♦ q✉❛rt✐s✱

❝♦♥té♠ 50% ❞❛s ♦❜s❡r✈❛çõ❡s✳ ❖ ♠í♥✐♠♦ ❡ ♦ ♠á①✐♠♦ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s s❡❣♠❡♥t♦s

✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r ✭✏♦s ❜✐❣♦❞❡s✑✮ ❞❡s❡♥❤❛❞♦s ♥♦ ❡①t❡r✐♦r ❞♦ r❡❝tâ♥❣✉❧♦ ❡ s❡r✈❡♠ ♣❛r❛

✐❞❡♥t✐✜❝❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ♦✉t❧✐❡rs✱ ♥♦ ❝❛s♦ ❞❡ ♦s ❜✐❣♦❞❡s s❡r❡♠✱ r❡❧❛t✐✈❛♠❡♥t❡ à ❝❛✐①❛✱

♠✉✐t♦ ❣r❛♥❞❡s✳

  ❖s ♦✉t❧✐❡rs sã♦ ✐❞❡♥t✐✜❝❛❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ✶✳ x i é ✉♠ ♦✉t❧✐❡r s❡✈❡r♦ s❡✿ x < Q − 3(Q − Q ) ou x > Q + 3(Q − Q ); i

  1

  3 1 i

  3

  3

  1 i ✷✳ x é ✉♠ ♦✉t❧✐❡r ♠♦❞❡r❛❞♦ s❡✿ Q − 3(Q − Q ) < x < Q − 1.5(Q − Q )

  1

  3 1 i

  1

  3

  1 ou Q + 1.5(Q − Q ) < x < Q + 3(Q − Q ).

  3

  3 1 i

  3

  3

  1 ❆ ❋✐❣✉r❛ r❡❧❛t✐✈❛♠❡♥t❡ ❛ ❡st❛ ✈❛r✐á✈❡❧✳

❈♦♠♦ s❡ ♣♦❞❡ ♦❜s❡r✈❛r✱ ❡①✐st❡ ✉♠❛ ♥♦tór✐❛ ❛ss✐♠❡tr✐❛ ❡♥tr❡ ♦s ♠♦♥t❛♥t❡s ❞❡ ❝ré❞✐t♦

❡ ✈❡r✐✜❝❛✲s❡✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♦✉t❧✐❡rs ♠♦❞❡r❛❞♦s ❡ s❡✈❡r♦s✱ ♣♦✐s ♦ t❛♠❛♥❤♦ ❞♦ ❜✐❣♦❞❡

s✉♣❡r✐♦r é s✉♣❡r✐♦r ❛ ✉♠❛ ✈❡③ ❡ ♠❡✐❛ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ q✉❛rt✐s ❡ s✉♣❡r✐♦r ❛ ✸ ✈❡③❡s ❛

♠❡s♠❛ ❞✐stâ♥❝✐❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  Caixa−e−bigodes − Crédito Concedido Max.

  3rd Qu.

  Median 1st Qu.

  Min.

  Crédito Concedido

❋✐❣✉r❛ ✸✳✷✿ ❈❛✐①❛✲❡✲❇✐❣♦❞❡s✿ ❱❛r✐á✈❡❧ ❈ré❞✐t♦

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✹✾

❆ ❋✐❣✉r❛ ✐❧✉str❛ ❛s ❝❛✐①❛s ❞❡ ❜✐❣♦❞❡s ♣❛r❛ ❛s Pr❡st❛çõ❡s ❚♦t❛✐s ❡ Pr❡st❛çõ❡s

P❛❣❛s✳ P❛r❛ ❛♠❜❛s ❛s ✈❛r✐á✈❡✐s é ♣♦ssí✈❡❧ ♦❜s❡r✈❛r✲s❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♦✉t❧✐❡rs ♠♦❞❡r❛❞♦s

♠❛s ❛♣❡♥❛s ♣❛r❛ ❛ ✈❛r✐á✈❡❧ Pr❡st❛çõ❡s P❛❣❛s s❡ ✈❡r✐✜❝❛ ✉♠❛ ❧✐❣❡✐r❛ ❛ss✐♠❡tr✐❛ ❡♥tr❡ ♦s

❞❛❞♦s✳

Prestações

  600 578 420 360 300 279 194 124

  60 Pagas Ptotal Prestações Totais e Prestações Pagas

  

❋✐❣✉r❛ ✸✳✸✿ ❈❛✐①❛✲❡✲❇✐❣♦❞❡s✿ Pr❡st❛çõ❡s ❚♦t❛✐s ❡ Pr❡st❛çõ❡s P❛❣❛s

❖s ❤✐st♦❣r❛♠❛s ❞❛ ❋✐❣✉r❛ r❡♣r❡s❡♥t❛♠ ❛ ❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛ ❞❛s ✈❛r✐á✈❡✐s q✉❛♥✲ t✐t❛t✐✈❛s✱ ❈ré❞✐t♦ ❡ Pr❡st❛çõ❡s ❚♦t❛✐s ❡ Pr❡st❛çõ❡s P❛❣❛s✳

Crédito Concedido Nº Total de Prestações Nº Prestações Pagas

  25

  35

  20

  30

  20

  25

  15

  15

  20

  (%) (%) (%)

  10

  15

  10

  10

  5

  5

  5 1e+05 3e+05 5e+05 7e+05 60 180 300 420 540 60 180 300 420 540

  Cred Ptotal Pagas

  

❋✐❣✉r❛ ✸✳✹✿ ❍✐st♦❣r❛♠❛s ❱❛r✐á✈❡✐s ◗✉❛♥t✐t❛t✐✈❛s

❚❛❧ ❝♦♠♦ ♥❛s ❋✐❣✉r❛s ♦ ❤✐st♦❣r❛♠❛s ❞❛ ✈❛r✐á✈❡❧ ❈ré❞✐t♦ ❡ ❞❛ ✈❛r✐á✈❡❧

  

✺✵ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ ♣r❡st❛çõ❡s ♣❛❣❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖❜s❡r✈❛✲s❡ t❛♠❜é♠ ❛ ❧✐❣❡✐r❛

s✐♠❡tr✐❛ ❡♥tr❡ ♦s ✈❛❧♦r❡s ❞❛s ♣r❡st❛çõ❡s t♦t❛✐s✱ q✉❡ s❡ ❥✉st✐✜❝❛ ♣❡❧❛ ❢♦r♠❛ ❝♦♠♦ ❡st❛

✈❛r✐á✈❡❧ ❢♦✐ s✐♠✉❧❛❞❛✳

  P❛r❛ ❛♥❛❧✐s❛r ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❉❡❢❛✉❧t ❡ ❛s ✈❛r✐á✈❡✐s q✉❛♥t✐t❛t✐✈❛s

❈r❡❞✐t♦✱ Pr❡st❛çõ❡s ❚♦t❛✐s ❡ Pr❡st❛çõ❡s P❛❣❛s ❝♦♥str✉ír❛♠✲s❡ ♦s ❣rá✜❝♦s ❞❛ ❋✐❣✉r❛

❡ ❞❛ ❋✐❣✉r❛

  6.7e+05 600 578 420 360 agas 300 otal de Prestações

  279 Prestações P Crédito Concedido

Nº T

  194 2e+05 1.5e+05

  124 1.1e+05 7.5e+04

  60

  60

  1

  1

  1 Default Default Default

❋✐❣✉r❛ ✸✳✺✿ ❈❛✐①❛✲❡✲❇✐❣♦❞❡s✿ ❱❛r✐á✈❡❧ ❉❡❢❛✉❧t ✈s Pr❡st❛çõ❡s

◆♦t❡✲s❡ q✉❡ ♥❛ ❋✐❣✉r❛ ♥♦ ❣rá✜❝♦ ✏❉❡❢❛✉❧t✴❈ré❞✐t♦ ❈♦♥❝❡❞✐❞♦✑✱ ♦s ❝❧✐❡♥t❡s q✉❡

  

❡♥tr❛♠ ❡♠ ❞❡❢❛✉❧t✱ r❡❣r❛ ❣❡r❛❧✱ sã♦ ❝❧✐❡♥t❡s ❝✉❥♦s ❝ré❞✐t♦s sã♦ ♠♦♥t❛♥t❡s ♠❛✐s ❜❛✐①♦s✱

❡①✐st❡♠ ♣♦✉❝♦s ❝❧✐❡♥t❡s ❝♦♠ ❝ré❞✐t♦ s✉♣❡r✐♦r ❛ e✸✵✵✳✵✵✵ q✉❡ ❡♥tr❛♠ ❡♠ ✐♥❝✉♠♣r✐♠❡♥t♦✳

❊♠ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❣rá✜❝♦ r❡s♣❡❝t✐✈♦ ❞❛ ❋✐❣✉r❛ ❛s ♦❜s❡r✈❛çõ❡s sã♦ s❡♠❡❧❤❛♥t❡s✱

❡①✐st❡♠ ❛❧❣✉♥s ❝❧✐❡♥t❡s ❝♦♠ ❝ré❞✐t♦s ❡❧❡✈❛❞♦s ✭♦✉t❧✐❡rs✮ q✉❡ ❡♥tr❛♠ ❡♠ ✐♥❝✉♠♣r✐♠❡♥t♦

  

♣r♦✈♦❝❛♥❞♦ ♦s ❞♦✐s ♣✐❝♦s ❛ ✈❡r❞❡ ❡s❝✉r♦✱ ♠❛s ♦s r❡st❛♥t❡s ✐♥❝✉♠♣r✐♠❡♥t♦s s✐t✉❛♠✲s❡

♥✉♠❛ ③♦♥❛ ♦♥❞❡ ♦ ♠♦♥t❛♥t❡ ❞♦ ❝ré❞✐t♦ é ✐♥❢❡r✐♦r✳ ❖❜s❡r✈❛♥❞♦ ❛❣♦r❛ ♦s ❣rá✜❝♦s ❉❡❢❛✉❧t✴P❛❣❛s ❞❛s ♠❡s♠❛s ✜❣✉r❛s✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ❛

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✺✶

  1.0

  1.0

  1.0

  1

  0.8

  0.8

  0.8

  1

  1

  0.6

  0.6

  0.6 ault ault ault Def Def Def

  0.4

  0.4

  0.4

  0.2

  0.2

  0.2

  0.0

  0.0

  0.0

1e+05 4e+05 100 300 500 100 300 500

Cred Ptotal Pagas

  

❋✐❣✉r❛ ✸✳✻✿ ❈❞P❧♦t ✲ ❘❡❧❛çã♦ ❡♥tr❡ ❉❡❢❛✉❧t ❡ ❛s ❱❛r✐á✈❡✐s ◗✉❛♥t✐t❛t✐✈❛s

r❛♥❞♦ ♦s ❣rá✜❝♦s ❝♦♠ ♦s ✈❛❧♦r❡s ❞❛ ❚❛❜❡❧❛ ♠❛✐s ✉♠❛ ✈❡③ s❡ ✈❡r✐✜❝❛ q✉❡ ♦s ❣rá✜❝♦s

r❡✢❡❝t❡♠ ♦s ♣❡s♦s ❞❛❞♦s às ✈❛r✐á✈❡✐s✳ ◆❡st❡ ❝❛s♦ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❢♦✐ ❞❛❞♦ ♣❡s♦ 1 à ♠❡♥♦r

♣❡r❝❡♥t❛❣❡♠ ❞❡ ♣r❡st❛çõ❡s ♣❛❣❛s ❡ 0.25 q✉❛♥❞♦ ❛s ♣r❡st❛çõ❡s ♣❛❣❛s ✉❧tr❛♣❛ss❛♠ ✼✺✪

❞♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♣r❡st❛çõ❡s✳ ❊st❡s ♣❡s♦s ❢♦r❛♠ ❞❛❞♦s✱ ❝♦♠ ❜❛s❡ ♥❛ ✐❞❡✐❛ ❞❡ q✉❡✱

❛❧❣✉é♠ q✉❡ ❛✐♥❞❛ só ♣❛❣♦✉ ❛❧❣✉♠❛s ♣r❡st❛çõ❡s ♠❛✐s ❢❛❝✐❧♠❡♥t❡ ❛❜❛♥❞♦♥❛ ♦ ❝♦♠♣r♦✲

♠✐ss♦ ❞❡ ❝ré❞✐t♦✱ ❡♠ ❝❛s♦ ❞❡ ❞✐✜❝✉❧❞❛❞❡✱ ♣♦✐s ♥ã♦ ♣❡r❞❡ t❛♥t♦ ❝♦♠♦ s❡ ❥á ❡st✐✈❡ss❡ ♥♦

✜♠ ❞♦ ❡♠♣rést✐♠♦✳

  P❛r❛ ♦ ◆➸ ❚♦t❛❧ ❞❡ Pr❡st❛çõ❡s ✈❡r✐✜❝❛✲s❡ ✉♠❛ s✐t✉❛çã♦ ♦♣♦st❛✱ ♦✉ s❡❥❛✱ q✉❛♥t♦

♠❛✐♦r ♦ ◆➸ ❚♦t❛❧ ❞❡ Pr❡st❛çõ❡s ❡♠ q✉❡ ♦ ❝ré❞✐t♦ ♣♦❞❡ s❡r ♣❛❣♦✱ ♠❛✐♦r ❛ ♣r♦❜❛❜✐❧✐✲

❞❛❞❡ ❞❡ ♦❝♦rr❡r ❉❡❢❛✉❧t✱ ❡ ♥♦✈❛♠❡♥t❡ ❡st❛s ♦❜s❡r✈❛çõ❡s sã♦ ❥✉st✐✜❝á✈❡✐s ♣♦✐s q✉❛♥t♦

♠❛✐♦r ♦ ◆➸ ❚♦t❛❧ ❞❡ Pr❡st❛çõ❡s ♠❛✐♦r é ♦ t❡♠♣♦ ❞✐s♣♦♥í✈❡❧ ♣❛r❛ ♦❝♦rr❡r❡♠ ❡✈❡♥t♦s

q✉❡ ♣♦ss❛♠ ❧❡✈❛r ❛♦ ✐♥❝✉♠♣r✐♠❡♥t♦✱ t❛✐s ❝♦♠♦ ❞❡s❡♠♣r❡❣♦ ♦✉ ❞♦❡♥ç❛✱ ❡♥tr❡ ♦✉tr♦s✳

◆♦✈❛♠❡♥t❡✱ ♦s ❣rá✜❝♦s sã♦ ♦ r❡✢❡①♦ ❞♦s ♣❡s♦s ❛tr✐❜✉í❞♦s às ✈❛r✐á✈❡✐s ✭r❡✈❡r ❚❛❜❡❧❛

  ❯♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❛♥❛❧✐s❛r ❛ ✈❛r✐á✈❡❧ ❉❡❢❛✉❧t ❡♠ r❡❧❛çã♦ às Pr❡st❛çõ❡s P❛❣❛s é

♦ r❡❝✉rs♦ ❛♦s s❝❛tt❡r♣❧♦ts ❞♦s ❞❛❞♦s✱ ❞❡s❡♥❤❛♥❞♦ ✉♠❛ ❧✐♥❤❛ ❛tr❛✈és ❞♦s ♠❡s♠♦s✳ ❊st❛

❧✐♥❤❛ é ❞❡s❡♥❤❛❞❛ ❛tr❛✈és ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❡ ✉t✐❧✐③❛ ✐♥❢♦r♠❛çã♦ ♣r♦✈❡♥✐❡♥t❡ ❞♦ ♠♦❞❡❧♦

❛❥✉st❛❞♦ ♣❛r❛ ♣r♦❞✉③✐r ✏❢✉♥çõ❡s s✉❛✈❡s✑✳ ❆❥✉st❛r❛♠✲s❡ ♦s ♣❛r❡s ❞❡ ✈❛r✐á✈❡✐s ❉❡❢❛✉❧t ✈s

❈ré❞✐t♦ ❡ ❉❡❢❛✉❧t ✈s Pr❡st❛çõ❡s P❛❣❛s ❡ ♦❜t✐✈❡r❛♠✲s❡ ♦s ❣rá✜❝♦s ❞❛ ❋✐❣✉r❛ ❖❜s❡r✈❛✲s❡ q✉❡ ❛ ❝✉r✈❛ ❞❡ ❛❥✉st❛♠❡♥t♦ ❞❛ ✈❛r✐á✈❡❧ ❈ré❞✐t♦✱ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠

❛ ❝✉r✈❛ ❞❡ ❛❥✉st❛♠❡♥t♦ ❞❛ ✈❛r✐á✈❡❧ Pr❡st❛çõ❡s P❛❣❛s✱ ❡stá ♠✉✐t♦ ♠❛✐s ♣ró①✐♠❛ ❞♦s

✈❛❧♦r❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦ ❉❡❢❛✉❧t ✐❣✉❛❧ ❛ ③❡r♦✳ P♦❞❡ ❞✐③❡r✲s❡ q✉❡✱ ❡♠❜♦r❛ ❛ ✈❛r✐á✈❡❧

❈ré❞✐t♦ s❡❥❛ s✐❣♥✐✜❝❛t✐✈❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ♦❝♦rrê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦✱ ❛

✈❛r✐á✈❡❧ Pr❡st❛çõ❡s P❛❣❛s é ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛✳ ❖ ❞❡❝❧✐✈❡ ♠❛✐s ❛❝❡♥t✉❛❞♦ ❞❛ ❝✉r✈❛ ❞❡

❛❥✉st❛♠❡♥t♦ ❞❛ ✈❛r✐á✈❡❧ Pr❡st❛çõ❡s P❛❣❛s s✉❣❡r❡ q✉❡ ❡st❛ ✈❛r✐á✈❡❧ t❡♥❤❛ ♠❛✐s ♣❡s♦

♥❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✳ ❆ ✈❛r✐á✈❡❧ ❈ré❞✐t♦ ♣❛r❡❝❡ ❡①♣❧✐❝❛r ♠❡❧❤♦r ❛ ♦❝♦rrê♥❝✐❛ ❞❡

❝✉♠♣r✐♠❡♥t♦ ❞♦ q✉❡ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦✳

  

✺✷ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

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  ❞❡ ❝❧✐❡♥t❡s q✉❡ ❡♥tr♦✉ ❡♠ ✐♥❝✉♠♣r✐♠❡♥t♦ ❡ ♦s q✉❡ ❝✉♠♣r✐r❛♠ ♦ ❝♦♥tr❛t♦✳ ❈♦♥❝❧✉í❞❛ ❛ ❛♥á❧✐s❡ ♣r❡❧✐♠✐♥❛r ❞♦s ❞❛❞♦s ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ é ✐♠♣♦rt❛♥t❡ ❞❡s✲

❝r❡✈❡r ♦s ♠ét♦❞♦s ❡ ♠♦❞❡❧♦s ❡st❛tíst✐❝♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❡st✉❞❛r ❞❡ ❢♦r♠❛

♠❛✐s ✐♥t❡♥s✐✈❛ ♦s ❞❛❞♦s ❡✱ ❞❡st❛ ❢♦r♠❛✱ ❞❡t❡r♠✐♥❛r q✉❛✐s ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ♠❛✐s

s✐❣♥✐✜❝❛t✐✈❛s ♣❛r❛ ❡①♣❧✐❝❛r ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦ ♣❛r❛ ♦s ❝❧✐❡♥t❡s ❞❡st❛ ❝❛r✲

t❡✐r❛✳

  1

❋✐❣✉r❛ ✸✳✼✿ ❆❥✉st❛♠❡♥t♦ ❯s❛♥❞♦ ❛ ❋✉♥çã♦ Pr❡❞✐❝t

❝❛❞❛ ✈❛r✐á✈❡❧ q✉❛❧✐t❛t✐✈❛ ❡ r❡s♣❡❝t✐✈❛ ❝❛t❡❣♦r✐❛ ♦❝♦rr❡♠ ❞✐❢❡r❡♥❝✐❛♥❞♦ ❛✐♥❞❛ ♦ ♥ú♠❡r♦

  Def ault 100 200 300 400 500 578

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  ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ●

  1e+05 3e+0.5 5e+0.5 6.7e+05

  Crédito Concedido Def ault

  ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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  ● ● ● ● ● ●● ● ● ● ● ● ● ● ●

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  ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ●

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  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●

  ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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  ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●

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  ◗✉❛♥❞♦ ♦ ♦❜❥❡❝t✐✈♦ é ❡st✉❞❛r ❛ r❡❧❛çã♦ ❡♥tr❡ ✈❛r✐á✈❡✐s ♦✉ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱

❛♥❛❧✐s❛r ❛ ✐♥✢✉ê♥❝✐❛ q✉❡ ✉♠❛ ✈❛r✐á✈❡❧ ♦✉ ♠❛✐s ✈❛r✐á✈❡✐s ✭❡①♣❧✐❝❛t✐✈❛s✮✱ ♠❡❞✐❞❛s ❡♠

✐♥❞✐✈í❞✉♦s ♦✉ ♦❜❥❡❝t♦s✱ tê♠ s♦❜r❡ ✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ✐♥t❡r❡ss❡ ✭✈❛r✐á✈❡❧ r❡s♣♦st❛✮✱ ❣❡✲

r❛❧♠❡♥t❡ ♦ ♣r♦❜❧❡♠❛ é ❛❜♦r❞❛❞♦ ❛tr❛✈és ❞♦ ❡st✉❞♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ q✉❡

r❡❧❛❝✐♦♥❡ ❡ss❛ ✈❛r✐á✈❡❧ ❞❡ ✐♥t❡r❡ss❡ ❝♦♠ ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✳ P♦r ❡st❛ r❛③ã♦✱ ♥❡st❡

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✺✸

✵ ✶

  

Pr♦❢ ❚✐♣♦✸ ✸✺✵✶ ✸✺✱✵✶✪ ✶✶✺ ✶✱✶✺✪

❚✐♣♦✹ ✶✹✶✻ ✶✹✱✶✻✪ ✹✹ ✵✱✹✹✪ ❆♥t✐❣♦ ✽✾✾✵ ✽✾✱✾✵✪ ✸✵✺ ✸✱✵✺✪

  

❚❛❜❡❧❛ ✸✳✺✿ ❉❡❢❛✉❧t ✈s ❱❛r✐á✈❡✐s ◗✉❛❧✐t❛t✐✈❛s

✸✳✷✳✷ ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s

❖s ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s ✭▼▲●✮ ✈ê♠ ✉♥✐✜❝❛r ♠♦❞❡❧♦s ❛♥t❡r✐♦r❡s ❞❡s❡♥✈♦❧✈✐✲

❞♦s ♣❛r❛ ❛ ♠♦❞❡❧❛çã♦ ❡st❛tíst✐❝❛ ❡ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞♦s ♣♦r

◆♦s ▼▲●✬s ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡♥tr♦ ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❞✐s✲

tr✐❜✉✐çõ❡s ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ♠✉✐t♦ ❡s♣❡❝í✜❝❛s✿ ❛ ❋❛♠í❧✐❛ ❊①♣♦♥❡♥❝✐❛❧✳

  ✸✻✵ ✷✺✷✻ ✷✺✱✷✻✪ ✻✻ ✵✱✻✻✪ ✹✷✵ ✷✵✶✼ ✷✵✱✶✼✪ ✽✽ ✵✱✽✽✪ ✹✽✵ ✶✶✽✼ ✶✶✱✽✼✪ ✾✷ ✵✱✾✷✪ ✺✹✵ ✸✼✶ ✸✱✼✶✪ ✹✷ ✵✱✹✷✪ ✻✵✵ ✺✷ ✵✱✺✷✪ ✽ ✵✱✵✽✪

  ✶✽✵ ✹✸✾ ✹✱✸✾✪ ✵ ✵✱✵✵✪ ✷✹✵ ✶✵✺✺ ✶✵✱✺✺✪ ✷ ✵✱✵✷✪

Pt♦t❛❧ ✸✵✵ ✶✾✶✻ ✶✾✱✶✻✪ ✷✾ ✵✱✷✾✪

  

❊♥tr ▼❡♥♦s✶✵ ✺✸✶✹ ✺✸✱✶✹✪ ✷✾✸ ✷✱✾✸✪

✻✵ ✶✷ ✵✱✶✷✪ ✵ ✵✱✵✵✪ ✶✷✵ ✾✽ ✵✱✾✽✪ ✵ ✵✱✵✵✪

  

❘❡♥❞ ▼❡❞✐♦ ✶✾✾✺ ✶✾✱✾✺✪ ✶ ✵✱✵✶✪

▼✐♥✐♠♦ ✸✻✾✼ ✸✻✱✾✼✪ ✸✵✻ ✸✱✵✻✪ ▼❛✐s✶✵ ✹✸✺✾ ✹✸✱✺✾✪ ✸✹ ✵✱✸✹✪

  

❆♥t❣ ❘❡❝❡♥t❡ ✻✽✸ ✻✱✽✸✪ ✷✷ ✵✱✷✷✪

❆❧t♦ ✸✵✷ ✸✱✵✷✪ ✵ ✵✱✵✵✪ ❇❛✐①♦ ✸✻✼✾ ✸✻✱✼✾✪ ✷✵ ✵✱✷✵✪

  ❩♦♥❛✹ ✻✷✵✸ ✻✷✱✵✸✪ ✷✻✻ ✷✱✻✻✪ ❚✐♣♦✶ ✶✽✽✻ ✶✽✱✽✻✪ ✼✵ ✵✱✼✵✪ ❚✐♣♦✷ ✷✽✼✵ ✷✽✱✼✵✪ ✾✽ ✵✱✾✽✪

  

❱❛r✐á✈❡❧ ❈❛t❡❣♦r✐❛ ❋rq✳ ❆❜s✳ ❋rq✳ ❘❡❧✳ ❋rq✳ ❆❜s✳ ❋rq✳ ❘❡❧✳

❆❞✉❧t♦s ✽✸✸✵ ✽✸✱✸✵✪ ✷✸✻ ✷✱✸✻✪

  ❩♦♥❛✷ ✶✵✵✺ ✶✵✱✵✺✪ ✾ ✵✱✵✾✪

P♦st❛❧ ❩♦♥❛✸ ✷✵✶✸ ✷✵✱✶✸✪ ✺✶ ✵✱✺✶✪

  

❈✐✈✐❧ ❉✐✈ ✷✹✼ ✷✱✹✼✪ ✶✺ ✵✱✶✺✪

❙♦❧ ✹✻✵✹ ✹✻✱✵✹✪ ✶✾✾ ✶✱✾✾✪ ❩♦♥❛✶ ✹✺✷ ✹✱✺✷✪ ✶ ✵✱✵✶✪

  

■❞❝❛s❛ ❯s❛❞❛ ✶✾✽✸ ✶✾✱✽✸✪ ✼✼ ✵✱✼✼✪

❱❡❧❤❛ ✹✺✵ ✹✱✺✵✪ ✶✻ ✵✱✶✻✪ ❈❛s ✹✽✷✷ ✹✽✱✷✷✪ ✶✶✸ ✶✱✶✸✪

  

❍❛❜ ❈♦♠❡r❝✐♦ ✼✾✽ ✼✱✾✽✪ ✸✷ ✵✱✸✷✪

❊♠♣r❡s❛ ✸✸✼ ✸✱✸✼✪ ✶✶ ✵✱✶✶✪ ◆♦✈❛ ✼✷✹✵ ✼✷✱✹✵✪ ✷✸✹ ✷✱✸✹✪

  

❉❡♣ ▼✉✐t♦s ✹✺✸ ✹✱✺✸✪ ✼ ✵✱✵✼✪

P♦✉❝♦s ✼✼✵✻ ✼✼✱✵✻✪ ✷✾✹ ✷✱✾✹✪ ❈❛s❛ ✽✺✸✽ ✽✺✱✸✽✪ ✷✽✹ ✷✱✽✹✪

  

■❞❛❞❡ ■❞♦s♦s ✷✵✹ ✷✱✵✹✪ ✸ ✵✱✵✸✪

❏♦✈❡♥s ✶✶✸✾ ✶✶✱✸✾✪ ✽✽ ✵✱✽✽✪ ❆❧❣✉♥s ✶✺✶✹ ✶✺✱✶✹✪ ✷✻ ✵✱✷✻✪

  ❖s ♠♦❞❡❧♦s q✉❡ s❡ ❡♥✉♠❡r❛♠ ❛ s❡❣✉✐r sã♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r❡s ❞♦s ♠♦❞❡❧♦s ❧✐♥❡❛r❡s ❣❡♥❡r❛❧✐③❛❞♦s✿ ❼ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r ❝❧áss✐❝♦✱

  

✺✹ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❼ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛✱ ❼ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❞❡ P♦✐ss♦♥✱ ❼ ♠♦❞❡❧♦s ❧♦❣✲❧✐♥❡❛r❡s ♣❛r❛ t❛❜❡❧❛s ❞❡ ❝♦♥t✐♥❣ê♥❝✐❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛✐s✱ ❼ ♠♦❞❡❧♦ ♣r♦❜✐t ♣❛r❛ ❡st✉❞♦s ❞❡ ♣r♦♣♦rçõ❡s✱ ❡t❝ ◆♦t❛çã♦ ❡ ❚❡r♠✐♥♦❧♦❣✐❛

  

❈♦♥s✐❞❡r❡✲s❡✱ ✈❡r ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ Y ✱ ❞❡ ✐♥t❡r❡ss❡ ♣r✐♠ár✐♦✱

T

  , . . . , x ) ❡ q✉❡ ❞❡s✐❣♥❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ♦✉ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✱ ❡ ✉♠ ✈❡❝t♦r ① = (x 1 k

  

❞❡ k ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✱ t❛♠❜é♠ ❞❡s✐❣♥❛❞❛s ♣♦r ❝♦✈❛r✐á✈❡✐s ♦✉ ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥✲

❞❡♥t❡s✱ q✉❡ s❡ ❝rê ❡①♣❧✐❝❛r❡♠ ♣❛rt❡ ❞❛ ✈❛r✐❛❜✐❧✐❞❛❞❡ ✐♥❡r❡♥t❡ ❛ Y ✳

❆ ✈❛r✐á✈❡❧ r❡s♣♦st❛ Y ♣♦❞❡ s❡r ❝♦♥tí♥✉❛✱ ❞✐s❝r❡t❛ ♦✉ ❞✐❝♦tó♠✐❝❛✳ ❆s ❝♦✈❛r✐á✈❡✐s✱ ❞❡✲

t❡r♠✐♥íst✐❝❛s ♦✉ ❡st♦❝ást✐❝❛s✱ ♣♦❞❡♠ s❡r t❛♠❜é♠ ❞❡ q✉❛❧q✉❡r ♥❛t✉r❡③❛✿ ❝♦♥tí♥✉❛s✱

❞✐s❝r❡t❛s✱ q✉❛❧✐t❛t✐✈❛s ❞❡ ♥❛t✉r❡③❛ ♦r❞✐♥❛❧ ♦✉ ❞✐❝♦tó♠✐❝❛s✳ ❆ss✉♠❡✲s❡ q✉❡ ♦s ❞❛❞♦s

tê♠ ❛ ❢♦r♠❛

  

(y , x ), i = 1, . . . , n,

i i

  ✭✸✳✷✮

r❡s✉❧t❛♥t❡s ❞❛ r❡❛❧✐③❛çã♦ ❞❡ (Y, ①) ❡♠ n ✐♥❞✐✈í❞✉♦s ♦✉ ✉♥✐❞❛❞❡s ❡①♣❡r✐♠❡♥t❛✐s✱ s❡♥❞♦

T

  , . . . , Y ) i i n

  

❛s ❝♦♠♣♦♥❡♥t❡s Y ❞♦ ✈❡❝t♦r ❛❧❡❛tór✐♦ ❨ = (Y ✐♥❞❡♣❡♥❞❡♥t❡s✳ P♦❞❡✲s❡ r❡✲

♣r❡s❡♥t❛r ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✱     y

  1 x 11 x 11 · · · x 1k     y x x · · · x

  2

  21 22 2k     .

  ❨ = ❳ = ✭✸✳✸✮    

      ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ y x x · · · x n n1 n2 nk

  

❖s ♠♦❞❡❧♦s ❧✐♥❡❛r❡s ❣❡♥❡r❛❧✐③❛❞♦s sã♦ ✉♠❛ ❡①t❡♥sã♦ ❞♦ ♠♦❞❡❧♦ ❧✐♥❡❛r ❝❧áss✐❝♦✱

❨ = ❩β + ε ou

  • β 1 x 1 + . . . + β i x i + ε.

  ❨ = β ✭✸✳✹✮ ♦♥❞❡ ❩ é ✉♠❛ ♠❛tr✐③ ❞❡ ❞✐♠❡♥sã♦ n × p ❞❡ ❡s♣❡❝✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦✱ ❛ss♦❝✐❛❞❛ ❛

  T , . . . , β )

  1 p ✉♠ ✈❡❝t♦r β = (β ❞❡ ♣❛râ♠❡tr♦s ❡ ε ✉♠ ✈❡❝t♦r ❞❡ ❡rr♦s ❛❧❡❛tór✐♦s✳ ❆ ❡s❝♦❧❤❛ ❞❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❞❡♣❡♥❞❡ ❞♦ t✐♣♦ ❞❡ r❡s♣♦st❛ ❡ ❞♦ ❡st✉❞♦ ♣❛rt✐❝✉❧❛r

q✉❡ s❡ ❡stá ❛ ❢❛③❡r✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ❞❛❞♦s ❜✐♥ár✐♦s ✉t✐❧✐③❛✲s❡ ❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦

▲♦❣✐t q✉❡ s❡rá ✈✐st❛ ♠❛✐s ❛❞✐❛♥t❡ ♥❛ ❡①♣❧✐❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ▲♦❣íst✐❝❛✳ ❋❛♠í❧✐❛ ❊①♣♦♥❡♥❝✐❛❧ ❉❡✜♥✐çã♦ ✸✳✶✳ ❋❛♠í❧✐❛ ❊①♣♦♥❡♥❝✐❛❧

❉✐③✲s❡ q✉❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ Y t❡♠ ❞✐str✐❜✉✐çã♦ ♣❡rt❡♥❝❡♥t❡ à ❢❛♠í❧✐❛ ❡①♣♦♥❡♥❝✐❛❧

s❡ ❛ s✉❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✭❢✳❞✳♣✳✮ ♦✉ ❢✉♥çã♦ ♠❛ss❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✺✺

n o yθ−b(θ) f (y|θ, φ) = exp + c(y, φ) ✱ a(φ)

  ♦♥❞❡ θ ❡ φ sã♦ ♣❛râ♠❡tr♦s ❡s❝❛❧❛r❡s✱ a(·)✱ b(·) ❡ c(·) sã♦ ❢✉♥çõ❡s r❡❛✐s ❝♦♥❤❡❝✐❞❛s✳ ◆❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛✲s❡ φ = 1✳ P❛r❛ ❛♣❧✐❝❛r ❛ ♠❡t♦❞♦❧♦❣✐❛ ❞♦s ▼▲●✬s ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❤á ♥❡❝❡ss✐❞❛❞❡✱ ❛♣ós

❛ ❢♦r♠✉❧❛çã♦ ❞♦ ♠♦❞❡❧♦ q✉❡ s❡ ♣❡♥s❛ ❛❞❡q✉❛❞♦✱ ❞❡ ♣r♦❝❡❞❡r à r❡❛❧✐③❛çã♦ ❞❡ ✐♥❢❡rê♥❝✐❛s

s♦❜r❡ ❡ss❡ ♠♦❞❡❧♦✳ ❆ ✐♥❢❡rê♥❝✐❛ ❡♠ ▼▲●✬s ❜❛s❡✐❛✲s❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ ♥❛ ✈❡r♦s✐♠✐❧❤❛♥ç❛✳

❚❡st❡s ❞❡ ❍✐♣ót❡s❡s

◗✉❛♥❞♦ ❡st❛♠♦s ♣❡r❛♥t❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ s❡❧❡❝çã♦ ❞❡ ❝♦✈❛r✐á✈❡✐s ❡ q✉❡r❡♠♦s t❡s✲

t❛r s❡ ✉♠ s✉❜♠♦❞❡❧♦ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ♦r✐❣✐♥❛❧ é ❝♦♠✉♠ ✉t✐❧✐③❛r ❛ ❊st❛tís✲

t✐❝❛ ❞❡ ❲❛❧❞✱ ❛ ❊st❛tíst✐❝❛ ❞❡ ❲✐❧❦s ♦✉ ❊st❛tíst✐❝❛ ❞❡ ❘❛③ã♦ ❞❡ ❱❡r♦s✐♠✐❧❤❛♥ç❛s✱ ✈❡r

❊st❛s ❡st❛tíst✐❝❛s sã♦ ❞❡❞✉③✐❞❛s ❛ ♣❛rt✐r ❞❛s ❞✐str✐❜✉✐çõ❡s ❛s✲

s✐♠♣tót✐❝❛s ❞♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛ ✈❡r♦s✐♠✐❧❤❛♥ç❛ ❡ ❞❡ ❢✉♥çõ❡s ❛❞❡q✉❛❞❛s ❞❡ss❡s

❡st✐♠❛❞♦r❡s✳

  ❈♦♥s✐❞❡r❛✲s❡ ♦ t❡st❡ ❞❡ ❤✐♣ót❡s❡s ❞❛ ❢♦r♠❛✿ H : 1 :

  ❈β = ξ ✈❡rs✉s H ❈β 6= ξ, ✭✸✳✺✮

♦♥❞❡ ❈ é ✉♠❛ ♠❛tr✐③ q × p✱ ❝♦♠ q ≤ p✱ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❝♦♠♣❧❡t❛ q ❡ ξ é ✉♠ ✈❡❝t♦r

❞❡ ❞✐♠❡♥sã♦ q ♣r❡✈✐❛♠❡♥t❡ ❡s♣❡❝✐✜❝❛❞♦✳ ❙❡❥❛ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r✿

  H : j = 0 1 : j 6= 0, ❈β ✈❡rs✉s H ❈β ✭✸✳✻✮

♣❛r❛ ❛❧❣✉♠ j✱ s❡♥❞♦ q = 1 ❡ ❈ = (0, ..., 0, 1, 0, ..., 0)✱ ✉♠ ✈❡❝t♦r ❝♦♠ t♦❞❛s ❛s ❝♦♠♣♦✲

♥❡♥t❡s ♥✉❧❛s ❡①❝❡♣t♦ ❛ ❥✲és✐♠❛ q✉❡ s❡rá ✐❣✉❛❧ ❛ ✶ ❡ ξ = 0

  ◆♦ ❝❛s♦ ❡♠ q✉❡ ✉♠❛ ✈❛r✐á✈❡❧ é ♣♦❧✐❝♦tó♠✐❝❛ ❡ q✉❡ t♦♠❛ r + 1 ✈❛❧♦r❡s ❞✐st✐♥t♦s✱ é

❛❝♦♥s❡❧❤á✈❡❧ ❝♦♥str✉✐r r ✈❛r✐á✈❡✐s ❞✐❝♦tó♠✐❝❛s ♣❛r❛ ❛s r❡♣r❡s❡♥t❛r ❤❛✈❡♥❞♦✱ ♥❡ss❡ ❝❛s♦✱

′ r s

♣❛râ♠❡tr♦s β q✉❡ ❧❤❡ ❡stã♦ ❛ss♦❝✐❛❞♦s✳ ❊♥tã♦✱ ♣❛r❛ ❛✈❡r✐❣✉❛r s❡ ❡ss❛ ✈❛r✐á✈❡❧ ❞❡✈❡

♦✉ ♥ã♦ s❡r ✐♥❝❧✉í❞❛ ♥♦ ♠♦❞❡❧♦✱ ✐♥t❡r❡ss❛ t❡st❛r s❡ ♦s r ♣❛râ♠❡tr♦s sã♦ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡

❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦✳

  ❚❡st❡ ❞❡ ❲❛❧❞ ➔

❆ ❊st❛tíst✐❝❛ ❞❡ ❲❛❧❞ ✭✈❡r ❜❛s❡✐❛✲s❡ ♥❛ ♥♦r♠❛❧✐❞❛❞❡ ❛ss✐♠♣✲

tót✐❝❛ ❞♦ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡r♦s✐♠✐❧❤❛♥ç❛✱ bβ ✳ ❙✉♣õ❡✲s❡ q✉❡ ❤✐♣ót❡s❡ ♥✉❧❛ ❡st❛❜❡❧❡❝❡ q✉❡ ❈β = ξ✱ ♦♥❞❡ ❈ é ✉♠❛ ♠❛tr✐③ q × p✱

❝♦♠ q ≤ p✱ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❝♦♠♣❧❡t❛ q✳ ❙❡❥❛ bβ ♦ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡r♦s✐♠✐❧❤❛♥ç❛

  −1 (β, ℑ (b β )) p

  

❞❡ β✱ ♦ q✉❛❧ t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❛ss✐♠♣tót✐❝❛ N ✭❛q✉✐ ♦ ✈❡❝t♦r β ❥á ❢♦✐

−1

  (β)

  

✺✻ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

✈❡❝t♦r ❈bβ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❞❡ bβ ❡♥tã♦✱ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞✐str✐❜✉✐çã♦

♥♦r♠❛❧ ♠✉❧t✐✈❛r✐❛❞❛✱ a −1 T

  

∼ N (b β )

q

  ❈bβ ❈β, ❈ℑ ❈ ❡ ♣♦rt❛♥t♦✱ s♦❜ ❛ ❤✐♣ót❡s❡ ♥✉❧❛✱ ❛ ❡st❛tíst✐❝❛ T h i −1 −1 T

  W = (b β ) ❈bβ − ξ ❈ℑ ❈ ❈bβ − ξ ✭✸✳✼✮

  2 t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❛ss✐♠♣tót✐❝❛ ❞❡ ✉♠ χ ❝♦♠ q ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳ ❆ ❡st❛tíst✐❝❛ W ❡♠ ❞❡s✐❣♥❛✲s❡ ♣♦r ❊st❛tíst✐❝❛ ❞❡ ❲❛❧❞✳ ii

  P❛r❛ ♦ ❚❡st❡ ❞❡ ❤✐♣ót❡s❡s r❡❢❡r✐❞♦ ❡♠ ❞❡s✐❣♥❛♥❞♦ ♣♦r σ ♦ ❥✲és✐♠♦ ❡❧❡♠❡♥t♦ −1

  (b β ) ❞❛ ❞✐❛❣♦♥❛❧ ❞❡ ℑ ✱ ❛ ❊st❛tíst✐❝❛ ❞❡ ❲❛❧❞ r❡s✉♠❡✲s❡ ❛✿

T

  

W = ( b β − β ) [σ ]( b β − β )

j j ii j j

  ❧♦❣♦ ✱ s♦❜ H ✱

  

2

b

β

j a

  2 W = ∼ χ .

  1

σ

ii

  ❆ss✐♠✱ r❡❥❡✐t❛✲s❡ ❛ ❤✐♣ót❡s❡ ♥✉❧❛✱ ❛ ✉♠ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛ α✱ s❡ ♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦

  2 ❞❛ ❊st❛tíst✐❝❛ ❞❡ ❲❛❧❞ ❢♦r s✉♣❡r✐♦r ❛♦ q✉❛♥t✐❧ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ 1 − α ❞❡ ✉♠ χ ✳ q

  ❊♠ ❣❡r❛❧✱ ❛ ❊st❛tíst✐❝❛ ❞❡ ❲❛❧❞ é ❛ q✉❡ s❡ ✉t✐❧✐③❛ ❝♦♠ ♠❛✐s ❢r❡q✉ê♥❝✐❛ ♣❛r❛ t❡st❛r

❤✐♣ót❡s❡s s♦❜r❡ ❝♦♠♣♦♥❡♥t❡s ✐♥❞✐✈✐❞✉❛✐s✱ ❡♠❜♦r❛ t❛♠❜é♠ s❡ ✉s❡ ♣❛r❛ t❡st❛r ❤✐♣ót❡s❡s

= 0

♥✉❧❛s ❞♦ t✐♣♦ β r q✉❛♥❞♦ ♦ s✉❜✈❡❝t♦r β r r❡♣r❡s❡♥t❛ ♦ ✈❡❝t♦r ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ✉♠❛

r❡❝♦❞✐✜❝❛çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ♣♦❧✐❝♦tó♠✐❝❛✳ ❊st❛ ❡st❛tíst✐❝❛ é ♠✉✐t♦ út✐❧ ♥❛ ❝♦♠♣❛r❛çã♦

❞❡ ♠♦❞❡❧♦s q✉❛♥❞♦ s❡ ❝♦♠❡ç❛ ♣♦r ❢♦r♠❛r ♦ ♠♦❞❡❧♦ ♠❛①✐♠❛❧ ❡ ❞❡♣♦✐s s❡ ❝♦♥s✐❞❡r❛♠

♠♦❞❡❧♦s ❛❧t❡r♥❛t✐✈♦s ♣❡❧❛ ❡①❝❧✉sã♦ ❞❡ ❝♦✈❛r✐á✈❡✐s ❞❡✈✐❞♦✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ à ✉t✐❧✐③❛çã♦

❞❛ ❡st✐♠❛t✐✈❛ ♥ã♦ r❡str✐t❛ ❞❡ ♠á①✐♠❛ ✈❡r♦s✐♠✐❧❤❛♥ç❛✳

  ➔ ❚❡st❡ ❞❡ ❘❛③ã♦ ❱❡r♦s✐♠✐❧❤❛♥ç❛s

❆ ❊st❛tíst✐❝❛ ❞❡ ❘❛③ã♦ ❞❡ ❱❡r♦s✐♠✐❧❤❛♥ç❛s✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ♣♦r ❡st❛tíst✐❝❛ ❞❡ ❲✐❧❦s✱

✭✈❡❥❛✲s❡ é ❞❡✜♥✐❞❛ ♣♦r✿ max L(β)

  H Λ = −2ln max L(β)

  H ∪H

  1

  n o β β

= −2 ℓ(e ) − ℓ(b ) ,

✭✸✳✽✮

  

♦♥❞❡ eβ✱ ♦ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡r♦s✐♠✐❧❤❛♥ç❛ r❡str✐t♦✱ é ♦ ✈❛❧♦r ❞❡ β q✉❡ ♠❛①✐♠✐③❛

❛ ✈❡r♦s✐♠✐❧❤❛♥ç❛ s✉❥❡✐t♦ às r❡str✐çõ❡s ✐♠♣♦st❛s ♣❡❧❛ ❤✐♣ót❡s❡ ❈β = ξ✳ ❖ ❚❡♦r❡♠❛ ❞❡ ❲✐❧❦s ❡st❛❜❡❧❡❝❡ q✉❡✱ s♦❜ ❝❡rt❛s ❝♦♥❞✐çõ❡s ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ✱ ❛ ❡st❛✲

  2

tíst✐❝❛ Λ t❡♠✱ s♦❜ H ✱ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❛ss✐♠♣tót✐❝❛ ❞❡ ✉♠ χ ♦♥❞❡ ♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s

∪ H

  1 ❞❡ ❧✐❜❡r❞❛❞❡ é ✐❣✉❛❧ à ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ ♣❛râ♠❡tr♦s ❛ ❡st✐♠❛r s♦❜ H

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✺✼

❆ss✐♠✱ s♦❜ H ✱ n o a

  2

Λ = −2 ℓ(e β ) − ℓ(b β ) ∼ χ

q

  :

❈♦♠ ❜❛s❡ ♥♦ ❚❡st❡ ❞❡ ❘❛③ã♦ ❞❡ ❱❡r♦s✐♠✐❧❤❛♥ç❛s r❡❥❡✐t❛✲s❡ ❛ ❤✐♣ót❡s❡ ♥✉❧❛ H

❈β = ξ✱ ❛ ✉♠ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛ α✱ s❡ ♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ❞❛ ❡st❛tíst✐❝❛ Λ ❢♦r

  2 s✉♣❡r✐♦r ❛♦ q✉❛♥t✐❧ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ 1 − α ❞❡ ✉♠ χ ✳ q

  ❆ ❊st❛tíst✐❝❛ ❞❡ ❘❛③ã♦ ❞❡ ❱❡r♦s✐♠✐❧❤❛♥ç❛s✱ ❝♦♠♦ r❡❢❡r❡♠

é ♠❛✐s ✉t✐❧✐③❛❞❛ q✉❛♥❞♦ é ♣r❡❝✐s♦ ❝♦♠♣❛r❛r ♠♦❞❡❧♦s q✉❡ ❡stã♦ ❡♥❝❛✐①❛❞♦s✱ ✐st♦ é✱

♠♦❞❡❧♦s ❡♠ q✉❡ ✉♠ é s✉❜♠♦❞❡❧♦ ❞♦ ♦✉tr♦✳ ◗✉❛❧✐❞❛❞❡ ❞❡ ❆❥✉st❛♠❡♥t♦

  ❋✉♥çã♦ ❉❡s✈✐♦ ➔

❙❡❣✉♥❞♦ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦ ♦✉ s❛t✉r❛❞♦ (S) é út✐❧ ♣❛r❛ ❛✈❛❧✐❛r

❛ q✉❛❧✐❞❛❞❡ ❞❡ ❛❥✉st❛♠❡♥t♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠♦❞❡❧♦ ❡♠ ✐♥✈❡st✐❣❛çã♦ (M)✱ ❛tr❛✈és

❞❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❞✐stâ♥❝✐❛ ❞♦s ✈❛❧♦r❡s ❛❥✉st❛❞♦s bµ ❝♦♠ ❡ss❡ ♠♦❞❡❧♦ ❡

❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛❧♦r❡s ♦❜s❡r✈❛❞♦s y✳ ❊st❛ ♠❡❞✐❞❛ ❞❡ ❞✐s❝r❡♣â♥❝✐❛ ❡♥tr❡ ♦ ♠♦❞❡❧♦

❝♦♠♣❧❡t♦ ❡ ♦ ♠♦❞❡❧♦ ❝♦rr❡♥t❡ ✭♠♦❞❡❧♦ ❝♦♠ ♠❡♥♦s ♣❛râ♠❡tr♦s q✉❡ ❡stá ❛ s❡r s✉❥❡✐t♦ ❛

✐♥✈❡st✐❣❛çã♦✮ ❜❛s❡✐❛✲s❡ ♥❛ ❡st❛tíst✐❝❛ ❞❡ r❛③ã♦ ❞❡ ✈❡r♦s✐♠✐❧❤❛♥ç❛s ❞❡ ❲✐❧❦s✳ ❖ ❧♦❣❛r✐t♠♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦s✐♠✐❧❤❛♥ç❛ ✭❢✉♥çã♦ ❧♦❣✲✈❡r♦s✐♠✐❧❤❛♥ç❛✮ ❞❡ ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦ é ❞❛❞♦ ♣♦r✿ h i n w i y i q(µ i ) − b q(µ i )

  X

lnL(β) = ℓ(β) = + c(y , φ, w )

i i

  ✭✸✳✾✮ φ i=1

  ) = θ i i i

  

♦♥❞❡ q(µ é ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❝❛♥ó♥✐❝❛✱ w é ✉♠ ♣❡s♦ ❝♦♥❤❡❝✐❞♦ ❛ss♦❝✐❛❞♦ à

✐✲és✐♠❛ ♦❜s❡r✈❛çã♦✳ ❈♦♠♣❛r❛♥❞♦ ♦ ♠♦❞❡❧♦ ❡♠ ✐♥✈❡st✐❣❛çã♦ M ❝♦♠ ♦ ♠♦❞❡❧♦ s❛t✉r❛❞♦ ❙ ❛tr❛✈és ❞❛ ❡st❛tíst✐❝❛ ❞❡ r❛③ã♦ ❞❡ ✈❡r♦s✐♠✐❧❤❛♥ç❛s ♦❜té♠✲s❡ ♦ ❞❡s✈✐♦ r❡❞✉③✐❞♦✿ h i

  ∗ µ β β D (y; ) = −2 ℓ M (b ) − ℓ S (b ) b M S h h

  X w i i i = −2 y i µ i µ i ) − y i q(y i ) − b q(y i )

q( b ) − b q( b

  φ i

  µ D(y; ) b =

  ✭✸✳✶✵✮ φ

❡ ❛ D(y; bµ) ❞á✲s❡ ♦ ♥♦♠❡ ❞❡ ❉❡s✈✐♦ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❡♠ ❡st✉❞♦✳ ❖ ❉❡s✈✐♦ é ❛♣❡♥❛s

  ✺✽ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦ ➱ ♣♦ssí✈❡❧ ❞❡✜♥✐r ❛ ❢✉♥çã♦ ❞❡s✈✐♦ ❝♦♠♦ X µ

  D(y; ) = 2w i y i q(y i µ i ) − b q(y i ) + b µ i ) b ) − q( b q( b i

  X = d i

  ✭✸✳✶✶✮ i i

  ❛ s♦♠❛ ❞❡ ♣❛r❝❡❧❛s d q✉❡ ♠❡❞❡♠ ❛ ❞✐❢❡r❡♥ç❛ ❞♦s ❧♦❣❛r✐t♠♦s ❞❛s ✈❡r♦s✐♠✐❧❤❛♥ç❛s ♦❜✲ s❡r✈❛❞❛ ❡ ❛❥✉st❛❞❛ ♣❛r❛ ❝❛❞❛ ♦❜s❡r✈❛çã♦✳ ❆ s♦♠❛ ❞❡st❛s ❝♦♠♣♦♥❡♥t❡s é ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❞✐s❝r❡♣â♥❝✐❛ t♦t❛❧ ❡♥tr❡ ❛s ❞✉❛s ❧♦❣✲✈❡r♦s✐♠✐❧❤❛♥ç❛s✳

❱❡r✐✜❝❛✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ ♦ ❉❡s✈✐♦ é s❡♠♣r❡ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ③❡r♦✱ s❡♥❞♦ q✉❡✱

♣❛r❛ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦✱ é ③❡r♦ ❡ ✈❛✐ ❝r❡s❝❡♥❞♦ à ♠❡❞✐❞❛ q✉❡ ❛s ❝♦✈❛r✐á✈❡✐s ✈ã♦ s❡♥❞♦ r❡t✐r❛❞❛s✳

  

❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ ❞♦ ❉❡s✈✐♦ é ❛ ❛❞✐t✐✈✐❞❛❞❡ ♣❛r❛ ♠♦❞❡❧♦s ❡♥❝❛✐①❛❞♦s✳

  1

  2

  2

  1 ❙❡❥❛♠ ❞♦✐s ♠♦❞❡❧♦s ✐♥t❡r♠é❞✐♦s M ❡ M ✱ ❝♦♠ M ❡stá ❡♥❝❛✐①❛❞♦ ❡♠ M ✱ ♦✉ s❡❥❛✱ sã♦

  2

  1 ♠♦❞❡❧♦s ❞♦ ♠❡s♠♦ t✐♣♦ ♠❛s ♦ ♠♦❞❡❧♦ M ❝♦♥té♠ ♠❡♥♦s ♣❛râ♠❡tr♦s q✉❡ ♦ ♠♦❞❡❧♦ M ✳ ) j , j = 1, 2 ❉❡s✐❣♥❛♥❞♦ D(y; bµ j ♦ ❞❡s✈✐♦ ❞♦ ♠♦❞❡❧♦ M ✱ ❡♥tã♦ ❛ ❡st❛tíst✐❝❛ ❞❛ r❛③ã♦ ❞❡ ✈❡r♦s✐♠✐❧❤❛♥ç❛s ♣❛r❛ ❝♦♠♣❛r❛r ❡st❡s ❞♦✐s ♠♦❞❡❧♦s r❡s✉♠❡✲s❡ ❛✿

  D(y; µ ) − D(y; µ ) b 2 b

  1 − 2 ℓ (β ) − ℓ (β ) = . M

  2 M

  1

  2

  1 φ

  1 ❙♦❜ ❛ ❤✐♣ót❡s❡ ❞♦ ♠♦❞❡❧♦ M s❡r ✈❡r❞❛❞❡✐r♦✱ t❡♠✲s❡ D(y; µ ) − D(y; µ ) a b 2 b

  1

  2 ∼ χ p

  1 −p

  2

  φ j j

  ♦♥❞❡ p r❡♣r❡s❡♥t❛ ❛ ❞✐♠❡♥sã♦ ❞♦ ✈❡❝t♦r β ♣❛r❛ ♠♦❞❡❧♦ M ✳ ❆ ❝♦♠♣❛r❛çã♦ ❞❡ ♠♦❞❡✲ ❧♦s ❡♥❝❛✐①❛❞♦s✱ ♣♦❞❡✱ ❛ss✐♠✱ s❡r ❢❡✐t❛ ❝♦♠ ❜❛s❡ ♥❛ ❞✐❢❡r❡♥ç❛ ❞♦s ❉❡s✈✐♦s ❞❡ ❝❛❞❛ ♠♦❞❡❧♦✳ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦

  ➔ ❙✉♣♦♥❤❛✲s❡ q✉❡ s❡ t❡♠ ✉♠ ♠♦❞❡❧♦ ♣❛r❛ ♦ q✉❛❧ s❡ ♣r❡t❡♥❞❡ ❡①♣❧✐❝❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡

✉♠❛ ✈❛r✐á✈❡❧ Y ❡♠ ❢✉♥çã♦ ❞❡ ❝❡rt❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✳ ❙❡❣✉♥❞♦

❡st✐♠❛❞♦s ♦s ♣❛râ♠❡tr♦s ❝♦♠ ❜❛s❡ ♥✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠ét♦❞♦ ♦❜tê♠✲s❡ ♦s ✈❛❧♦r❡s ❛❥✉s✲ i i t❛❞♦s✱ ˆy ✱ ❞❛s ♦❜s❡r✈❛çõ❡s ❞❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✱ y ✳

❯♠❛ ❢♦r♠❛ ❞❡ ❛✈❛❧✐❛r ❛ ❛❞❡q✉❛❜✐❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r ♠✉❧t✐♣❧❛

❡st✐♠❛❞♦ ❛♦s ❞❛❞♦s ❝♦♥s✐st❡ ❡♠ ❞✐s♣♦r ❞❡ ✉♠ ✐♥❞✐❝❛❞♦r q✉❡ ♣❡r♠✐t❛ ♠❡❞✐r ♦ ✏❣r❛✉ ❞❡ (i = 1, 2, . . . , n) ❛❥✉st❛♠❡♥t♦✑ ❡♥tr❡ ♦s y i ❡ ♦s ˆy i ✳ ❖ ✐♥❞✐❝❛❞♦r ❤❛❜✐t✉❛❧♠❡♥t❡ ♣r♦♣♦st♦ é i

  ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ♦❜s❡r✈❛çõ❡s ❞❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✱ y ✱ ❡ ♦s r❡s♣❡❝t✐✈♦s i

  ✈❛❧♦r❡s ❛❥✉st❛❞♦s✱ ˆy ✳ i

  

❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞❡t❡r♠✐♥❛çã♦ é ♦ q✉❛❞r❛❞♦ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ♦s y

i (i = 1, 2, . . . , n)

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✺✾

P

  2 (y − ¯ y)(ˆ y − ¯ˆ y) i i

  2

r = ,

y ˆ y P P ✭✸✳✶✷✮

  2

  2 (y − ¯ y) (ˆ y − ¯ˆ y) i i

  ♦♥❞❡ ¯y ❡ ¯ˆy sã♦ ❛s ♠é❞✐❛s ❞♦s y i ❡ ❞♦s ˆy i r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  2 ≤ 1 ❈♦♠♦ 0 ≤ r y ˆ y ✱ ♣♦❞❡ ❝♦♥❝❧✉✐r✲s❡ q✉❡ q✉❛♥t♦ ♠❛✐s ♣ró①✐♠♦ ❞❡ 1 ❡st✐✈❡r ♦

  

❝♦❡✜❝✐❡♥t❡ ❞❡ ❞❡t❡r♠✐♥❛çã♦ ♠❡❧❤♦r é ♦ ✏❣r❛✉ ❞❡ ❛❥✉st❛♠❡♥t♦✑✱ ♦✉ s❡❥❛✱ ♠❛✐♦r é ❛

i i

  ✏♣r♦①✐♠✐❞❛❞❡✑ ❡♥tr❡ ♦s y ❡ ♦s ˆy ✳

❆ ❡q✉❛çã♦ ♦❜t❡♥❞♦✲s❡✿

  ✭✸✳✶✸✮ SQT SQT ♦♥❞❡✱ ❼ ❙◗❘ é ❛ ❙♦♠❛ ❞♦s ◗✉❛❞r❛❞♦s ❞♦s ❘❡sí❞✉♦s✱

  ❼ ❙◗❚ é ❛ ❙♦♠❛ ❞♦s ◗✉❛❞r❛❞♦s ❚♦t❛❧✱ ❼ ❙◗❊ é ❛ ❙♦♠❛ ❞♦s ◗✉❛❞r❛❞♦s ❊①♣❧✐❝❛❞❛✳ ❆♦ ❛❞✐❝✐♦♥❛r ❛♦ ♠♦❞❡❧♦ ✉♠❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✱ q✉❛❧q✉❡r q✉❡ ❡❧❛ s❡❥❛✱ ♦ ❝♦❡✜❝✐✲

  2

❡♥t❡ ❞❡ ❞❡t❡r♠✐♥❛çã♦ R ♥✉♥❝❛ ❞❡❝r❡s❝❡✳ ❉❡st❡ ♠♦❞♦✱ q✉❛♥❞♦ s❡ ♣❛ss❛ ❞❡ k r❡❣r❡ss♦r❡s

♣❛r❛ k + 1 ✭❛❝r❡s❝❡♥t❛♥❞♦ ✉♠ r❡❣r❡ss♦r ❛♦ ♠♦❞❡❧♦✮✱ ❛ ✐♥tr♦❞✉çã♦ ❞❡ss❛ ✈❛r✐á✈❡❧ ❛❞✐✲

  2

❝✐♦♥❛❧ ✐♠♣❧✐❝❛ ♥❡❝❡ss❛r✐❛♠❡♥t❡ q✉❡ R ♥✉♥❝❛ ❞✐♠✐♥✉❛✱ ♣♦✐s ❙◗❘ ♥✉♥❝❛ ♣♦❞❡ ❝r❡s❝❡r✳

❖✉ s❡❥❛✱

  2 SQE ↑⇒ R ↑ P♦r ✐ss♦✱ ❝♦♥s✐❞❡r❛✲s❡ ✉♠ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞❡t❡r♠✐♥❛çã♦ ❛❧t❡r♥❛t✐✈♦ q✉❡ ❞❡s❝♦♥t❛ ♦

  2 R

❡❢❡✐t♦ ❞❡ ✉♠ ❡❧❡✈❛❞♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✳ ❊st❡ ❝♦❡✜❝✐❡♥t❡✱ ¯ ✱ ❞❡s✐❣♥❛✲s❡

♣♦r ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦ ♦✉ ❝♦rr✐❣✐❞♦ ♣❡❧♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳

  SQR/(T − n − 1)

  2 ¯

R = 1 − ,

  ✭✸✳✶✹✮ SQT /(T − 1) ♦♥❞❡ T é ♦ t❛♠❛♥❤♦ ❞❛ ❛♠♦str❛ ❡ n é ♦ ♥➸ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳ ✸✳✷✳✸ ❙❡❧❡❝çã♦ ❞❡ ▼♦❞❡❧♦s

❙❡❣✉♥❞♦ ❛ ♠❡t♦❞♦❧♦❣✐❛ ✉t✐❧✐③❛❞❛ ✐♥✐❝✐❛❧♠❡♥t❡ ♥❛ ❝♦♥str✉çã♦ ❞❡ ♠♦❞❡❧♦s

❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❡r❛ ❜❛s❡❛❞❛ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❡♠ ❥✉❧❣❛♠❡♥t♦s ❢❡✐t♦s ♣❡❧♦s ❛✈❛❧✐st❛s ❞❡

❝ré❞✐t♦✳ ◆❡st❡s ♠♦❞❡❧♦s✱ ❛s ✈❛r✐á✈❡✐s q✉❡ ❝♦♠♣õ❡♠ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ❝❧✐❡♥t❡s ❡ ♦s ♣❡s♦s

❞❛s ♠❡s♠❛s sã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧♦s ❣❡st♦r❡s ❞❡ ❝ré❞✐t♦ ❞❛ ✐♥st✐t✉✐çã♦ ❜❛♥❝ár✐❛✳ ❊♠❜♦r❛

❛✐♥❞❛ s❡ ✉t✐❧✐③❡♠ ❡st❡s ♠♦❞❡❧♦s ❡♠ ❛❧❣✉♠❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s✱ ❛❝t✉❛❧♠❡♥t❡ ❛

♠❛✐♦r✐❛ ❞♦s ♠♦❞❡❧♦s ✉t✐❧✐③❛❞♦s sã♦ ❝♦♥str✉í❞♦s ❝♦♠ ❜❛s❡ ❡♠ té❝♥✐❝❛s ❡st❛tíst✐❝❛s ❝♦♠♦

  

✻✵ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

  ♣❛rt✐r ❞❡ ♠♦❞❡❧♦s ❞❡ ✐♥t❡❧✐❣ê♥❝✐❛ ❛rt✐✜❝✐❛❧✱ ❝♦♠♦ ❛s r❡❞❡s ♥❡✉r♦♥❛✐s ✳ ❆ té❝♥✐❝❛ ❞❡ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐♥❛♥t❡ ❢♦✐ ❛ ♣r✐♠❡✐r❛ té❝♥✐❝❛ ❡st❛tíst✐❝❛ ✉t✐❧✐③❛❞❛ ♥❛

❝r✐❛çã♦ ❞❡ ♠♦❞❡❧♦s ❞❡ ❇❡❤❛✈✐♦✉r❛❧ ❙❝♦r✐♥❣✱ ♥♦ ❡♥t❛♥t♦✱ ❛ ✐♥❞ústr✐❛ ❞❡ ❝ré❞✐t♦ ♥ã♦

❝♦♥s✐❞❡r♦✉ ❞❡ ❢♦r♠❛ sér✐❛ ♦s ♠♦❞❡❧♦s ❞❡ ♣♦♥t✉❛çã♦ ❞❡ ❝ré❞✐t♦ ❛té ♠❡❛❞♦s ❞❛ ❞é❝❛❞❛

❞❡ s❡ss❡♥t❛✳ ❙ó ❛ ♣❛rt✐r ❞❛ ❞é❝❛❞❛ ❞❡ ♦✐t❡♥t❛ é q✉❡ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ❛ té❝♥✐❝❛ ❞❡ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛✱

❡ ♠❛✐s r❡❝❡♥t❡♠❡♥t❡ ❢♦r❛♠ ✐♠♣❧❛♥t❛❞♦s ♠ét♦❞♦s ❞❡ ❛♥á❧✐s❡ ❞❡ ❝ré❞✐t♦ ❜❛s❡❛❞♦s ❡♠

r❡❞❡s ♥❡✉r♦♥❛✐s✳ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛

❆ r❡❣r❡ssã♦ ▲♦❣íst✐❝❛ ♦✉ ❛♥á❧✐s❡ ▲♦❣✐t é ✉♠❛ té❝♥✐❝❛ ❡st❛tíst✐❝❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ♣r♦❞✉③✐r✱

❛ ♣❛rt✐r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦❜s❡r✈❛çõ❡s✱ ✉♠ ♠♦❞❡❧♦ q✉❡ ♣❡r♠✐t❛ ❛ ♣r❡❞✐çã♦ ❞♦s ✈❛❧♦r❡s

t♦♠❛❞♦s ♣♦r ✉♠❛ ✈❛r✐á✈❡❧ ❝❛t❡❣ór✐❝❛✱ ❢r❡q✉❡♥t❡♠❡♥t❡ ❜✐♥ár✐❛✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ sér✐❡ ❞❡

✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❝♦♥tí♥✉❛s ❡✴♦✉ ❜✐♥ár✐❛s✳ ❆ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ é ❢r❡q✉❡♥t❡♠❡♥t❡ ✉t✐❧✐③❛❞❛ ❡♠ ❝✐ê♥❝✐❛s ♠é❞✐❝❛s ❡ s♦❝✐❛✐s✱ ❡

t❡♠ ♦✉tr❛s ❞❡s✐❣♥❛çõ❡s✱ ❝♦♠♦ ♠♦❞❡❧♦ ❧♦❣íst✐❝♦✱ ♠♦❞❡❧♦ ❧♦❣✐t✱ ❡ ❝❧❛ss✐✜❝❛❞♦r ❞❡ ♠á①✐♠❛

❡♥tr♦♣✐❛✳

  ❖ ê①✐t♦ ❞❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ ❛ss❡♥t❛ s♦❜r❡t✉❞♦ ♥❛s ♥✉♠❡r♦s❛s ❢❡rr❛♠❡♥t❛s q✉❡

♣❡r♠✐t❡♠ ✐♥t❡r♣r❡t❛r✱ ❞❡ ♠♦❞♦ ❛♣r♦❢✉♥❞❛❞♦✱ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳ ❈♦♠♣❛r❛♥❞♦ ❝♦♠

❛s té❝♥✐❝❛s ❝♦♥❤❡❝✐❞❛s ❡♠ r❡❣r❡ssã♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ r❡❣r❡ssã♦ ❧✐♥❡❛r✱ ❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛

❞✐st✐♥❣✉❡✲s❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ ♣❡❧♦ ❢❛❝t♦ ❞❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ s❡r ❝❛t❡❣ór✐❝❛✳

  ❚r❛t❛✲s❡ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ♣❛r❛ ✈❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡s ✭♦✉ ❞❡ r❡s♣♦st❛✮ ∼ B(1, π ) i i

  

❜✐♥♦♠✐❛❧♠❡♥t❡ ❞✐str✐❜✉í❞❛s✱ Y ✳ ➱ út✐❧ ♣❛r❛ ♠♦❞❡❧❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠

❡✈❡♥t♦ ♦❝♦rr❡r✱ ❝♦♠♦ ❢✉♥çã♦ ❞❡ ♦✉tr♦s ❢❛❝t♦r❡s✳ ➱ ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦ q✉❡

✉s❛ ❝♦♠♦ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❛ ❢✉♥çã♦ ❧♦❣✐t✿ π

  

θ = ln .

  1 − π ❈♦♥❢♦r♠❡ r❡ss❛❧t❛ ❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ é ✉♠❛ té❝♥✐❝❛ ❡st❛tíst✐❝❛

✉t✐❧✐③❛❞❛ ❛q✉❛♥❞♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ❞♦✐s ❣r✉♣♦s✱ ❡ ✈✐s❛ ♦❜t❡r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛

♦❜s❡r✈❛çã♦ ♣❡rt❡♥❝❡r ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝♦♥❥✉♥t♦✱ ❡♠ ❢✉♥çã♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s

✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s✳ ➱ ❢r❡q✉❡♥t❡♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❝♦♠ r❡s✲

♣♦st❛ ❜✐♥ár✐❛ ♦✉ ❞✐❝♦tó♠✐❝❛ ❡ ❝♦♥s✐st❡ ❡♠ r❡❧❛❝✐♦♥❛r✱ ❛tr❛✈és ❞❡ ✉♠ ♠♦❞❡❧♦✱ ❛ ✈❛r✐á✈❡❧

  ✶ ❆s r❡❞❡s ♥❡✉r♦♥❛✐s t❡♥t❛♠ ❝♦♥str✉✐r r❡♣r❡s❡♥t❛çõ❡s ✐♥t❡r♥❛s ❞❡ ♠♦❞❡❧♦s ♦✉ ♣❛❞rõ❡s ❞❡t❡❝t❛❞♦s

♥♦s ❞❛❞♦s✱ ♦s q✉❛✐s ❣❡r❛❧♠❡♥t❡ ♥ã♦ sã♦ ✈✐sí✈❡✐s s❡♠ ✉♠❛ ❛♥á❧✐s❡ ❝✉✐❞❛❞❛✳ ❆s r❡❞❡s ♥❡✉r♦♥❛✐s ✉t✐❧✐③❛♠

  

✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ✭♥ós✮ ❛♥á❧♦❣♦s ❛♦s ♥❡✉ró♥✐♦s✳ ❊ss❡s ❡❧❡♠❡♥t♦s ❞❡ ♣r♦❝❡s✲

s❛♠❡♥t♦ sã♦ ✐♥t❡r❧✐❣❛❞♦s ♥✉♠❛ r❡❞❡ q✉❡ ♣♦❞❡ ✐❞❡♥t✐✜❝❛r ♣❛❞rõ❡s ♥♦s ❞❛❞♦s✱ ♦✉ s❡❥❛✱ ❛ r❡❞❡ ❛♣r❡♥❞❡

❛tr❛✈és ❞❛ ❡①♣❡r✐ê♥❝✐❛✳ ❊♠ ❣❡r❛❧✱ ❡♠ ❡st✉❞♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ✉t✐❧✐③❛♠✲s❡ ❛s r❡❞❡s ♥❡✉r♦♥❛✐s ▼✉❧t✐

♣❧❛②❡r P❡r❝❡♣❝tr♦♥ ❛tr❛✈és ❞♦ ❛❧❣♦r✐t♠♦ ❇❛❝❦♣r♦♣❛❣❛t✐♦♥✳

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✻✶

r❡s♣♦st❛ ✭✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❜✐♥ár✐❛✮ ❝♦♠ ❢❛❝t♦r❡s q✉❡ ✐♥✢✉❡♥❝✐❛♠ ♦✉ ♥ã♦ ❛ ♣r♦❜❛❜✐✲

❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡t❡r♠✐♥❛❞♦ ❡✈❡♥t♦ ✭✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s✮✳ ❆ss✐♠✱ ♥❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛✱ ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✱ ✉♠❛ ✈❡③ q✉❡ ♣♦ss✉✐ ❝❛rá❝t❡r

♥ã♦✲♠étr✐❝♦✱ é ✐♥s❡r✐❞❛ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ✈❛r✐á✈❡✐s ❞✉♠♠② ✭❞✐❝♦tó♠✐❝❛ ♦✉ ❜✐♥ár✐❛✮✱

q✉❡ ❛ss✉♠❡♠ ✈❛❧♦r ✵ ♣❛r❛ ✐♥❞✐❝❛r ❛ ❛✉sê♥❝✐❛ ❞❡ ✉♠ ❛tr✐❜✉t♦ ❡ ✶ ♣❛r❛ ✐♥❞✐❝❛r ❛ ♣r❡✲

s❡♥ç❛ ❞❡ ✉♠ ❛tr✐❜✉t♦✳ ❆ ♣❛rt✐r ❞❡ss❡ ✈❛❧♦r ❞✐❝♦tó♠✐❝♦✱ ❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ ❡st✐♠❛ ❛

♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ss❡ ❡✈❡♥t♦ ♦❝♦rr❡r ♦✉ ♥ã♦✳

  ◆♦ â♠❜✐t♦ ❞❛ ❛♣❧✐❝❛çã♦ ❡♠ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ❛ té❝♥✐❝❛ ❞❡ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ é ✉t✐✲

❧✐③❛❞❛ ♣❛r❛ ❛ ❛✈❛❧✐❛çã♦ ❞❡ ❞❡t❡r♠✐♥❛❞♦ ❣r✉♣♦ ❞❡ ❝❧✐❡♥t❡s ❡♠ r❡❧❛çã♦ ❛♦ ❝✉♠♣r✐♠❡♥t♦

❞❛s s✉❛s ♦❜r✐❣❛çõ❡s ❡♥q✉❛♥t♦ ❞❡✈❡❞♦r ❞❡ ❝ré❞✐t♦✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❛ té❝♥✐❝❛ ❛ss✉♠❡ q✉❡

❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t é ❧♦❣✐st✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛✱ ❝♦♠ ✉♠ r❡s✉❧t❛❞♦ ❜✐♥♦♠✐❛❧ 0

♦✉ 1✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❛♣❧✐❝❛çã♦ ❞❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ ✐♠♣❧✐❝❛ ❝♦✲

♥❤❡❝❡r ❛ ♦❝♦rrê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❞❡t❡r♠✐♥❛❞♦ ❡✈❡♥t♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ s✐t✉❛çã♦ ❞❡

✐♥❝✉♠♣r✐♠❡♥t♦ ♦✉ ♥ã♦ ❞❡ ✉♠ ❝❧✐❡♥t❡ ♦✉ ❛ s✐t✉❛çã♦ ❞❡ ✐♥s♦❧✈ê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ✉♠❛

❡♠♣r❡s❛✳ P♦❞❡♠♦s s✉♣♦r✱ ✈❡r q✉❡ t❡♠♦s n ✈❛r✐á✈❡✐s r❡s♣♦st❛ ✐♥❞❡✲

  ∼ B(1, π ) ∼ Ber(π ) i i i i

  ♣❡♥❞❡♥t❡s Y ♦✉ Y ♦✉ s❡❥❛✱ y i 1−y i

f (y |π ) = π (1 − π ) , y = 0, 1

i i i i i i

  

❡ q✉❡ ❛ ❝❛❞❛ ✐♥❞✐✈í❞✉♦ i ❡stá ❛ss♦❝✐❛❞♦ ✉♠ ✈❡❝t♦r ❞❡ ❡s♣❡❝✐✜❝❛çã♦ z ✱ q✉❡ r❡s✉❧t❛ ❞♦

i , i = 1, . . . , n

  

✈❡❝t♦r ❞❡ ❝♦✈❛r✐á✈❡✐s x ✳ ❆ ❞✐str✐❜✉✐çã♦ ❇✐♥♦♠✐❛❧ ♣❡rt❡♥❝❡ à ❋❛♠í❧✐❛

  θ

  e

  i

  ] = θ i

  ❊①♣♦♥❡♥❝✐❛❧ ❞♦♥❞❡ t❡♠♦s q✉❡ E[Y ✳ 1+e

  i

  i ] = π i ❈♦♠♦ E[Y ♣❛r❛ ♦ ♠♦❞❡❧♦ ❜✐♥♦♠✐❛❧✱ ❡♥tã♦ t❡♠♦s π i

θ = ln .

i

  1 − π i T

  = η = z β i i

  ❆♦ ❢❛③❡r θ i ❝♦♥❝❧✉✐✲s❡ q✉❡ ❛ ❛ss♦❝✐❛çã♦ ❡♥tr❡ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❛

✈❛r✐á✈❡❧ r❡s♣♦st❛ ❡ ❛s ❝♦✈❛r✐á✈❡✐s é ❢❡✐t❛ ❛tr❛✈és ❞❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❝❛♥ó♥✐❝❛✱ ❢✉♥çã♦

= P [Y = 1|X = x ] i i i

  

❧♦❣✐t✳ ❆ss✐♠ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s✉❝❡ss♦✱ π ✱ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠

i

  ♦ ✈❡❝t♦r z ❛tr❛✈és ❞❡ T exp(z β ) i

  π = i

  ✭✸✳✶✺✮ T

  1 + exp(z β ) i π i

  θ ) = ln = ln(e ) = θ i i

  ❆ss✐♠ s❡♥❞♦✱ Logit(π i ❡ 1−π i

  Logit(π i ) = β + β

  ✭✸✳✶✻✮ T

  

1

X 1 + . . . + β i X i .

  β i

  ❈♦♠♦ ♦s ✈❛❧♦r❡s ♣♦ssí✈❡✐s ❞❡ π s❡ s✐t✉❛♠ ♥♦ ✐♥t❡r✈❛❧♦ [0, 1]✱ ♦ ✈❛❧♦r ❞❡ z i é ❢r❡✲

q✉❡♥t❡♠❡♥t❡ ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✳ ❆ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❡st✐♠❛r

❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ✐♥❞✐✈✐❞✉❛✐s ❝♦♥st✐t✉✐ ❛ ♣r✐♥❝✐♣❛❧ ✈❛♥t❛❣❡♠ ❞❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛✳ ❆

  

✻✷ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

♠♦❞❡❧♦s ❞❡ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐♥❛♥t❡ é s✐♠✐❧❛r✱ ❧♦❣♦ ♥❡♥❤✉♠❛ ❞❛s té❝♥✐❝❛s ❡st❛tíst✐❝❛s é ♣r❡✲

❞♦♠✐♥❛♥t❡ s♦❜r❡ ❛ ♦✉tr❛✳ ◆❛ r❡❣r❡ssã♦ ▲♦❣íst✐❝❛ ♥ã♦ é ♥❡❝❡ssár✐♦ ✈❡r✐✜❝❛r ❛s s✉♣♦s✐çõ❡s

❜ás✐❝❛s ❞❛ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐♥❛♥t❡✱ t❛✐s ❝♦♠♦ ❛ ♥♦r♠❛❧✐❞❛❞❡ ❞❛s ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥✲

t❡s ❡ ❛ ✐❣✉❛❧❞❛❞❡ ❞❛s ♠❛tr✐③❡s ❞❡ ✈❛r✐â♥❝✐❛✴❝♦✈❛r✐â♥❝✐❛ ♣❛r❛ ♦s ❣r✉♣♦s ✭❛✉sê♥❝✐❛ ❞❡

❤❡t❡r♦❝❡❞❛st✐❝✐❞❛❞❡✮✳

  ❙❡❣✉♥❞♦ ❞❡✈❡r✲s❡✲✐❛ t❛♠❜é♠ ❛♥❛❧✐s❛r ❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❛s r❡❧❛çõ❡s

❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❡ ❛ ❛✉sê♥❝✐❛ ❞❡ ♠✉❧t✐❝♦❧✐♥❡❛r✐❞❛❞❡ ❡♥tr❡ ❛s ♠❡s♠❛s✱ ♥♦ ❡♥t❛♥t♦✱ ♥♦

♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ♥❡♥❤✉♠❛ ❞❡st❛s s✉♣♦s✐çõ❡s s❡rã♦ ✈❡r✐✜❝❛❞❛s ❛♦ ♣♦r♠❡♥♦r✱ ♣r✐♥❝✐♣❛❧✲

♠❡♥t❡ ❞❡✈✐❞♦ ❛♦ ❢❛❝t♦ ❞❡ ❛ ❝❛rt❡✐r❛ t❡r s✐❞♦ ♦❜t✐❞❛ ❛tr❛✈és ❞❛ s✐♠✉❧❛çã♦✱ ♦ q✉❡ ♣♦❞❡rá

✐♥tr♦❞✉③✐r r❡❧❛çõ❡s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ♣♦✉❝♦ r❡❛❧✐st❛s ♦✉ ♦♠✐t✐r ♣♦ssí✈❡✐s r❡❧❛çõ❡s q✉❡

♣♦ss❛♠ ❡①✐st✐r ♥✉♠❛ ❝❛rt❡✐r❛ r❡❛❧✳

  ❖ ♠ét♦❞♦ ❡st❛tíst✐❝♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ s❡❧❡❝çã♦ ❞❛s ✈❛r✐á✈❡✐s ♠❛✐s r❡❧❡✈❛♥t❡s é ♦

st❡♣✇✐s❡ q✉❡ ✈❡r✐✜❝❛ ❛ ❛✉sê♥❝✐❛ ❞❡ ♠✉❧t✐❝♦❧✐♥❡❛r✐❞❛❞❡ ❡ ❝♦rr✐❣❡✱ ❛✉t♦♠❛t✐❝❛♠❡♥t❡✱ ❛s

♣♦ssí✈❡✐s ❞✐st♦rçõ❡s ❝❛✉s❛❞❛s ♣❡❧❛ ♠✉❧t✐❝♦❧✐♥❡❛r✐❞❛❞❡✳ ❆♦ ✐♥❝❧✉✐r ♦✉ ❡①❝❧✉✐r ❛s ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s ♥♦ ♠♦❞❡❧♦✱ ✉♠❛ ❞❡ ❝❛❞❛ ✈❡③✱ ❝♦♠

❜❛s❡ ♥♦ s❡✉ ♣♦❞❡r ❞✐s❝r✐♠✐♥❛tór✐♦✱ ♦ ♠ét♦❞♦ st❡♣✇✐s❡ ❡①❝❧✉✐ ❛s ✈❛r✐á✈❡✐s q✉❡ ❡st❡❥❛♠

❛❧t❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛ ✈❛r✐á✈❡❧ q✉❡ ❡stá ❛ s❡r ✐♥s❡r✐❞❛ ♥❛ ❢✉♥çã♦✳

  ❋✐♥❛❧♠❡♥t❡✱ é ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ t♦❞♦s ❡ss❛s s✉♣♦s✐çõ❡s sã♦ ♥❡❝❡ssár✐❛s ❛♣❡♥❛s

♣❛r❛ ❛ té❝♥✐❝❛ ❞❡ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐♥❛♥t❡✳ ❆ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛✱ ♣♦r s❡r ✉♠❛ té❝♥✐❝❛ ♠❛✐s

r♦❜✉st❛✱ ♥ã♦ ❡①✐❣❡ ❛ ✈❡r✐✜❝❛çã♦ ❞❡st❛s s✉♣♦s✐çõ❡s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ♥♦r♠❛❧✐❞❛❞❡✳ ❆ss✐♠✱ ♥❛ ❛♥á❧✐s❡ ▲♦❣✐t✱ q✉❛♥❞♦ ♥ã♦ sã♦ t✐❞♦s ❡♠ ❝♦♥t❛ ♦s ♣r❡ss✉♣♦st♦s ♥ã♦ ✐♠♣❧✐❝❛ q✉❡ ❤❛❥❛ ❞✐st♦rçõ❡s ❞♦s ♣❛râ♠❡tr♦s✳ ◆❡st❡ tr❛❜❛❧❤♦ ❛♣❡♥❛s s❡ ✉t✐❧✐③❛ ❛ r❡❣r❡ssã♦ ▲♦❣íst✐❝❛ ♦✉ ▲♦❣✐t ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r

♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ❈r❡❞✐t ❙❝♦r✐♥❣✱ ❡ ♣♦r ❡st❛ r❛③ã♦✱ ♦♣t♦✉✲s❡ ♣♦r ♦♠✐t✐r ❛

❞❡s❝r✐çã♦ ♣♦r♠❡♥♦r✐③❛❞❛ ❞❛ ❛♥á❧✐s❡ ❞✐s❝r✐♠✐♥❛♥t❡ ❡ ❞❛s r❡❞❡s ♥❡✉r♦♥❛✐s✳ ❱❛❧❡ ❛ ♣❡♥❛

r❡ss❛❧t❛r q✉❡ ❡st❛ té❝♥✐❝❛ é ✉t✐❧✐③❛❞❛ ❡♠ ❡st✉❞♦s q✉❡ t❡♥❤❛♠ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ s❡♣❛r❛r

❞♦✐s ❣r✉♣♦s ❞✐st✐♥t♦s✱ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s ♣ré✲

❞❡t❡r♠✐♥❛❞❛s✳ ❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ✐❞❡♥t✐✜❝❛r ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❢♦r❛♠ ✉t✐❧✐✲

③❛❞♦s ♦s ♠ét♦❞♦s ❡st❛tíst✐❝♦s ❙t❡♣✇✐s❡ ❡ ❆■❈✱ ♦s q✉❛✐s s❡rã♦ ❞❡t❛❧❤❛❞♦s ❡♠ s❡❣✉✐❞❛✳

▼ét♦❞♦ ❞❡ ❡s❝♦❧❤❛ ❞❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ✲ ❙t❡♣✇✐s❡

❆s ✈❛r✐á✈❡✐s ❡s❝♦❧❤✐❞❛s ✐♥✐❝✐❛❧♠❡♥t❡ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ sã♦ ❛♣❡♥❛s

♣♦t❡♥❝✐❛✐s ✈❛r✐á✈❡✐s ❞♦ ♠♦❞❡❧♦ ✜♥❛❧✱ ♣♦✐s ♥❛ ♣rát✐❝❛✱ ♥❡♠ t♦❞❛s ❢♦r♥❡❝❡♠ ✐♥❢♦r♠❛çã♦

r❡❧❡✈❛♥t❡ ♣❛r❛ ❡①♣❧✐❝❛r ❛ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦✳ ❚♦r♥♦✉✲s❡ ❛ss✐♠✱ ♥❡❝❡ssár✐♦ ❡s❝♦❧❤❡r

❞❡♥tr♦ ❞❡ss❡ ❣r✉♣♦ ❞❡ ✈❛r✐á✈❡✐s✱ ❛s ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛s ♣❛r❛ ❡①♣❧✐❝❛r ❛ ◗✉❛❧✐❞❛❞❡ ❞❡

❈ré❞✐t♦✱ s❡♥❞♦ q✉❡ ❡ss❛s ❝♦♠♣õ❡♠ ♦ ♠♦❞❡❧♦ ✜♥❛❧✳ ❯♠ ❞♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛

r❡❛❧✐③❛r ❡st❛ t❛r❡❢❛ ❢♦✐ ♦ ♠ét♦❞♦ st❡♣✇✐s❡✳

  

✸✳✷ ▼♦❞❡❧♦ ❡ ❚é❝♥✐❝❛s ❊st❛tíst✐❝❛s ❯t✐❧✐③❛❞❛s ✻✸

q✉❡r ❝♦♥s✐❞❡r❛r✱ ❛♦ ✐♥í❝✐♦✱ ✉♠ ♥ú♠❡r♦ r❡❧❛t✐✈❛♠❡♥t❡ ❡❧❡✈❛❞♦ ❞❡ ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥✲

t❡s ♣❛r❛ ✐♥❝❧✉✐r ♥❛ ❢✉♥çã♦✱ ❡♠❜♦r❛ s❡❥❛ ✉t✐❧✐③❛❞♦ ❢r❡q✉❡♥t❡♠❡♥t❡ ❡♠ ♠♦❞❡❧♦s ✐♥✐❝✐❛✐s

♥✉❧♦s✳ ❆ s❡❧❡❝çã♦ ❞❛ ✈❛r✐á✈❡❧ ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛ ♥❛ ❡①♣❧✐❝❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ é

s❡q✉❡♥❝✐❛❧✳ ❊♠ ❝❛❞❛ ♣❛ss♦ ❛s ✈❛r✐á✈❡✐s ♠❡♥♦s út❡✐s✱ ♥❛ ❞❡s❝r✐♠✐♥❛çã♦ ❡♥tr❡ ♦s ❣r✉♣♦s✱

sã♦ ❡❧✐♠✐♥❛❞❛s✱ ♦✉ ♥♦ ❝❛s♦ ❞❡ ✐♥✐❝✐❛r ❝♦♠ ♦ ♠♦❞❡❧♦ ♥✉❧♦✱ ❛s ✈❛r✐á✈❡✐s ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛s

sã♦ ❛❞✐❝✐♦♥❛❞❛s✱ ❡ ❛♣❡♥❛s sã♦ r❡t✐❞❛s ✉♠ ♥ú♠❡r♦ r❡❞✉③✐❞♦ ❞❡ ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s✳

❊ss❡ ❝♦♥❥✉♥t♦ r❡❞✉③✐❞♦✱ ❣❡r❛❧♠❡♥t❡✱ é tã♦ ❜♦♠ ✭♦✉ ♠❡❧❤♦r✮ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡

✈❛r✐á✈❡✐s✱ ❝♦♥❢♦r♠❡ ❛❝r❡s❝❡♥t❛ q✉❡ ❛ ❡st✐♠❛çã♦

st❡♣✇✐s❡ é ❢r❡q✉❡♥t❡♠❡♥t❡ ✉t✐❧✐③❛❞❛ ❡♠ s✐t✉❛çõ❡s ♥❛s q✉❛✐s ❛s ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥✲

t❡s ✐♠♣♦rt❛♥t❡s ♥ã♦ sã♦ ❝♦♥❤❡❝✐❞❛s✱ ❜❡♠ ❝♦♠♦ ♥ã♦ sã♦ ❝♦♠♣r❡❡♥❞✐❞❛s ❛s r❡s♣❡❝t✐✈❛s

r❡❧❛çõ❡s ❝♦♠ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ♦✉ ❞❡♣❡♥❞❡♥t❡✳

  ❖s ♠ét♦❞♦s st❡♣✇✐s❡ ♣♦❞❡♠ t❡r ❞✉❛s ❞✐♥â♠✐❝❛s ❞✐❢❡r❡♥t❡s ❛ ❢♦r✇❛r❞ st❡♣✇✐s❡ ❡

❜❛❝❦✇❛r❞ st❡♣✇✐s❡✳ ❙✉❝✐♥t❛♠❡♥t❡✱ ♦ ♠ét♦❞♦ ❜❛❝❦✇❛r❞ st❡♣✇✐s❡ ♣❛rt❡ ❞❡ ✉♠ ♠♦❞❡❧♦

✐♥✐❝✐❛❧ ❝♦♠ t♦❞❛s ❛s ♣♦ssí✈❡✐s ✈❛r✐á✈❡✐s✱ q✉❡ ✈ã♦ s❡♥❞♦ ❡❧✐♠✐♥❛❞❛s ❛ ❝❛❞❛ ♣❛ss♦✱ ❛té

❝❤❡❣❛r ❛♦ ♠♦❞❡❧♦ ✜♥❛❧✳ ❖ ♠ét♦❞♦ ❢♦r✇❛r❞ st❡♣✇✐s❡ ✐♥✐❝✐❛✲s❡ ❝♦♠ ✉♠ ♠♦❞❡❧♦ s❡♠

♥❡♥❤✉♠❛ ✈❛r✐á✈❡❧ ❡①♣❧✐❝❛t✐✈❛ ❡ ❛ ❝❛❞❛ ♣❛ss♦ sã♦ ✐♥❝❧✉í❞❛s ❛s ✈❛r✐á✈❡✐s r❡❧❡✈❛♥t❡s ❛té ❛

♦❜t❡♥çã♦ ❞♦ ♠♦❞❡❧♦ ✜♥❛❧✳

  ❖ ♠ét♦❞♦ ❞❡ s❡❧❡❝çã♦ st❡♣✇✐s❡✱ ✈❡r ❜❛s❡✐❛✲s❡ ♥♦ ✈❛❧♦r ❞♦s

♣✲✈❛❧✉❡s r❡❧❛t✐✈♦s ❛♦s t❡st❡s ❞❡ r❛③ã♦ ❞❡ ✈❡r♦s✐♠✐❧❤❛♥ç❛s ❞❡ ❲✐❧❦s ❡♥tr❡ ♠♦❞❡❧♦s ❝♦♠

✐♥❝❧✉sã♦ ♦✉ ❡①❝❧✉sã♦ ❞❡ ❝♦✈❛r✐á✈❡✐s ♣❛r❛ ❞❡❝✐❞✐r q✉❛✐s ❛s ❝♦✈❛r✐á✈❡✐s q✉❡ ❞❡✈❡♠ s❡r

✐♥❝❧✉í❞❛s ♥♦ ♠♦❞❡❧♦ ✜♥❛❧✳ ❈♦♠❡ç❛ ♣♦r ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞♦ ♣✲✈❛❧✉❡ ❞❛❞♦ ♣❡❧♦ t❡st❡ ❞❡

❲❛❧❞ ❡✱ ❝♦♠ ❜❛s❡ ♥❡st❡✱ ❡s❝♦❧❤❡ q✉❛❧ ❛ ✈❛r✐á✈❡❧ q✉❡✱ ❡♠ ♣r✐♠❡✐r❛ ❛♥á❧✐s❡✱ ❞❡✈❡ s❛✐r ✭♦✉

❡♥tr❛r✮ ♥♦ ♠♦❞❡❧♦ ✜♥❛❧✳ ◗✉❛♥t♦ ♠❡♥♦r ✭♦✉ ♠❛✐♦r✮ ❢♦r ♦ ✈❛❧♦r ❞♦ ♣✲✈❛❧✉❡ ❞❛❞♦ ♣❡❧♦

t❡st❡ ❞❡ ❲❛❧❞ ♠❛✐s ✭♠❡♥♦s✮ ✐♠♣♦rt❛♥t❡ é ❝♦♥s✐❞❡r❛❞❛ ❛ ❝♦✈❛r✐á✈❡❧✳ ❆♣ós ❛ ❡s❝♦❧❤❛

❞❛ ❝♦✈❛r✐á✈❡❧✱ ❢❛③✲s❡ ✉♠❛ s❡❣✉♥❞❛ ❛♥á❧✐s❡ ❛♦ s❡✉ ❣r❛✉ ❞❡ ✐♠♣♦rtâ♥❝✐❛ ❛tr❛✈és ❞♦ ✈❛❧♦r

❞♦ ♣✲✈❛❧✉❡ ❞♦ t❡st❡ ❞❡ r❛③ã♦ ❞❡ ✈❡r♦s✐♠✐❧❤❛♥ç❛s ❡♥tr❡ ♦s ♠♦❞❡❧♦s q✉❡ ❛ ✐♥❝❧✉❡♠ ❡ ❛

❡①❝❧✉❡♠✱ ❡ ❛ss✐♠ t♦♠❛ ❛ ❞❡❝✐sã♦ ✜♥❛❧ s♦❜r❡ ❛ ❡①❝❧✉sã♦ ✭♦✉ ✐♥❝❧✉sã♦✮ ❞❛ ✈❛r✐á✈❡❧ ♥♦

♠♦❞❡❧♦ ✜♥❛❧✳ ❖♣t♦✉✲s❡✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ♣❡❧❛ ✉t✐❧✐③❛çã♦ ❞♦ ♠ét♦❞♦ ❜❛❝❦✇❛r❞ st❡♣✇✐s❡✳

  ❇❛❝❦✇❛r❞ ❙t❡♣✇✐s❡ ➔

◆♦ â♠❜✐t♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ s❡❧❡❝çã♦ ❞♦ ♠♦❞❡❧♦ ❜❛❝❦✇❛r❞✱ ❛s ✈❛r✐á✈❡✐s sã♦ r❡♠♦✈✐❞❛s

s❡q✉❡♥❝✐❛❧♠❡♥t❡ ❞♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦ ✭q✉❡ ❝♦♥té♠ t♦❞❛s ❛s ❝♦✈❛r✐á✈❡✐s ❞♦ ❡st✉❞♦✮✳ ■♥✐❝✐❛❧♠❡♥t❡ ❛❥✉st❛✲s❡ ♦ ♠♦❞❡❧♦ ❝♦♥t❡♥❞♦ t♦❞❛s ❛s ❝♦✈❛r✐á✈❡✐s ❡ ❡s❝♦❧❤❡✲s❡✱ ❝♦♠

❜❛s❡ ♥♦ t❡st❡ ❞❡ ❲❛❧❞✱ ❛ ❝♦✈❛r✐á✈❡❧ ♠❡♥♦s s✐❣♥✐✜❝❛t✐✈❛✱ ✐st♦ é✱ ❛ q✉❡ t❡♠ ♦ ♠❛✐♦r ♣✲

✈❛❧✉❡✳ ❉❡♣♦✐s✱ ❛tr❛✈és ❞♦ t❡st❡ ❞❡ r❛③ã♦ ❞❡ ✈❡r♦s✐♠✐❧❤❛♥ç❛s✱ ❝♦♠♣❛r❛✲s❡ ♦ ❛❥✉st❡ ❞♦

♠♦❞❡❧♦ ❧♦❣íst✐❝♦ ❝♦♠♣❧❡t♦ ❝♦♠ ♦ ♠♦❞❡❧♦ q✉❡ ❞❡❧❡ r❡s✉❧t❛ ♣❡❧❛ ❡①❝❧✉sã♦ ❞❛ ❝♦✈❛r✐á✈❡❧

♠❡♥♦s s✐❣♥✐✜❝❛t✐✈❛✳ ❙❡ ♦ ♣✲✈❛❧✉❡ r❡s✉❧t❛♥t❡ ❞♦ t❡st❡ ❞❡ ❲✐❧❦s✱ ❢♦r ✐♥❢❡r✐♦r ❛ 0.05 ✈♦❧t❛ ❛

❝♦❧♦❝❛r✲s❡ ❛ ❝♦✈❛r✐á✈❡❧ ♥♦ ♠♦❞❡❧♦✱ ♣♦✐s ♦ ♠♦❞❡❧♦ s❡♠ ❛ ❝♦✈❛r✐á✈❡❧ é ♠❡♥♦s ❡①♣❧✐❝❛t✐✈♦ ❞♦

q✉❡ q✉❛♥❞♦ ❛ ❝♦♥té♠✳ ❈❛s♦ ❝♦♥trár✐♦ ❡❧❛ é r❡♠♦✈✐❞❛✳ ◆❛s ❡t❛♣❛s s❡❣✉✐♥t❡s✱ ❝♦♥s✐❞❡r❛✲

s❡ ♦ ♠♦❞❡❧♦ r❡s✉❧t❛♥t❡ ❞❛ ❡t❛♣❛ ❛♥t❡r✐♦r ❝♦♠♦ ♠♦❞❡❧♦ ✐♥✐❝✐❛❧ ❡ ❛♣❧✐❝❛♠✲s❡ ♦s ♠❡s♠♦s

♣❛ss♦s✳ ❈❛s♦ ♥❛ ❡t❛♣❛ ❛♥t❡r✐♦r ❛ ✈❛r✐á✈❡❧ ♥ã♦ t❡♥❤❛ s✐❞♦ r❡♠♦✈✐❞❛ ❡ ❛✐♥❞❛ ❤❛❥❛ ✈❛r✐á✈❡✐s

  

✻✹ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

♣❡r♠❛♥❡❝❡✉ ♥♦ ♠♦❞❡❧♦✳ ◗✉❛♥❞♦ ♥✉♠❛ ❡t❛♣❛ t♦❞❛s ❛s ✈❛r✐á✈❡✐s✱ ♣❡❧♦ t❡st❡ ❞❡ ❲❛❧❞✱

❢♦r❡♠ s✐❣♥✐✜❝❛t✐✈❛s✱ s❡♠ t❡r♠♦s ❡♠ ❝♦♥t❛ ❛s q✉❡ ❛✐♥❞❛ ❛ss✐♠ ♣❡r♠❛♥❡❝❡r❛♠ ♥♦ ♠♦❞❡❧♦✱

♦ ♣r♦❝❡ss♦ t❡r♠✐♥❛ ❡ ♦ ♠♦❞❡❧♦ ✜♥❛❧ é ❝♦♥st✐t✉í❞♦ ♣♦r t♦❞❛s ❛s ❝♦✈❛r✐á✈❡✐s ❞❡ss❛ ❡t❛♣❛✳

▼ét♦❞♦ ❞❡ ❡s❝♦❧❤❛ ❞❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ✲ ❆■❈

❖ ❈r✐tér✐♦ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❆❦❛✐❦❡ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ❡♠ ✶✾✼✶ ♣♦r ❍✐r♦ts✉❣✉ ❆❦❛✐❦❡ s♦❜

♦ ♥♦♠❡ ✏❆❦❛✐❦❡ ■♥❢♦r♠❛t✐♦♥ ❈r✐t❡r✐♦♥✑ ✭❆■❈✮ ❡ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❆❦❛✐❦❡ ❡♠ ✶✾✼✹✳ ➱

✉♠❛ ♠❡❞✐❞❛ ❞♦ ♠❡❧❤♦r ❛❥✉st❡✱ ♣❛r❛ ✉♠ ♠♦❞❡❧♦ ❡st❛tíst✐❝♦ ❡st✐♠❛❞♦✳ ➱ ❢✉♥❞❛♠❡♥t❛❞♦

♥♦ ❝♦♥❝❡✐t♦ ❞❡ ❡♥tr♦♣✐❛✱ ❡ ♦❢❡r❡❝❡ ✉♠❛ ♠❡❞✐❞❛ r❡❧❛t✐✈❛ ❞❛ ✐♥❢♦r♠❛çã♦ ♣❡r❞✐❞❛ q✉❛♥❞♦

✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠♦❞❡❧♦ é ✉t✐❧✐③❛❞♦✳ P♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ❡q✉✐❧í❜r✐♦ ❡♥tr❡

❛ ✈❛r✐â♥❝✐❛ ❡ ❛ t❡♥❞❡♥❝✐♦s✐❞❛❞❡ ✭❜✐❛s ♦✉ ✈✐és✮ ❞❛ ❝♦♥str✉çã♦ ❞♦ ♠♦❞❡❧♦✱ ♦✉ ♣♦r ♦✉tr❛s

♣❛❧❛✈r❛s ❛ ♣r❡❝✐sã♦ ❡ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦✱ ✈❡r

  ❖ ❆■❈ ♥ã♦ é ✉♠ t❡st❡ ❛♦ ♠♦❞❡❧♦ ♥♦ s❡♥t✐❞♦ ❞❡ t❡st❛r ❤✐♣ót❡s❡s✱ ♠❛s s✐♠ ✉♠ t❡st❡

❡♥tr❡ ♠♦❞❡❧♦s✱ ♦✉ s❡❥❛✱ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣❛r❛ s❡❧❡❝❝✐♦♥❛r ✉♠ ♠♦❞❡❧♦ ❞❡ ❡♥tr❡ ✉♠

❝♦♥❥✉♥t♦ ❞❡ ♠♦❞❡❧♦s✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✱ ❡ ✈ár✐♦s ♠♦❞❡❧♦s ♣❛r❛ ♦s ♠❡s♠♦s✱

♦ ❆■❈ ❝❧❛ss✐✜❝❛✲♦s ❡ ♦ q✉❡ t✐✈❡r ♦ ♠❡♥♦r ❆■❈ ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❡❧❤♦r ♠♦❞❡❧♦✳

  ❊st❡ ❝r✐tér✐♦ ❞❡ s❡❧❡❝çã♦ ❜❛s❡✐❛✲s❡ ♥❛ ❢✉♥çã♦ ▲♦❣✲✈❡r♦s✐♠✐❧❤❛♥ç❛✱ ❝♦♠ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠ ❢❛❝t♦r ❞❡ ❝♦rr❡❝çã♦ ❝♦♠♦ ♠♦❞♦ ❞❡ ♣❡♥❛❧✐③❛çã♦ ❞❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦✳ ❆ ❡st❛tíst✐❝❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❡♠ H é ✭✈❡r

AIC = −2ℓ(e β , 0, e φ) + 2r

  1 ✱ ) ♦♥❞❡ r = dim(β

  1 ✳ ❆ ❡①♣r❡ssã♦ r❡♣r❡s❡♥t❛ ❛ r❡❧❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ♦ ❆■❈ ❡ ♦ ❞❡s✈✐♦ r❡❞✉③✐❞♦

r❡❧❛t✐✈♦ ❛♦ ♠♦❞❡❧♦ ❡s♣❡❝✐✜❝❛❞♦ ♣♦r H ✭s✉♣õ❡✲s❡ q✉❡ ♦ ♣❛râ♠❡tr♦ φ é ❝♦♥❤❡❝✐❞♦ ♦✉ é

s✉❜st✐t✉í❞♦ ♣♦r ✉♠❛ ❡st✐♠❛t✐✈❛ ❝♦♥s✐st❡♥t❡✱ ♥❡st❡ ❝❛s♦ ❞❡ ❡st✉❞♦ ❡♠ ♣❛rt✐❝✉❧❛r✱ s❡rá

❝♦♥s✐❞❡r❛❞♦ φ = 1✮✳

  β β β

AIC r = −2ℓ(e , 0) + 2ℓ(b ) − 2ℓ(b ) + 2r

  1 S S ∗ = D + 2r − 2(ℓ(b β )), r S ✭✸✳✶✼✮

  

♦♥❞❡ r é ♦ í♥❞✐❝❡ q✉❡ ❡s♣❡❝í✜❝❛ ♦ ♠♦❞❡❧♦ q✉❡ s❡ ❡stá ❛ ❝♦♥s✐❞❡r❛r ❡ S r❡❢❡r❡✲s❡ ❛♦

♠♦❞❡❧♦ ❝♦♠♣❧❡t♦✳ ➱ ❛✐♥❞❛ s✉❣❡r✐❞❛ ✉♠❛ ♠♦❞✐✜❝❛çã♦ ♥♦ ♠♦❞❡❧♦ ❞❡ ❆❦❛✐❦❡ ♣❛r❛ s❡❧❡❝❝✐♦♥❛r ♠♦❞❡❧♦s✳ ❙❡❥❛✿

  ∗ ∗

C = D + 2r − n = AIC + 2ℓ(b β ) − n.

r r r S

  ∗ ❙❡ s❡ ❞❡s❡♥❤❛r ♦ ❣rá✜❝♦ ❞❡ C r ❝♦♥tr❛ r ♦❜té♠✲s❡ ✉♠❛ ❜♦❛ ✐♥❞✐❝❛çã♦ ♣❛r❛ ❝♦♠♣❛✲

  ∗ r❛çã♦ ❞❡ ♠♦❞❡❧♦s✳ ❙❡ ♦ ♠♦❞❡❧♦ ❢♦r ✈❡r❞❛❞❡✐r♦ ❡s♣❡r❛✲s❡ q✉❡ C s❡❥❛ ♣ró①✐♠♦ ❞❡ r✳ r

  1

  2

  1

  2

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✻✺ > r

  1

  2 ♣❛râ♠❡tr♦s ❡ r t❡♠✲s❡✿

  ∗ ∗ ∗ ∗

AIC − AIC = C − C = D − D + 2(r − r ),

r

  1 r

  2

  1

  2 r

  1 r 2 r 1 r

  2

  ❡ s✉♣♦♥❞♦ q✉❡ ♦ ♠♦❞❡❧♦ M 2 é ✈❡r❞❛❞❡✐r♦✱ t❡♠✲s❡ −1

E [AIC − AIC ] = r − r + 0(n ).

r1 r2

  1

  2 ◆❛ ❝♦♠♣❛r❛çã♦ ❞❡ ♠♦❞❡❧♦s s✉❝❡ss✐✈❛♠❡♥t❡ ♠❛✐s r✐❝♦s✱ ♦ ❞❡❝❧✐✈❡ ❡s♣❡r❛❞♦ ❞♦ s❡❣✲ r1 − AIC r2

  

♠❡♥t♦ ❞❡ r❡❝t❛ q✉❡ ✉♥❡ AIC ❞❡✈❡ ❡st❛r ♣ró①✐♠♦ ❞❡ ✶ s✉♣♦♥❞♦ ♦ ♠♦❞❡❧♦

  2

♠❡♥♦r M ✈❡r❞❛❞❡✐r♦✳ P❛r❡s ❞❡ ♠♦❞❡❧♦s q✉❡ ❡①✐❜❡♠ ❞❡❝❧✐✈❡ ♠❛✐♦r ❞♦ q✉❡ ✶ ✐♥❞✐❝❛♠

q✉❡ ♦ ♠♦❞❡❧♦ ♠❛✐♦r ♥ã♦ é s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ♠❡♥♦r✳

  ❈♦♥❢♦r♠❡ ❛✉♠❡♥t❛r ♦ ♥ú♠❡r♦ ❞❡ ♣❛râ♠❡tr♦s ❧✐✈r❡s ❛ s❡r❡♠

❡st✐♠❛❞♦s ♠❡❧❤♦r❛ ❛ q✉❛❧✐❞❛❞❡ ❞♦ ❛❥✉st❡✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❛râ♠❡tr♦s

❧✐✈r❡s ♥♦ ♣r♦❝❡ss♦ ❞❡ ❣❡r❛çã♦ ❞❡ ❞❛❞♦s✳ ❱❡r✐✜❝❛✲s❡ ❡♥tã♦ q✉❡ ♦ ❆■❈ ♥ã♦ só ♠❡❞❡ ❛ ♠❡❧❤♦r✐❛ ❞♦ ♠♦❞❡❧♦✱ ❝♦♠♦ t❛♠❜é♠ ✐♥❝❧✉✐

✉♠❛ ♣❡♥❛❧✐③❛çã♦✱ q✉❡ é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❛râ♠❡tr♦s ❡st✐♠❛❞♦s✳

❊st❛ ♣❡♥❛❧✐❞❛❞❡ ❞✐♠✐♥✉✐ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ♦✈❡r✜tt✐♥❣✳ ❖ ♠♦❞❡❧♦ ♣r❡❢❡r✐❞♦ é ❛q✉❡❧❡ ❝♦♠

♦ ♠❡♥♦r ✈❛❧♦r ❞❡ ❆■❈✳ ❆ ♠❡t♦❞♦❧♦❣✐❛ ❆■❈ t❡♠ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ❡♥❝♦♥tr❛r ♦ ♠♦❞❡❧♦ q✉❡

♠❡❧❤♦r ❡①♣❧✐❝❛ ♦s ❞❛❞♦s ❝♦♠ ✉♠ ♠í♥✐♠♦ ❞❡ ♣❛râ♠❡tr♦s ❧✐✈r❡s✳

  ❖ ❆■❈ ❝❧❛ss✐✜❝❛ ✉♠ ♠♦❞❡❧♦ ♣❡❧❛ ♣r♦①✐♠✐❞❛❞❡ ❞♦s s❡✉s ✈❛❧♦r❡s ❝♦♠ ♦s ✈❡r❞❛❞❡✐r♦s

✈❛❧♦r❡s✱ ❡♠ t❡r♠♦s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✈❛❧♦r ❡s♣❡r❛❞♦✳ ▼❛s é ✐♠♣♦rt❛♥t❡ ♣❡r❝❡❜❡r

q✉❡ ♦ ✈❛❧♦r ❞❡ ❆■❈ ❛tr✐❜✉í❞♦ ❛ ✉♠ ♠♦❞❡❧♦ s❡r✈❡ ❛♣❡♥❛s ♣❛r❛ ❝❧❛ss✐✜❝❛r ♦s ♠♦❞❡❧♦s

❝♦♥❝♦rr❡♥t❡s ❡ ❞✐③❡r q✉❛❧ é ♦ ♠❡❧❤♦r ❡♥tr❡ ❛s ❛❧t❡r♥❛t✐✈❛s ❞❛❞❛s✳ ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s

◆❡st❛ s❡❝çã♦✱ r❡❝♦rr❡♥❞♦ ❛♦ s♦❢t✇❛r❡ R✱ ❛♣❧✐❝❛♠✲s❡ ♦s ♠ét♦❞♦s ❡st❛tíst✐❝♦s ❡st✉❞❛❞♦s

♣❛r❛ ❛ ♠♦❞❡❧❛çã♦ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❡ ♦s r❡s♣❡❝t✐✈♦s r❡s✉❧t❛❞♦s✳

  ❙❡rá ❛♣❧✐❝❛❞❛ ❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ ♣❛r❛ ❛❥✉st❛r ♦s ♠♦❞❡❧♦s ❡ ♦s ♠ét♦❞♦s ❙t❡♣✇✐s❡✲

❜❛❝❦✇❛r❞ ❡ ❆■❈ ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❛s ✈❛r✐á✈❡✐s q✉❡ ♠❡❧❤♦r ❡①♣❧✐❝❛♠ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛

❡ q✉❡ ❝♦♥st✐t✉❡♠ ♦ ♠♦❞❡❧♦ ✜♥❛❧✳

✸✳✸✳✶ ❆❥✉st❛♠❡♥t♦ ❞♦s ❉❛❞♦s ❆tr❛✈és ❞❛ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛

➱ ✐♠♣♦rt❛♥t❡ r❡❧❡♠❜r❛r q✉❡ ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r t❡♠ ❛ ❢♦r♠❛✿

Y = β + β x + . . . + β x + ε,

  1 1 i i ✭✸✳✶✽✮ i i

s❡♥❞♦ β ♦ ❝♦❡✜❝✐❡♥t❡ ❞✐s❝r✐♠✐♥❛♥t❡ ♣❛r❛ ❛ ✈❛r✐á✈❡❧ x ❡ ε ✉♠ ✈❡❝t♦r ❞❡ ❡rr♦s ❛❧❡❛tór✐♦s✳

❈♦♥s✐❞❡r❛✲s❡✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s✿ ■❞❛❞❡✱ ◆➸ ❉❡♣❡♥❞❡♥✲

  

✻✻ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❆♥t✐❣✉✐❞❛❞❡ Pr♦✜ssã♦✱ ❘❡♥❞✐♠❡♥t♦ ▲íq✉✐❞♦✱ ❈ré❞✐t♦✱ P❡r❝❡♥t❛❣❡♠ ❞❡ ❊♥tr❛❞❛✱ ◆➸ ❚♦✲

t❛❧ Pr❡st❛çõ❡s ❡ ◆➸ ❞❡ Pr❡st❛çõ❡s P❛❣❛s✳ ❯t✐❧✐③❛♥❞♦ ♦ s♦❢t✇❛r❡ R ❝♦♥s❡❣✉❡✲s❡ r❡❛❧✐③❛r t♦❞♦ ♦ ❡st✉❞♦ q✉❡ s❡ s❡❣✉❡✳

  Pr✐♠❡✐r♦ ❝❛rr❡❣❛♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✿ ❃ ❞❛t❛ ❁✲ r❡❛❞✳t❛❜❧❡✭✧❉❆❉❖❙❈❘✳t①t✧✱ ❤❡❛❞❡r ❂ ❚❘❯❊✮ ❃ ❛tt❛❝❤✭❞❛t❛✮

  ❖ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦ s❡rá ♦ ♠♦❞❡❧♦ q✉❡ ❝♦♥t✐✈❡r ❛s ❝♦✈❛r✐á✈❡✐s✿ ■❞❛❞❡✱ ❉❡♣✱ ❍❛❜✱ ■❞❝❛s❛✱ ❈✐✈✐❧✱ P♦st❛❧✱ Pr♦❢✱ ❆♥t❣✱ ❘❡♥❞✱ ❈r❡❞✱ ❊♥tr✱ P❚♦t❛❧ ❡ P❛❣❛s ❆ ❝♦✈❛r✐á✈❡❧ ❈ó❞✐❣♦ P♦st❛❧ ✐♥✐❝✐❛❧♠❡♥t❡✱ t❛❧ ❝♦♠♦ r❡❢❡r✐❞♦ ♥❛ ❙❡❝çã♦ é ✉♠❛

✈❛r✐á✈❡❧ ❝❛t❡❣ór✐❝❛ ❞✐✈✐❞✐❞❛ ❡♠ ✹ ❝❛t❡❣♦r✐❛s ❞✐st✐♥t❛s✿ ✏❩♦♥❛ ✶✑✱ ✏❩♦♥❛ ✷✑✱ ✏❩♦♥❛ ✸✑ ❡

  ✏❩♦♥❛ ✹✑✳ ◆♦ ❡♥t❛♥t♦✱ ❛♣ós ♦ ❛❥✉st❛♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ ❡ ❛ ❛♥á❧✐s❡ ❞❛ s✐❣♥✐✜❝â♥❝✐❛ ❞❛s ✈❛r✐á✈❡✐s

❡ r❡s♣❡❝t✐✈❛s ❝❛t❡❣♦r✐❛s✱ ♦ t❡st❡ ❞❡ ❲❛❧❞ s✉❣❡r❡ q✉❡ ❛ ❝❛t❡❣♦r✐❛s ✏❩♦♥❛ ✷✑ ♥ã♦ é s✐❣♥✐✜✲

❝❛t✐✈❛♠❡♥t❡ ❞✐❢❡r❡♥t❡ ❞❛ ❝❛t❡❣♦r✐❛ ✏❩♦♥❛ ✶✑✱ ♣❡❧♦ q✉❡ s❡ ❞❡✈❡rã♦✱ ❞❡ ❢♦r♠❛ ❛ ♦♣t✐♠✐③❛r

♦ ♠♦❞❡❧♦✱ ❛❣r✉♣❛r ❡st❛s ❞✉❛s ❝❛t❡❣♦r✐❛s ❞❛ ✈❛r✐á✈❡❧ ❈ó❞✐❣♦ P♦st❛❧✳ ❖s r❡s✉❧t❛❞♦s ❞♦ ♠♦❞❡❧♦ ✐♥✐❝✐❛❧ ❝♦♠ q✉❛tr♦ ♥í✈❡✐s ♥❛ ✈❛r✐á✈❡❧ ❈ó❞✐❣♦ P♦st❛❧ ♣♦❞❡✲ rã♦ s❡r ✈✐st♦s ♥♦s ♦✉t♣✉ts ❞♦ s♦❢t✇❛r❡ R q✉❡ s❡ ❡♥❝♦♥tr❛♠ ♥♦ ❆♥❡①♦ ◆♦ q✉❡ s❡ s❡❣✉❡✱ ❡ ♣❛r❛ ♥ã♦ ❛❧♦♥❣❛r ❛ ❡①♣♦s✐çã♦ ❞♦s r❡s✉❧t❛❞♦s✱ ❛♥❛❧✐s❛✲s❡ ♦ ♠♦❞❡❧♦

❛ ♣❛rt✐r ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❈ó❞✐❣♦ P♦st❛❧ ❝♦♠ ❛♣❡♥❛s ✸ ❝❛t❡❣♦r✐❛s✿ ✏❩♦♥❛ ✶✑✱ ✏❩♦♥❛ ✸✑ ❡

✏❩♦♥❛ ✹✑✳

  ▼♦❞❡❧♦ ❈♦♠♣❧❡t♦ ➔ ◆♦ s♦❢t✇❛r❡ R✱ ❛ ❢✉♥çã♦ ❣❧♠ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛❥✉st❛r ♠♦❞❡❧♦s ❧✐♥❡❛r❡s ❣❡♥❡r❛❧✐③❛❞♦s✱

  

♦♥❞❡ s❡ ❡s♣❡❝✐✜❝❛ q✉❛✐s ❛s ✈❛r✐á✈❡✐s ♣r❡❞✐t♦r❛s ❞♦ ♠♦❞❡❧♦ ❡ ❛ r❡s♣❡❝t✐✈❛ ❞✐str✐❜✉✐çã♦ ❞♦s

❡rr♦s✳ ❯t✐❧✐③❛r✲s❡✲á ❝♦♠♦ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❛ ❢✉♥çã♦ ▲♦❣✐t ✭❧✐♥❦ ❂ ✧❧♦❣✐t✧✮✱ q✉❡ s❡

♦♠✐t✐rá ♥♦s ❛❥✉st❛♠❡♥t♦s s❡❣✉✐♥t❡s ♣♦✐s é ❛ ❢✉♥çã♦ ✉t✐❧✐③❛❞❛✱ ♣♦r ❞❡❢❡✐t♦✱ ♣❡❧♦ s♦❢t✇❛r❡

R

  ♣❛r❛ ♦ ♠♦❞❡❧♦ ❝♦♠ ❡rr♦s ❜✐♥♦♠✐❛✐s✳ ❃ ▼♦❞❢✉❧❧ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✾✶❡✰✵✵ ✲✼✳✷✾❡✲✵✸ ✲✹✳✵✷❡✲✵✹ ✲✶✳✹✵❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✾❡✰✵✶ ✻✳✵✼❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✷ ■❞❛❞❡■❞♦s♦s ✲✷✳✵✼❡✰✵✵ ✽✳✼✸❡✲✵✶ ✲✷✳✸✼ ✵✳✵✶✽ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✾✷❡✰✵✵ ✸✳✶✽❡✲✵✶ ✶✷✳✸✶ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✸✵❡✰✵✵ ✻✳✽✽❡✲✵✶ ✶✳✽✾ ✵✳✵✺✽ ✳ ❉❡♣P♦✉❝♦s ✷✳✽✹❡✰✵✵ ✸✳✺✸❡✲✵✶ ✽✳✵✻ ✼✳✾❡✲✶✻ ✯✯✯ ❍❛❜❈♦♠❡r❝✐♦ ✶✳✷✻❡✲✵✷ ✸✳✻✽❡✲✵✶ ✵✳✵✸ ✵✳✾✼✸ ❍❛❜❊♠♣r❡s❛ ✲✸✳✷✸❡✲✵✶ ✺✳✾✾❡✲✵✶ ✲✵✳✺✹ ✵✳✺✾✵ ■❞❝❛s❛❯s❛❞❛ ✲✻✳✾✷❡✲✵✷ ✷✳✻✵❡✲✵✶ ✲✵✳✷✼ ✵✳✼✾✵

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✻✼

  ■❞❝❛s❛❱❡❧❤❛ ✹✳✼✾❡✲✵✶ ✹✳✾✵❡✲✵✶ ✵✳✾✽ ✵✳✸✷✽ ❈✐✈✐❧❉✐✈ ✸✳✸✽❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✻✸ ✶✳✽❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✻✵❡✰✵✵ ✷✳✷✽❡✲✵✶ ✼✳✵✷ ✷✳✸❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✺✽❡✰✵✵ ✻✳✺✼❡✲✵✶ ✺✳✹✺ ✺✳✶❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✼✽❡✰✵✵ ✻✳✻✺❡✲✵✶ ✽✳✻✾ ❁ ✷❡✲✶✻ ✯✯✯ Pr♦❢❚✐♣♦✷ ✲✸✳✼✷❡✲✵✶ ✷✳✾✻❡✲✵✶ ✲✶✳✷✻ ✵✳✷✵✾ Pr♦❢❚✐♣♦✸ ✲✺✳✹✾❡✲✵✶ ✷✳✽✹❡✲✵✶ ✲✶✳✾✹ ✵✳✵✺✸ ✳ Pr♦❢❚✐♣♦✹ ✲✻✳✻✷❡✲✵✶ ✸✳✻✷❡✲✵✶ ✲✶✳✽✸ ✵✳✵✻✽ ✳ ❆♥t❣❘❡❝❡♥t❡ ✲✺✳✷✶❡✲✵✶ ✸✳✽✶❡✲✵✶ ✲✶✳✸✼ ✵✳✶✼✷ ❘❡♥❞❇❛✐①♦ ✶✳✼✷❡✰✵✶ ✻✳✵✼❡✰✵✷ ✵✳✵✸ ✵✳✾✼✼ ❘❡♥❞▼❡❞✐♦ ✽✳✼✾❡✰✵✵ ✻✳✵✼❡✰✵✷ ✵✳✵✶ ✵✳✾✽✽ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✼❡✰✵✶ ✻✳✵✼❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✺❡✲✵✺ ✶✳✻✻❡✲✵✻ ✲✻✳✸✹ ✷✳✸❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✶✶❡✰✵✵ ✸✳✼✸❡✲✵✶ ✶✸✳✼✶ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✺❡✲✵✷ ✷✳✷✼❡✲✵✸ ✶✻✳✶✶ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✻✼❡✲✵✷ ✸✳✻✷❡✲✵✸ ✲✶✺✳✻✼ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✹✻✳✾ ♦♥ ✾✾✼✻ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✾✹✳✾ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾

  ❘❡❛❧✐③♦✉✲s❡ ❛ ♠❡s♠❛ ❛♥á❧✐s❡ ♠❛s ♣❛r❛ ♦ ♠♦❞❡❧♦ ♥✉❧♦ ✭♥ã♦ ❝♦♥té♠ ♥❡♥❤✉♠❛ ❝♦✈❛✲

r✐á✈❡❧✮ ❡ ❛✐♥❞❛✱ ❛tr❛✈és ❞♦ ❚❡st❡ ❘❛③ã♦ ❞❡ ❱❡r♦s✐♠✐❧❤❛♥ç❛s✱ ❝♦♠♣❛r♦✉✲s❡ q✉❛❧ ❞♦s ❞♦✐s

♠♦❞❡❧♦s s❡r✐❛ ♦ q✉❡ ♠❡❧❤♦r ❡①♣❧✐❝❛✈❛ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛✳ ➔ ▼♦❞❡❧♦ ◆✉❧♦ ❃ ▼♦❞♥✉❧❧ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✷✳✻✶✺ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✸✳✸✽✼✶ ✵✳✵✺✻✷ ✲✻✵✳✷ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✷✽✽✷ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✻

  ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ❞♦✐s ♠♦❞❡❧♦s ➔ ❈♦♥s✐❞❡r❡✲s❡ ♦ ♠♦❞❡❧♦ ▼♦❞♥✉❧❧ ❡♥❝❛✐①❛❞♦ ♥♦ ♠♦❞❡❧♦ ▼♦❞❢✉❧❧✳ ◆♦ s♦❢t✇❛r❡ R✱ ❛ ❢✉♥çã♦

❛♥♦✈❛✱ ❝♦♠ ✐♥❞✐❝❛çã♦ ❞❡ t❡st❡ t❡st ❂ ✧❈❤✐sq✧ r❡❛❧✐③❛ ♦ ❚❡st❡ ❘❛③ã♦ ❞❡ ❱❡r♦s✐♠✐❧❤❛♥ç❛s✱

t❡st❛♥❞♦ s❡ ♦ ♠♦❞❡❧♦ ❝♦♠ ♠❡♥♦s ❝♦✈❛r✐á✈❡✐s é s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ❝♦♠

♠❛✐s ❝♦✈❛r✐á✈❡✐s✳ ◆❛ ♣rát✐❝❛✱ ❡stá✲s❡ ❛ ❛♥❛❧✐s❛r✿

  

✻✽ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

H ✿ ▼♦❞❡❧♦ ▼♦❞❢✉❧❧ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ▼♦❞♥✉❧❧

  

✈❡rs✉s

H

1 ✿ ▼♦❞❡❧♦ ▼♦❞♥✉❧❧ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ▼♦❞❢✉❧❧

  ❃ ❛♥♦✈❛✭▼♦❞♥✉❧❧✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ✶ ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s

  ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮ ✶ ✾✾✾✾ ✷✽✽✵ ✷ ✾✾✼✻ ✻✹✼ ✷✸ ✷✷✸✸ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶

  ❖❜s❡r✈❛✲s❡ q✉❡ ♦ ✈❛❧♦r ❞♦ ♣✲✈❛❧✉❡ ♣❡❧♦ ❚❡st❡ ❞❡ ❲✐❧❦s é ♠✉✐t♦ ♣❡q✉❡♥♦ ✭✐♥❢❡r✐♦r ❛ −16

  2 × 10 ✮ ❧♦❣♦✱ ♣♦❞❡ ❞✐③❡r✲s❡ q✉❡ s❡ ❞❡✈❡ r❡❥❡✐t❛r✱ ❝♦♠ ❛❧❣✉♠❛ ❡✈✐❞ê♥❝✐❛✱ ❛ ❤✐♣ót❡s❡

❞❡ q✉❡ ♦ ♠♦❞❡❧♦ ♥✉❧♦ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦✱ ♦✉ s❡❥❛✱ r❡❥❡✐t❛✲s❡ ❛ ❤✐♣ót❡s❡

❞❡ q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❛❝r❡s❝✐❞♦s s❡❥❛♠ ✐❣✉❛✐s ❛ ③❡r♦✱ ♦✉ s❡❥❛✱ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s

s❡rá s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❆❧é♠ ❞♦ ❚❡st❡ ❞❡ ❲✐❧❦s✱ ♣♦❞❡♠✲s❡ ♦❜s❡r✲

✈❛r ♦s ✈❛❧♦r❡s ❞♦ ❈r✐tér✐♦ ❞❡ ■♥❢♦r♠❛çã♦ ❞❡ ❆❦❛✐❦❡ ✭❆■❈✮ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s ♠♦❞❡❧♦s✳

❱❡r✐✜❝❛✲s❡ q✉❡ ♦ ❆■❈ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦ ✭✻✾✹✳✾✮ é ❜❛st❛♥t❡ ♠❡♥♦r q✉❡ ♦ ❞♦

♠♦❞❡❧♦ ♥✉❧♦ ✭✷✽✽✷✳✶✮ ❞♦♥❞❡✱ ♠❛✐s ✉♠❛ ✈❡③ s❡ r❡❥❡✐t❛ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ♦ ♠♦❞❡❧♦ ♥✉❧♦

é s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦✳ ❙❡❧❡❝çã♦ ❞❡ ❝♦✈❛r✐á✈❡✐s ❛tr❛✈és ❞♦ ▼ét♦❞♦ ❙t❡♣✇✐s❡ ✲ ❇❛❝❦✇❛r❞

❖ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦ ❞❡✜♥✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡ é ♦ ♠♦❞❡❧♦ q✉❡ s❡ ✉t✐❧✐③❛ ♣❛r❛ ✐♥✐❝✐❛❧✐③❛r ♦

♠ét♦❞♦ ❙t❡♣✇✐s❡ ✲ ❇❛❝❦✇❛r❞✳ ❊st❡ ♠ét♦❞♦✱ t❛❧ ❝♦♠♦ r❡❢❡r✐❞♦ ♥❛ ❙❡❝çã♦ ❝♦♥s✐st❡

❡♠ r❡t✐r❛r ❛s ✈❛r✐á✈❡✐s q✉❡ ♦ ❚❡st❡ ❞❡ ❲❛❧❞ ❝♦♥s✐❞❡r❛ ♠❡♥♦s s✐❣♥✐✜❝❛t✐✈❛s✳ ❖❜s❡r✈❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞♦ ❛❥✉st❛♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦✱ é ♣♦ssí✈❡❧ ✐❞❡♥t✐✜❝❛r

❛❧❣✉♠❛s ❝♦✈❛r✐á✈❡✐s q✉❡ ♥ã♦ tê♠ ♦s ❝ó❞✐❣♦s ❞❡ s✐❣♥✐✜❝â♥❝✐❛✳ ❊ss❛s sã♦ ❛s ✈❛r✐á✈❡✐s q✉❡

♦ ❚❡st❡ ❞❡ ❲❛❧❞ ❝♦♥s✐❞❡r❛ ♠❡♥♦s s✐❣♥✐✜❝❛t✐✈❛s✱ ❡♠ ♦♣♦s✐çã♦ àq✉❡❧❛s q✉❡ tê♠ ❵✯✯✯✬✱ ♣♦r

❡①❡♠♣❧♦✱ q✉❡ sã♦ ❝♦♥s✐❞❡r❛❞❛s ❛s ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛s✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♠ét♦❞♦ ❙t❡♣✇✐s❡✱ ❛ ♣r✐♠❡✐r❛ ✈❛r✐á✈❡❧ ❛ s❡r r❡t✐r❛❞❛ ❞♦ ♠♦❞❡❧♦ é ❛

q✉❡ t❡♠ ♠❛✐♦r ♣✲✈❛❧✉❡✱ ❧♦❣♦ ♦♣t❛✲s❡ ♣♦r ❛♥❛❧✐s❛r ♦ ♠♦❞❡❧♦ ❛♣ós ❛ ❡①❝❧✉sã♦ ❞❛ ✈❛r✐á✈❡❧

❘❡♥❞✐♠❡♥t♦ ✭❘❡♥❞✮✳ ❙❡❣✉✐❞❛♠❡♥t❡ ❝♦♠♣❛r❛✲s❡ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦ ❝♦♠ ♦ ♠♦❞❡❧♦ s❡♠

❛ ❝♦✈❛r✐á✈❡❧ ❘❡♥❞ ✭❛❴✶✮ r❡❝♦rr❡♥❞♦ ❛♦ ❚❡st❡ ❘❛③ã♦ ❞❡ ❱❡r♦s✐♠✐❧❤❛♥ç❛s✳ ◆♦ q✉❡ s❡ s❡❣✉❡✱ ✐♥t❡r❡ss❛ t❡st❛r✿

  H ✿ ▼♦❞❡❧♦ ▼♦❞❢✉❧❧ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ❛❴✶

✈❡rs✉s

H

  1 ✿ ▼♦❞❡❧♦ ❛❴✶ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ▼♦❞❢✉❧❧✳

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✻✾

  ❃ ❛❴✶ ❁✲ ✭▼♦❞❢✉❧❧ ✲ ❘❡♥❞✮ ❃ ❛♥♦✈❛✭❛❴✶✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰

  ❆♥t❣ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✼✾ ✶✹✽✹ ✷ ✾✾✼✻ ✻✹✼ ✸ ✽✸✼ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶

  ❆♥❛❧✐s❛♥❞♦ ♦ ♣✲✈❛❧✉❡ ❞♦ t❡st❡ ❝♦♥❝❧✉í♠♦s q✉❡✱ ❡♠❜♦r❛ ♦ ❚❡st❡ ❞❡ ❲❛❧❞ ❝♦♥s✐❞❡r❡

❛ ❝♦✈❛r✐á✈❡❧ ❘❡♥❞ ♣♦✉❝♦ s✐❣♥✐✜❝❛t✐✈❛✱ ♦ ♠♦❞❡❧♦ ❝♦♠ ❡st❛ ❝♦✈❛r✐á✈❡❧ r❡✈❡❧❛✲s❡ s✐❣♥✐✜❝❛✲

t✐✈❛♠❡♥t❡ ♠❡❧❤♦r✳ ❆ss✐♠✱ ♣❡❧♦ ❚❡st❡ ❞❡ ❲✐❧❦s ❡①✐st❡ ❡✈✐❞ê♥❝✐❛ ❡♠ r❡❥❡✐t❛r ❛ ❤✐♣ót❡s❡

♥✉❧❛✱ ♦✉ s❡❥❛ r❡❥❡✐t❛r ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ♦ ♠❡❧❤♦r ♠♦❞❡❧♦ é ♦ ♠♦❞❡❧♦ s❡♠ ❛ ❝♦✈❛r✐á✈❡❧

❘❡♥❞✳ P❛r❛ ♣r♦ss❡❣✉✐r ❝♦♥s✐❞❡r❛✲s❡ ♦ ♠♦❞❡❧♦ ❝♦♠ ❛ ❝♦✈❛r✐á✈❡❧ ❘❡♥❞ ✐♥❝❧✉í❞❛ ✭▼♦❞❢✉❧❧✮

❡ r❡t✐r❛✲s❡ ♦✉tr❛ ❝♦✈❛r✐á✈❡❧✱ ♥♦♠❡❛❞❛♠❡♥t❡ ❛ ❝♦✈❛r✐á✈❡❧ q✉❡ t❡♠ ♠❡♥♦s s✐❣♥✐✜❝â♥❝✐❛

❛ s❡❣✉✐r à ❝♦✈❛r✐á✈❡❧ ❘❡♥❞✱ ♣♦r ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✈♦❧t❛♥❞♦ ❛ ♦❜s❡r✈❛r ♦s ♣✲✈❛❧✉❡s ❞♦s

t❡st❡s ❞❡ ❲❛❧❞ ♣❛r❛ ❛ s✐❣♥✐✜❝â♥❝✐❛ ❞♦s ♣❛râ♠❡tr♦s ❛ss♦❝✐❛❞♦s às ❝♦✈❛r✐á✈❡✐s ♥♦ ♠♦❞❡❧♦

❝♦♠♣❧❡t♦✱ ❛ ✈❛r✐á✈❡❧ ♠❡♥♦s s✐❣♥✐✜❝❛t✐✈❛ ❞❡♣♦✐s ❞❛ ✈❛r✐á✈❡❧ ❘❡♥❞✐♠❡♥t♦ é ❛ ✈❛r✐á✈❡❧

❚✐♣♦ ❞❡ ❍❛❜✐t❛çã♦ ✭❍❛❜✮✳ ❊♥tã♦ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❢❛③✲s❡✿ ❃ ❛❴✷ ❁✲ ✭▼♦❞❢✉❧❧ ✲ ❍❛❜✮ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✾✵✽❡✰✵✵ ✲✼✳✸✷✵❡✲✵✸ ✲✹✳✵✵✹❡✲✵✹ ✲✶✳✹✵✶❡✲✵✺ ✷✳✼✾✻❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✽✽❡✰✵✶ ✻✳✵✼✸❡✰✵✷ ✲✵✳✵✼✷ ✵✳✾✹✷✹ ■❞❛❞❡■❞♦s♦s ✲✷✳✵✾✸❡✰✵✵ ✽✳✼✵✷❡✲✵✶ ✲✷✳✹✵✺ ✵✳✵✶✻✷ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✾✷✾❡✰✵✵ ✸✳✶✽✵❡✲✵✶ ✶✷✳✸✺✸ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✸✶✽❡✰✵✵ ✻✳✽✻✹❡✲✵✶ ✶✳✾✷✶ ✵✳✵✺✹✽ ✳ ❉❡♣P♦✉❝♦s ✷✳✽✹✸❡✰✵✵ ✸✳✺✷✼❡✲✵✶ ✽✳✵✻✷ ✼✳✺✸❡✲✶✻ ✯✯✯ ■❞❝❛s❛❯s❛❞❛ ✲✼✳✼✹✷❡✲✵✷ ✷✳✺✾✶❡✲✵✶ ✲✵✳✷✾✾ ✵✳✼✻✺✶ ■❞❝❛s❛❱❡❧❤❛ ✹✳✺✻✹❡✲✵✶ ✹✳✽✻✺❡✲✵✶ ✵✳✾✸✽ ✵✳✸✹✽✷ ❈✐✈✐❧❉✐✈ ✸✳✸✽✵❡✰✵✵ ✺✳✾✾✼❡✲✵✶ ✺✳✻✸✻ ✶✳✼✹❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✺✾✽❡✰✵✵ ✷✳✷✽✶❡✲✵✶ ✼✳✵✵✾ ✷✳✹✶❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✺✾✺❡✰✵✵ ✻✳✺✹✺❡✲✵✶ ✺✳✹✾✷ ✸✳✾✼❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✼✾✹❡✰✵✵ ✻✳✻✷✸❡✲✵✶ ✽✳✼✹✽ ❁ ✷❡✲✶✻ ✯✯✯ Pr♦❢❚✐♣♦✷ ✲✸✳✼✶✵❡✲✵✶ ✷✳✾✺✽❡✲✵✶ ✲✶✳✷✺✹ ✵✳✷✵✾✼ Pr♦❢❚✐♣♦✸ ✲✺✳✺✹✹❡✲✵✶ ✷✳✽✸✵❡✲✵✶ ✲✶✳✾✺✾ ✵✳✵✺✵✶ ✳ Pr♦❢❚✐♣♦✹ ✲✻✳✻✺✷❡✲✵✶ ✸✳✻✷✶❡✲✵✶ ✲✶✳✽✸✼ ✵✳✵✻✻✷ ✳ ❆♥t❣❘❡❝❡♥t❡ ✲✺✳✶✽✻❡✲✵✶ ✸✳✽✵✽❡✲✵✶ ✲✶✳✸✻✷ ✵✳✶✼✸✷ ❘❡♥❞❇❛✐①♦ ✶✳✼✶✾❡✰✵✶ ✻✳✵✼✸❡✰✵✷ ✵✳✵✷✽ ✵✳✾✼✼✹ ❘❡♥❞▼❡❞✐♦ ✽✳✼✽✸❡✰✵✵ ✻✳✵✼✸❡✰✵✷ ✵✳✵✶✹ ✵✳✾✽✽✺ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✼✷❡✰✵✶ ✻✳✵✼✸❡✰✵✷ ✵✳✵✸✾ ✵✳✾✻✽✽ ❈r❡❞ ✲✶✳✵✺✶❡✲✵✺ ✶✳✻✺✵❡✲✵✻ ✲✻✳✸✼✶ ✶✳✽✽❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✶✶✸❡✰✵✵ ✸✳✼✸✹❡✲✵✶ ✶✸✳✻✾✸ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✹✾❡✲✵✷ ✷✳✷✻✹❡✲✵✸ ✶✻✳✶✶✹ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✻✼✵❡✲✵✷ ✸✳✻✶✹❡✲✵✸ ✲✶✺✳✻✽✻ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲

  

✼✵ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮

  ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✹✼✳✷ ♦♥ ✾✾✼✽ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✾✶✳✷ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ❙❡♥❞♦ ♦ ♠♦❞❡❧♦ ❛❴✷ ❡♥❝❛✐①❛❞♦ ♥♦ ♠♦❞❡❧♦ ▼♦❞❢✉❧❧ ❝♦♥s✐❞❡r❛✲s❡ ♦ t❡st❡✿ H

  ✿ ▼♦❞❡❧♦ ▼♦❞❢✉❧❧ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ❛❴✷

✈❡rs✉s

H

  1 ✿ ▼♦❞❡❧♦ ❛❴✷ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ▼♦❞❢✉❧❧✳ ❆♣❧✐❝❛♥❞♦ ♦ ❚❡st❡ ❞❡ ❲✐❧❦s✱ ✈❡r✐✜❝❛✲s❡ q✉❡✿

  ❃ ❛♥♦✈❛✭❛❴✷✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰

  ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✼✽ ✻✹✼ ✷ ✾✾✼✻ ✻✹✼ ✷ ✵✳✸✵✷ ✵✳✽✻ ◆❡st❡ ❝❛s♦✱ ❝♦♠♦ ♦ ♣✲✈❛❧✉❡ ♣❡❧♦ ❚❡st❡ ❞❡ ❲✐❧❦s é ❡❧❡✈❛❞♦ ❞❡✈❡ r❡❥❡✐t❛r✲s❡ ❛ ❤✐♣ót❡s❡

❞❡ q✉❡ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ❛❴✷ ❡ r❡t✐r❛✲s❡ ❞❡✜♥✐t✐✈❛♠❡♥t❡ ❛

  ❝♦✈❛r✐á✈❡❧ ❍❛❜✳ Pr♦ss❡❣✉❡✲s❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♠❛s ❛❣♦r❛ ♦ ♠♦❞❡❧♦ ❝♦♠ ♠❡♥♦s ❝♦✈❛r✐á✈❡✐s ✭❛❴✷✮

♣❛ss❛ ❛ s❡r ♦ ♠♦❞❡❧♦ ❛❝❡✐t❡ ♥♦ ♣❛ss♦ ❛♥t❡r✐♦r✱ ❞♦ q✉❛❧ ✐rá s❡r r❡t✐r❛❞❛ ❛ ♣ró①✐♠❛

❝♦✈❛r✐á✈❡❧ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛té q✉❡ s❡ ♦❜t❡♥❤❛ ✉♠ ♠♦❞❡❧♦ ❡♠ q✉❡ t♦❞❛s ❛s

❝♦✈❛r✐á✈❡✐s s❡❥❛♠ s✐❣♥✐✜❝❛t✐✈❛s✳ P❛r❛ ♥ã♦ ❛❧♦♥❣❛r ❛ ❛♥á❧✐s❡ ❞♦s ♠♦❞❡❧♦s ❡♥❝❛✐①❛❞♦s

✐♥❞✐❝❛♠✲s❡✱ ❛♣❡♥❛s ♦s ♣❛ss♦s ✐♥t❡r♠é❞✐♦s s❡♠ ❞❡♠❛s✐❛❞♦ ❞❡t❛❧❤❡✳ P❛r❛ ✉♠❛ ❛♥á❧✐s❡

♠❛✐s ❝✉✐❞❛❞❛ ❞♦s t❡st❡s r❡❧❛t✐✈♦s à ❡s❝♦❧❤❛ ❞❛s ❝♦✈❛r✐á✈❡✐s✱ ♣♦❞❡ ❝♦♥s✉❧t❛r✲s❡ ♦ ❆♥❡①♦

❃ ❛❴✸ ❁✲ ✭❛❴✷ ✲ ■❞❝❛s❛✮ ❃ ❛♥♦✈❛✭❛❴✸✱ ❛❴✷✱ t❡st ❂ ✧❈❤✐sq✧✮

  ♣✲✈❛❧✉❡ ❂ P✭❃⑤❈❤✐⑤✮❂✵✳✻✵✹✽ ❆❝❡✐t❛✲s❡ ❡①❝❧✉✐r ❛ ✈❛r✐á✈❡❧ ■❞❝❛s❛ ❡ ♣r♦ss❡❣✉❡✲s❡

  ❃ ❛❴✹ ❁✲ ✭❛❴✸ ✲ Pr♦❢✮ ❃ ❛♥♦✈❛✭❛❴✹✱ ❛❴✸✱ t❡st ❂ ✧❈❤✐sq✧✮ ♣✲✈❛❧✉❡ ❂ P✭❃⑤❈❤✐⑤✮❂ ✵✳✶✼✸

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✼✶

  ❃ ❛❴✺ ❁✲ ✭❛❴✹ ✲ ❆♥t❣✮ ❃ s✉♠♠❛r②✭❛❴✺✮

❆♣ós ❛ ❡①❝❧✉sã♦ ❞❛s ❝♦✈❛r✐á✈❡✐s ❍❛❜✱ ■❞❝❛s❛✱ Pr♦❢✱ ❆♥t❣✱ ♦❜t✐✈❡r❛♠✲s❡ ♦s s❡❣✉✐♥t❡s

r❡s✉❧t❛❞♦s✿ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✽✾❡✰✵✵ ✲✼✳✾✷❡✲✵✸ ✲✹✳✺✶❡✲✵✹ ✲✶✳✻✼❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✼❡✰✵✶ ✻✳✶✹❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✸ ■❞❛❞❡■❞♦s♦s ✲✷✳✶✺❡✰✵✵ ✽✳✼✶❡✲✵✶ ✲✷✳✹✻ ✵✳✵✶✹ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✽✺❡✰✵✵ ✸✳✶✷❡✲✵✶ ✶✷✳✸✺ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✷✼❡✰✵✵ ✻✳✽✻❡✲✵✶ ✶✳✽✺ ✵✳✵✻✺ ✳ ❉❡♣P♦✉❝♦s ✷✳✼✼❡✰✵✵ ✸✳✹✽❡✲✵✶ ✼✳✾✼ ✶✳✻❡✲✶✺ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✷✷❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✸✻ ✽✳✶❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✺✻❡✰✵✵ ✷✳✷✺❡✲✵✶ ✻✳✾✸ ✹✳✸❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✹✺❡✰✵✵ ✻✳✸✵❡✲✵✶ ✺✳✹✽ ✹✳✸❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✻✶❡✰✵✵ ✻✳✸✻❡✲✵✶ ✽✳✽✷ ❁ ✷❡✲✶✻ ✯✯✯ ❘❡♥❞❇❛✐①♦ ✶✳✼✶❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✸ ✵✳✾✼✽ ❘❡♥❞▼❡❞✐♦ ✽✳✼✻❡✰✵✵ ✻✳✶✹❡✰✵✷ ✵✳✵✶ ✵✳✾✽✾ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✺❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✷❡✲✵✺ ✶✳✻✸❡✲✵✻ ✲✻✳✷✽ ✸✳✸❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✵✼❡✰✵✵ ✸✳✻✾❡✲✵✶ ✶✸✳✼✸ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✵❡✲✵✷ ✷✳✷✷❡✲✵✸ ✶✻✳✷✸ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✺✼❡✲✵✷ ✸✳✺✷❡✲✵✸ ✲✶✺✳✽✶ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮

  ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✺✹✳✽✻ ♦♥ ✾✾✽✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✽✻✳✾ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ❃ ❛♥♦✈❛✭❛❴✺✱ ❛❴✹✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰

  Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✹ ✻✺✺ ✷ ✾✾✽✸ ✻✺✸ ✶ ✶✳✻✼ ✵✳✷✵

❖❜s❡r✈❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞♦ s♦❢t✇❛r❡ R ♣❛r❛ ♦ ♠♦❞❡❧♦✭❛❴✺✮ ✈❡r✐✜❝❛✲s❡ q✉❡ ❛ ú♥✐❝❛

❝♦✈❛r✐á✈❡❧ ❝♦♠ ♣♦✉❝❛ s✐❣♥✐✜❝â♥❝✐❛ é ❛ ✈❛r✐á✈❡❧ ❘❡♥❞✐♠❡♥t♦ ❝✉❥❛ r❛③ã♦ ♣❛r❛ s❡ ♠❛♥t❡r

♥♦ ♠♦❞❡❧♦ ✜♥❛❧ ❥á ❢♦✐ ❥✉st✐✜❝❛❞❛✳ ❆ss✐♠✱ ❝♦♥s✐❞❡r❛✲s❡ ♦ ♠♦❞❡❧♦ ❛❴✺ ❝♦♠♦ ♦ ♠♦❞❡❧♦

q✉❡ ♠❡❧❤♦r ❡①♣❧✐❝❛ ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✳

  ▼❛✐s ❝♦♥❝r❡t❛♠❡♥t❡✱ ♦ ♠❡❧❤♦r ♠♦❞❡❧♦ ❡♥❝♦♥tr❛❞♦ ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❙t❡♣✇✐s❡✲

❇❛❝❦✇❛r❞ é ❝♦♥st✐t✉í❞♦ ♣❡❧❛s ❝♦✈❛r✐á✈❡✐s ✿ ■❞❛❞❡✱ ❉❡♣✱ ❈✐✈✐❧✱ P♦st❛❧✱ ❈r❡❞✱ ❊♥tr✱ Pt♦✲

t❛❧✱ P❛❣❛s ❡ ❘❡♥❞✳ ❉❡st❛ ❢♦r♠❛ ♣♦❞❡♠♦s ❛❥✉st❛r ♦ ♠♦❞❡❧♦ ▲♦❣✐t ❛♦ ♠♦❞❡❧♦ ❛❴✺✱

❞❡s✐❣♥❛♥❞♦✲♦ ♣♦r ▼♦❞❙t♣✳

  

✼✷ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❃ ▼♦❞❙t♣

  ❖s ✈❛❧♦r❡s ❞♦ ♦✉t♣✉t ❞♦ s♦❢t✇❛r❡ R ♣❛r❛ ♦ ♠♦❞❡❧♦ ▼♦❞❙t♣ sã♦ ♦s ♠❡s♠♦s ❞❡s❝r✐t♦s ♣❡❧♦ ♦✉t♣✉t ♣❛r❛ ♦ ♠♦❞❡❧♦ ❛❴✺✳ ❙❡❧❡❝çã♦ ❞❡ ❝♦✈❛r✐á✈❡✐s ❛tr❛✈és ❞♦ ❈r✐tér✐♦ ❞❡ ■♥❢♦r♠❛çã♦ ❞❡ ❆❦❛✐❦❡

❯♠ ♦✉tr♦ ♠ét♦❞♦ ❞❡ s❡❧❡❝çã♦ ❞❡ ♠♦❞❡❧♦s é ♦ ❈r✐tér✐♦ ❞❡ ■♥❢♦r♠❛çã♦ ❞❡ ❆❦❛✐❦❡ ✭❆■❈✮

❥á ❞❡s❝r✐t♦ ❛♥t❡r✐♦r♠❡♥t❡ ♥❛ ❙❡❝çã♦ ◆❡st❛ s❡❝çã♦ r❡❝♦rr❡✲s❡ ❛ ❡st❡ ♠ét♦❞♦ ♣❛r❛

s❡❧❡❝❝✐♦♥❛r ❛s ❝♦✈❛r✐á✈❡✐s q✉❡ ♠❡❧❤♦r ❡①♣❧✐❝❛♠ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❡ ❛ss✐♠ ♦❜t❡r ♦ ♠❡✲

❧❤♦r ♠♦❞❡❧♦✳ ❖ s♦❢t✇❛r❡ R ❝♦♥té♠ ✉♠❛ ❢✉♥çã♦ q✉❡ ❡①❡❝✉t❛ t♦❞♦s ♦s ♣❛ss♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦❜t❡r ❛ ♠❡❧❤♦r ❡s❝♦❧❤❛ ❞❡ ❝♦✈❛r✐á✈❡✐s✱ ❛tr❛✈és ❞❡ ❝r✐tér✐♦ ❞❡ ❆❦❛✐❦❡✳ Pr✐♠❡✐r♦ ♣❛ss♦ ❆■❈✿ ❙t❛rt✿ ❆■❈❂✻✾✹✳✾ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰

  ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ✲ ❍❛❜ ✷ ✻✹✼✳✷✵ ✻✾✶✳✷✵ ✲ ■❞❝❛s❛ ✷ ✻✹✼✳✾✻ ✻✾✶✳✾✻ ✲ Pr♦❢ ✸ ✻✺✶✳✻✶ ✻✾✸✳✻✶ ✲ ❆♥t❣ ✶ ✻✹✽✳✽✹ ✻✾✹✳✽✹ ❁♥♦♥❡❃ ✻✹✻✳✾✵ ✻✾✹✳✾✵ ✲ ❈r❡❞ ✶ ✻✾✹✳✹✶ ✼✹✵✳✹✶ ✲ ❈✐✈✐❧ ✷ ✼✶✾✳✶✻ ✼✻✸✳✶✻ ✲ ❉❡♣ ✷ ✼✸✼✳✵✵ ✼✽✶✳✵✵ ✲ P♦st❛❧ ✷ ✽✹✻✳✼✸ ✽✾✵✳✼✸ ✲ ■❞❛❞❡ ✷ ✽✻✶✳✹✸ ✾✵✺✳✹✸ ✲ ❊♥tr ✶ ✶✵✹✽✳✻✵ ✶✵✾✹✳✻✵ ✲ Pt♦t❛❧ ✶ ✶✹✶✻✳✻✵ ✶✹✻✷✳✻✵ ✲ ❘❡♥❞ ✸ ✶✹✽✸✳✺✹ ✶✺✷✺✳✺✹ ✲ P❛❣❛s ✶ ✶✼✾✶✳✵✻ ✶✽✸✼✳✵✻ ❘❡t✐r❛ ❛ ❝♦✈❛r✐á✈❡❧ ❍❛❜ ❡ ♦❜té♠✿ ❙t❡♣✿ ❆■❈❂✻✾✶✳✷✵ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰

  ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❆■❈❂✻✽✽✳✷✱ ❞❡♣♦✐s

  ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ✈❛✐ r❡t✐r❛r ❛ ❝♦✈❛r✐á✈❡❧ ■❞❝❛s❛ ❞♦♥❞❡ r❡s✉❧t❛ ✉♠

❛ ❝♦✈❛r✐á✈❡❧ Pr♦❢ ♦❜t❡♥❞♦ ❆■❈❂✻✽✼✳✶✾✱ s❡❣✉✐❞❛ ❞❛ ❝♦✈❛r✐á✈❡❧ ❆♥t❣ ♦❜t❡♥❞♦✲s❡ ❆■❈❂✻✽✻✳✽✻

❞♦♥❞❡ r❡s✉❧t❛ ♦ ♠♦❞❡❧♦ ✜♥❛❧✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s

  ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ❁♥♦♥❡❃ ✻✺✹✳✽✻ ✻✽✻✳✽✻ ✰ ❆♥t❣ ✶ ✻✺✸✳✶✾ ✻✽✼✳✶✾ ✰ Pr♦❢ ✸ ✻✺✵✳✵✽ ✻✽✽✳✵✽ ✰ ■❞❝❛s❛ ✷ ✻✺✸✳✼✺ ✻✽✾✳✼✺ ✰ ❍❛❜ ✷ ✻✺✹✳✺✶ ✻✾✵✳✺✶ ✲ ❈r❡❞ ✶ ✼✵✶✳✷✷ ✼✸✶✳✷✷ ✲ ❈✐✈✐❧ ✷ ✼✷✸✳✼✼ ✼✺✶✳✼✼

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✼✸

  ✲ ❉❡♣ ✷ ✼✹✷✳✹✼ ✼✼✵✳✹✼ ✲ P♦st❛❧ ✷ ✽✺✶✳✻✻ ✽✼✾✳✻✻ ✲ ■❞❛❞❡ ✷ ✽✻✼✳✽✽ ✽✾✺✳✽✽ ✲ ❊♥tr ✶ ✶✵✺✻✳✺✾ ✶✵✽✻✳✺✾ ✲ Pt♦t❛❧ ✶ ✶✹✷✷✳✶✷ ✶✹✺✷✳✶✷ ✲ ❘❡♥❞ ✸ ✶✹✽✼✳✹✷ ✶✺✶✸✳✹✷ ✲ P❛❣❛s ✶ ✶✼✾✽✳✻✶ ✶✽✷✽✳✻✶ ❘❡s✐❞✉❛❧ ❉❡✈✐❛♥❝❡✿ ✻✺✹✳✾ ❆■❈✿ ✻✽✻✳✾

  ◆♦t❡✲s❡ q✉❡ ❛s ✈❛r✐á✈❡✐s q✉❡ ♦ ♠ét♦❞♦ ❆■❈ ♦♣t❛ ♣♦r r❡t✐r❛r sã♦ ❛s ♠❡s♠❛s q✉❡ ❢♦r❛♠ r❡t✐r❛❞❛s ♣❡❧♦ ♠ét♦❞♦ ❙t❡♣✇✐s❡✳ ❉❡s✐❣♥❛♥❞♦ ♦ ♠♦❞❡❧♦ ❞❡✜♥✐❞♦ ♣❡❧♦ ❝r✐tér✐♦ ❆■❈ ♣♦r ▼♦❞❆■❈ ❡ ❢❛③❡♥❞♦ ♦ s❡✉ ❛❥✉s✲ t❛♠❡♥t♦ t❡♠✲s❡✿

  ❃ ▼♦❞❆■❈ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✽✾❡✰✵✵ ✲✼✳✾✷❡✲✵✸ ✲✹✳✺✶❡✲✵✹ ✲✶✳✻✼❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✼❡✰✵✶ ✻✳✶✹❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✸ ■❞❛❞❡■❞♦s♦s ✲✷✳✶✺❡✰✵✵ ✽✳✼✶❡✲✵✶ ✲✷✳✹✻ ✵✳✵✶✹ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✽✺❡✰✵✵ ✸✳✶✷❡✲✵✶ ✶✷✳✸✺ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✷✼❡✰✵✵ ✻✳✽✻❡✲✵✶ ✶✳✽✺ ✵✳✵✻✺ ✳ ❉❡♣P♦✉❝♦s ✷✳✼✼❡✰✵✵ ✸✳✹✽❡✲✵✶ ✼✳✾✼ ✶✳✻❡✲✶✺ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✷✷❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✸✻ ✽✳✶❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✺✻❡✰✵✵ ✷✳✷✺❡✲✵✶ ✻✳✾✸ ✹✳✸❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✹✺❡✰✵✵ ✻✳✸✵❡✲✵✶ ✺✳✹✽ ✹✳✸❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✻✶❡✰✵✵ ✻✳✸✻❡✲✵✶ ✽✳✽✷ ❁ ✷❡✲✶✻ ✯✯✯ ❘❡♥❞❇❛✐①♦ ✶✳✼✶❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✸ ✵✳✾✼✽ ❘❡♥❞▼❡❞✐♦ ✽✳✼✻❡✰✵✵ ✻✳✶✹❡✰✵✷ ✵✳✵✶ ✵✳✾✽✾ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✺❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✷❡✲✵✺ ✶✳✻✸❡✲✵✻ ✲✻✳✷✽ ✸✳✸❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✵✼❡✰✵✵ ✸✳✻✾❡✲✵✶ ✶✸✳✼✸ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✵❡✲✵✷ ✷✳✷✷❡✲✵✸ ✶✻✳✷✸ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✺✼❡✲✵✷ ✸✳✺✷❡✲✵✸ ✲✶✺✳✽✶ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮

  ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✺✹✳✽✻ ♦♥ ✾✾✽✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✽✻✳✽✻ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾

  ❊♠ ❛♠❜♦s ♦s ♠ét♦❞♦s✱ ♦ ❚❡st❡ ❞❡ ❲❛❧❞ s✉❣❡r❡ q✉❡ ❛ ❝❛t❡❣♦r✐❛ ✏▼✉✐t♦s✑ ❞❛ ❝♦✈❛r✐á✲

✈❡❧ ❉❡♣ ✭◆➸ ❞❡ ❉❡♣❡♥❞❡♥t❡s✮ ♥ã♦ é s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❞✐❢❡r❡♥t❡ ❞❛ ❝❛t❡❣♦r✐❛ ✏❆❧❣✉♥s✑✳

❖♣t♦✉✲s❡✱ ♥♦ ❡♥t❛♥t♦✱ ♣♦r ♠❛♥t❡r ❛s ❝❛t❡❣♦r✐❛s s❡♣❛r❛❞❛s ♣♦rq✉❡ s❡ ❝♦♥s✐❞❡r♦✉ ✐♥✐✲

❝✐❛❧♠❡♥t❡✱ ♥❛ s✐♠✉❧❛çã♦ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✱ q✉❡ ♦s ❝❧✐❡♥t❡s ♥❛ ❝❛t❡❣♦r✐❛ ✏❏♦✈❡♥s✑

❞❛ ❝♦✈❛r✐á✈❡❧ ■❞❛❞❡ só ♣♦❞❡r✐❛♠ t❡r ✏P♦✉❝♦s✑ ♦✉ ✏❆❧❣✉♥s✑ ❞❡♣❡♥❞❡♥t❡s ❧♦❣♦ ❛❣r✉♣❛r ❛

❝❛t❡❣♦r✐❛ ✏❆❧❣✉♥s✑ ❝♦♠ ❛ ❝❛t❡❣♦r✐❛ ✏▼✉✐t♦s✑ ✐r✐❛ ❝♦♥tr❛ ♦ ♣r❡ss✉♣♦st♦ ✐♥✐❝✐❛❧ ❞❛ s✐♠✉✲

❧❛çã♦✳ ❃ ❛♥♦✈❛✭▼♦❞❆■❈✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮

  

✼✹ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰

  Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✹ ✻✺✹✳✽✻ ✷ ✾✾✼✻ ✻✹✻✳✾✵ ✽ ✼✳✾✻✸✺ ✵✳✹✸✼ ❖❜s❡r✈❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞❛ ❛♥♦✈❛ ❡♥tr❡ ♦ ▼♦❞❡❧♦ ❆■❈ ❡ ♦ ▼♦❞❡❧♦ ❈♦♠♣❧❡t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ▼♦❞❡❧♦ ❆■❈ ✭▼♦❞❆■❈✮ é s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♠❡❧❤♦r q✉❡ ▼♦❞❢✉❧❧✳ ◆♦ ❡♥t❛♥t♦ ❝♦♠♦ ❛ ❝♦✈❛r✐á✈❡❧ ❘❡♥❞ ♥ã♦ é s✐❣♥✐✜❝❛t✐✈❛ ♣❡❧♦ ❚❡st❡ ❞❡ ❲❛❧❞✱ ✈❛♠♦s

r❡t✐rá✲❧❛ ❡ ❛♥❛❧✐s❛r ❛ ❛♥♦✈❛ ❞♦s ❞♦✐s ♠♦❞❡❧♦s✳ ❙❡❥❛ ▼♦❞❆■❈✷ ♦ ▼♦❞❡❧♦ ❆■❈ s❡♠ ❛

  

✈❛r✐á✈❡❧ ❘❡♥❞✱ ♦✉ s❡❥❛✱ ♦ ♠♦❞❡❧♦ ▼♦❞❆■❈✷ ❡♥❝❛✐①❛❞♦ ♥♦ ♠♦❞❡❧♦ ▼♦❞❆■❈ ❡ r❡❛❧✐③❛✲s❡ ♦

t❡st❡✿ H ✿ ▼♦❞❡❧♦ ▼♦❞❆■❈ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ▼♦❞❆■❈✷

  

✈❡rs✉s

H

  1 ✿ ▼♦❞❡❧♦ ▼♦❞❆■❈✷ é ♠❡❧❤♦r q✉❡ ♦ ♠♦❞❡❧♦ ▼♦❞❆■❈✳

  ❃ ▼♦❞❆■❈✷ ❁✲✭▼♦❞❆■❈ ✲ ❘❡♥❞✮ ❃ ❛♥♦✈❛✭▼♦❞❆■❈✷✱ ▼♦❞❆■❈✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆■❈✿ ✶✺✶✸✳✹ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s

  ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮ ✶ ✾✾✽✼ ✶✹✽✼✳✹✷ ✷ ✾✾✽✹ ✻✺✺✳✽✻ ✸ ✽✸✷✳✺✻ ❁ ✷✳✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶

  ❱❡r✐✜❝❛✲s❡ ♥♦✈❛♠❡♥t❡ q✉❡✱ ❡♠❜♦r❛ ❛ ✈❛r✐á✈❡❧ ❘❡♥❞ ♥ã♦ s❡❥❛ s✐❣♥✐✜❝❛t✐✈❛ ♣❡❧♦ ❚❡st❡

❞❡ ❲❛❧❞✱ ♦ ♠♦❞❡❧♦ ❝♦♠ ❡st❛ ✈❛r✐á✈❡❧ ✐♥❝❧✉s❛ é s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♠❡❧❤♦r✳ ▲♦❣♦✱ ♥♦

♠♦❞❡❧♦ ✜♥❛❧ ♦♣t♦✉✲s❡ ♣♦r ✐♥❝❧✉✐r ❛ ✈❛r✐á✈❡❧ ❘❡♥❞✱ ❛té ♣♦rq✉❡✱ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛

♣rát✐❝♦ ♣❛r❡❝❡✲♥♦s q✉❡ ❛ ✐♥st✐t✉✐çã♦ ❜❛♥❝ár✐❛ ❝♦♥s✐❞❡r❛rá ♦ ✈❛❧♦r ❞♦ r❡♥❞✐♠❡♥t♦ ❞❡ ✉♠

❝❧✐❡♥t❡ ❝♦♠♦ ✉♠ ❢❛❝t♦r r❡❧❡✈❛♥t❡ ♥❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦✳ ❆ r❛③ã♦ q✉❡ ❧❡✈❛ ❛ ❡st❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞❡❝✐sã♦ ❡♥tr❡ ♦s ❞✐❢❡r❡♥t❡s t❡st❡s ❞❡✈❡✲s❡✱ ❡♠

♣r✐♥❝í♣✐♦✱ ❛♦ ❢❛❝t♦ ❞❛ ✈❛r✐á✈❡❧ ❘❡♥❞ t❡r s✐❞♦ s✐♠✉❧❛❞❛ ❝♦♠♦ ✉♠❛ ✈❛r✐á✈❡❧ ❝❛t❡❣ór✐❝❛ ❡

♥ã♦ ✉♠❛ ✈❛r✐á✈❡❧ ♥✉♠ér✐❝❛ à q✉❛❧✱ ♥♦ ❡♥t❛♥t♦✱ s❡ ❛tr✐❜✉✐✉ t❡♠ ♣❡s♦ ❜❛st❛♥t❡ ❡❧❡✈❛❞♦

✭✈❡r ♥♦✈❛♠❡♥t❡ ❚❛❜❡❧❛ ❞❡✈✐❞♦ às r❛③õ❡s r❡❢❡r✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡✳

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✼✺

  2 R ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦ ✭ ¯ ✮

❖ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦ ❞❡s❝r✐t♦ ♥❛ ❙❡❝çã♦ s❡rá ❝❛❧❝✉❧❛❞♦✱ ♥❡st❛

s❡❝çã♦✱ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❡st✐♠❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❞❡ ❢♦r♠❛ ❛ ❛♥❛❧✐s❛r ❛ q✉❛❧✐❞❛❞❡ ❞♦s

❛❥✉st❛♠❡♥t♦s ❝♦♠ ❜❛s❡ ♥♦✉tr♦ ❝r✐tér✐♦✳

  ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦✿ SQR/(T − n − 1)

  2 ¯

R = 1 − .

  ✭✸✳✶✾✮ SQT /(T − 1) ❆ ❚❛❜❡❧❛ ❝♦♥té♠ ♦s ✈❛❧♦r❡s ❞♦s ❈♦❡✜❝✐❡♥t❡s ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦s ❡ ❞♦s ❈r✐tér✐♦s ❞❡ ❆❦❛✐❦❡ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❛❥✉st❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✳

  2 ¯ R

  ▼♦❞❡❧♦ ❆■❈

▼♦❞❡❧♦ ❈♦♠♣❧❡t♦ ✵✳✼✼✺✼✻✸✷ ✻✾✹✳✽✾✻✼

  

▼♦❞❡❧♦ ❆■❈ ✭♦✉ ❙t❡♣✇✐s❡✮ ✵✳✼✼✷✷✽✺✻ ✻✽✻✳✽✻✵✷

▼♦❞❡❧♦ ❆■❈ s❡♠ ❝♦✈❛r✐á✈❡❧ ❘❡♥❞✐♠❡♥t♦ ✵✳✹✽✷✾✸✻✶ ✶✺✶✸✳✹✶✽

❚❛❜❡❧❛ ✸✳✻✿ ❈♦❡✜❝✐❡♥t❡s ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦s ❡ ❈r✐tér✐♦s ❞❡ ❆❦❛✐❦❡

❖❜s❡r✈❛♥❞♦ ❛ ❚❛❜❡❧❛ ❝♦♥❝❧✉✐✲s❡ q✉❡ ♦ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦

  

é ♠❡❧❤♦r ♣❛r❛ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡t♦ ❞♦ q✉❡ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❡st✐♠❛❞♦s ♣❡❧♦s ♠ét♦❞♦s

❙t❡♣✇✐s❡ ❡ ❈r✐tér✐♦ ❞❡ ❆❦❛✐❦❡ ✭♦s ♠♦❞❡❧♦s ❡st✐♠❛❞♦s ♣♦r ❡st❡s ❞♦✐s ♠ét♦❞♦s ❝♦♥tê♠ ❛s

♠❡s♠❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s✮✱ ❡♠❜♦r❛ ❝♦♠ ✉♠❛ ❞✐❢❡r❡♥ç❛ ♣❡q✉❡♥❛ ❝♦♠♣❛r❛❞❛ ❝♦♠ ♦

❢❛❝t♦ ❞❡ s❡ t❡r ♦❜t✐❞♦ ✉♠ ♠♦❞❡❧♦ ♠❛✐s s✐♠♣❧❡s✳ ❖ ✈❛❧♦r ❞♦ ❈r✐tér✐♦ ❞❡ ■♥❢♦r♠❛çã♦ ❞❡

❆❦❛✐❦❡ r❡✈❡❧❛ q✉❡ ❡st❡s ú❧t✐♠♦s ♠♦❞❡❧♦s sã♦ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♠❡❧❤♦r❡s q✉❡ ♦ ♠♦❞❡❧♦

❝♦♠♣❧❡t♦✳ ❊♠ r❡❧❛çã♦ ❛♦ ♠♦❞❡❧♦ ❡st✐♠❛❞♦ s❡♠ ❛ ❝♦✈❛r✐á✈❡❧ ❘❡♥❞✐♠❡♥t♦✱ ♦❜s❡r✈❛✲s❡

✉♠❛ ❞✐❢❡r❡♥ç❛ ❝♦♥s✐❞❡rá✈❡❧✱ t❛♥t♦ ♣❛r❛ ♦ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦ ❝♦♠♦

♣❛r❛ ♦ ❈r✐tér✐♦ ❞❡ ❆❦❛✐❦❡ ❡ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❡st❡ ♠♦❞❡❧♦ é ❜❛st❛♥t❡ ♣✐♦r q✉❡ ♦s r❡st❛♥t❡s✳

❈♦♥✜r♠❛çã♦ ❙❡❧❡❝çã♦ ❞❡ ❱❛r✐á✈❡✐s

P❛r❛ ❝♦♥✜r♠❛r ❛ s❡❧❡❝çã♦ ❞❡ ✈❛r✐á✈❡✐s ♣❡❧❛s q✉❛✐s s❡ ♦♣t♦✉✱ ✉t✐❧✐③❛r❛♠✲s❡ ❞✉❛s ❢✉♥çõ❡s

❞♦ s♦❢t✇❛r❡ R✳

  2 ❆tr❛✈és ❞♦ t❡st❡ χ ✱ ✉♠❛ ❞❛s ❢✉♥çõ❡s ❛♥❛❧✐s❛ q✉❛✐s ❛s ✈❛r✐á✈❡✐s ❞♦ ♠♦❞❡❧♦ q✉❡ sã♦

♠❛✐s s✐❣♥✐✜❝❛t✐✈❛s✳ ❖❜t✐✈❡r❛♠✲s❡ ❛s ♠❡s♠❛s ❝♦✈❛r✐á✈❡✐s q✉❡ ❥á ❤❛✈✐❛♠ s✐❞♦ ❡s❝♦❧❤✐❞❛s

  ❛tr❛✈és ♥♦s ♠ét♦❞♦s ❛♥t❡r✐♦r❡s ❝♦♥❢♦r♠❡ s❡ ♣♦❞❡ ✈❡r✐✜❝❛r ♥♦s ♦✉t♣✉ts ❞♦ ❆♥❡①♦ ❈♦♥s✐❞❡r♦✉✲s❡ ❛✐♥❞❛ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✐♥t❡r❛❝çõ❡s ❡♥tr❡ ❛❧❣✉♠❛s ❞❛s

❝♦✈❛r✐á✈❡✐s✳ P❛r❛ ❛♥❛❧✐s❛r ❡✈❡♥t✉❛✐s r❡❧❛çõ❡s ❞❡ ✷➟ ♦r❞❡♠ ❡♥tr❡ ❝♦✈❛r✐á✈❡✐s q✉❡ ♣✉❞❡s✲

s❡♠ ♠❡❧❤♦r❛r ♦ ♠♦❞❡❧♦ ✉t✐❧✐③♦✉✲s❡ ❛ ♦✉tr❛ ❢✉♥çã♦✳ ▼♦❞❡❧✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr

  ✰ Pt♦t❛❧ ✰ P❛❣❛s

  ✼✻ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦ ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ▲❘❚ Pr✭❈❤✐✮ ❁♥♦♥❡❃ ✻✹✻✳✾✵ ✻✾✹✳✾✵

  ■❞❛❞❡✿❉❡♣ ✸ ✻✹✺✳✻✽ ✻✾✾✳✻✽ ✶✳✷✶✼✼ ✵✳✼✹✽✼✼✸ ■❞❛❞❡✿❍❛❜ ✹ ✻✹✷✳✷✷ ✻✾✽✳✷✷ ✹✳✻✽✵✼ ✵✳✸✷✶✻✺✼ ■❞❛❞❡✿■❞❝❛s❛ ✹ ✻✹✺✳✵✽ ✼✵✶✳✵✽ ✶✳✽✶✾✻ ✵✳✼✻✽✽✾✺ ■❞❛❞❡✿❈✐✈✐❧ ✹ ✻✹✺✳✼✻ ✼✵✶✳✼✻ ✶✳✶✸✸✸ ✵✳✽✽✽✾✹✻ ■❞❛❞❡✿P♦st❛❧ ✹ ✻✹✸✳✽✸ ✻✾✾✳✽✸ ✸✳✵✻✾✻ ✵✳✺✹✻✷✹✹ ■❞❛❞❡✿Pr♦❢ ✻ ✻✹✹✳✼✺ ✼✵✹✳✼✺ ✷✳✶✹✺✻ ✵✳✾✵✺✽✸✺ ■❞❛❞❡✿❆♥t❣ ✷ ✻✹✻✳✹✾ ✻✾✽✳✹✾ ✵✳✹✵✹✶ ✵✳✽✶✼✵✺✻ ■❞❛❞❡✿❘❡♥❞ ✻ ✻✹✻✳✻✶ ✼✵✻✳✻✶ ✵✳✷✽✽✽ ✵✳✾✾✾✺✺✵ ■❞❛❞❡✿❈r❡❞ ✷ ✻✹✵✳✵✶ ✻✾✷✳✵✶ ✻✳✽✽✸✹ ✵✳✵✸✷✵✶✵ ✯ ■❞❛❞❡✿❊♥tr ✷ ✻✹✻✳✸✶ ✻✾✽✳✸✶ ✵✳✺✽✹✾ ✵✳✼✹✻✹✷✶ ■❞❛❞❡✿Pt♦t❛❧ ✷ ✻✹✻✳✵✸ ✻✾✽✳✵✸ ✵✳✽✻✷✻ ✵✳✻✹✾✻✼✾ ■❞❛❞❡✿P❛❣❛s ✷ ✻✹✺✳✶✷ ✻✾✼✳✶✷ ✶✳✼✼✾✵ ✵✳✹✶✵✽✺✾ ❉❡♣✿❍❛❜ ✹ ✻✸✼✳✶✶ ✻✾✸✳✶✶ ✾✳✼✽✸✽ ✵✳✵✹✹✷✸✶ ✯ ❉❡♣✿■❞❝❛s❛ ✹ ✻✹✹✳✹✻ ✼✵✵✳✹✻ ✷✳✹✸✼✵ ✵✳✻✺✺✾✺✶ ❉❡♣✿❈✐✈✐❧ ✹ ✻✹✸✳✼✼ ✻✾✾✳✼✼ ✸✳✶✷✽✾ ✵✳✺✸✻✹✽✾ ❉❡♣✿P♦st❛❧ ✹ ✻✹✹✳✺✶ ✼✵✵✳✺✶ ✷✳✸✽✽✽ ✵✳✻✻✹✻✹✹ ❉❡♣✿Pr♦❢ ✻ ✻✹✷✳✼✻ ✼✵✷✳✼✻ ✹✳✶✸✹✾ ✵✳✻✺✽✹✸✸ ❉❡♣✿❆♥t❣ ✷ ✻✹✻✳✺✻ ✻✾✽✳✺✻ ✵✳✸✸✶✼ ✵✳✽✹✼✶✺✽ ❉❡♣✿❘❡♥❞ ✻ ✻✹✺✳✽✽ ✼✵✺✳✽✽ ✶✳✵✶✼✺ ✵✳✾✽✹✾✹✶ ❉❡♣✿❈r❡❞ ✷ ✻✹✹✳✸✺ ✻✾✻✳✸✺ ✷✳✺✹✺✻ ✵✳✷✽✵✵✹✶ ❉❡♣✿❊♥tr ✷ ✻✹✸✳✷✼ ✻✾✺✳✷✼ ✸✳✻✷✸✺ ✵✳✶✻✸✸✻✽ ❉❡♣✿Pt♦t❛❧ ✷ ✻✹✶✳✺✸ ✻✾✸✳✺✸ ✺✳✸✻✾✸ ✵✳✵✻✽✷✹✺ ✳ ❉❡♣✿P❛❣❛s ✷ ✻✹✵✳✻✹ ✻✾✷✳✻✹ ✻✳✷✺✾✺ ✵✳✵✹✸✼✷✾ ✯ ❍❛❜✿■❞❝❛s❛ ✹ ✻✹✷✳✾✾ ✻✾✽✳✾✾ ✸✳✾✵✹✶ ✵✳✹✶✾✶✹✹ ❍❛❜✿❈✐✈✐❧ ✹ ✻✸✸✳✶✵ ✻✽✾✳✶✵ ✶✸✳✼✾✹✼ ✵✳✵✵✼✾✽✵ ✯✯ ❍❛❜✿P♦st❛❧ ✹ ✻✹✷✳✾✾ ✻✾✽✳✾✾ ✸✳✾✵✷✵ ✵✳✹✶✾✹✸✵ ❍❛❜✿Pr♦❢ ✻ ✻✸✽✳✵✷ ✻✾✽✳✵✷ ✽✳✽✼✼✸ ✵✳✶✽✵✺✾✺ ❍❛❜✿❆♥t❣ ✷ ✻✹✹✳✾✷ ✻✾✻✳✾✷ ✶✳✾✽✵✼ ✵✳✸✼✶✹✹✷ ❍❛❜✿❘❡♥❞ ✻ ✻✹✸✳✽✹ ✼✵✸✳✽✹ ✸✳✵✺✸✺ ✵✳✽✵✷✵✾✾ ❍❛❜✿❈r❡❞ ✷ ✻✹✻✳✷✶ ✻✾✽✳✷✶ ✵✳✻✽✼✶ ✵✳✼✵✾✷✻✻ ❍❛❜✿❊♥tr ✷ ✻✹✹✳✺✺ ✻✾✻✳✺✺ ✷✳✸✹✹✶ ✵✳✸✵✾✼✷✼ ❍❛❜✿Pt♦t❛❧ ✷ ✻✹✺✳✷✼ ✻✾✼✳✷✼ ✶✳✻✷✸✷ ✵✳✹✹✹✶✹✼ ❍❛❜✿P❛❣❛s ✷ ✻✹✸✳✷✹ ✻✾✺✳✷✹ ✸✳✻✺✾✻ ✵✳✶✻✵✹✹✸ ■❞❝❛s❛✿❈✐✈✐❧ ✹ ✻✹✷✳✼✺ ✻✾✽✳✼✺ ✹✳✶✹✸✷ ✵✳✸✽✻✾✼✸ ■❞❝❛s❛✿P♦st❛❧ ✹ ✻✹✸✳✸✹ ✻✾✾✳✸✹ ✸✳✺✺✸✹ ✵✳✹✻✾✽✵✷ ■❞❝❛s❛✿Pr♦❢ ✻ ✻✹✵✳✹✷ ✼✵✵✳✹✷ ✻✳✹✼✹✷ ✵✳✸✼✷✷✶✶ ■❞❝❛s❛✿❆♥t❣ ✷ ✻✹✺✳✾✼ ✻✾✼✳✾✼ ✵✳✾✷✺✵ ✵✳✻✷✾✼✵✸ ■❞❝❛s❛✿❘❡♥❞ ✻ ✻✹✸✳✾✵ ✼✵✸✳✾✵ ✷✳✾✾✺✺ ✵✳✽✵✾✹✵✽ ■❞❝❛s❛✿❈r❡❞ ✷ ✻✹✻✳✵✷ ✻✾✽✳✵✷ ✵✳✽✼✶✼ ✵✳✻✹✻✼✵✸ ■❞❝❛s❛✿❊♥tr ✷ ✻✹✺✳✸✷ ✻✾✼✳✸✷ ✶✳✺✼✶✽ ✵✳✹✺✺✼✵✵

  ■❞❝❛s❛✿Pt♦t❛❧ ✷ ✻✹✻✳✻✶ ✻✾✽✳✻✶ ✵✳✷✽✽✹ ✵✳✽✻✺✼✶✺ ■❞❝❛s❛✿P❛❣❛s ✷ ✻✹✻✳✵✺ ✻✾✽✳✵✺ ✵✳✽✹✾✹ ✵✳✻✺✸✾✺✼ ❈✐✈✐❧✿P♦st❛❧ ✹ ✻✸✻✳✶✽ ✻✾✷✳✶✽ ✶✵✳✼✶✻✾ ✵✳✵✷✾✾✸✼ ✯ ❈✐✈✐❧✿Pr♦❢ ✻ ✻✹✶✳✷✹ ✼✵✶✳✷✹ ✺✳✻✺✺✸ ✵✳✹✻✷✽✽✹ ❈✐✈✐❧✿❆♥t❣ ✷ ✻✹✻✳✵✽ ✻✾✽✳✵✽ ✵✳✽✷✵✵ ✵✳✻✻✸✻✹✻ ❈✐✈✐❧✿❘❡♥❞ ✻ ✻✹✹✳✻✵ ✼✵✹✳✻✵ ✷✳✷✾✹✺ ✵✳✽✾✵✼✷✹ ❈✐✈✐❧✿❈r❡❞ ✷ ✻✹✺✳✺✵ ✻✾✼✳✺✵ ✶✳✹✵✵✸ ✵✳✹✾✻✺✶✶ ❈✐✈✐❧✿❊♥tr ✷ ✻✹✸✳✽✵ ✻✾✺✳✽✵ ✸✳✶✵✵✸ ✵✳✷✶✷✷✶✼ ❈✐✈✐❧✿Pt♦t❛❧ ✷ ✻✹✸✳✽✷ ✻✾✺✳✽✷ ✸✳✵✽✵✹ ✵✳✷✶✹✸✸✻ ❈✐✈✐❧✿P❛❣❛s ✷ ✻✹✺✳✾✺ ✻✾✼✳✾✺ ✵✳✾✹✾✹ ✵✳✻✷✷✵✼✵ P♦st❛❧✿Pr♦❢ ✻ ✻✹✺✳✶✺ ✼✵✺✳✶✺ ✶✳✼✹✻✵ ✵✳✾✹✶✺✶✷ P♦st❛❧✿❆♥t❣ ✷ ✻✹✺✳✵✸ ✻✾✼✳✵✸ ✶✳✽✻✻✽ ✵✳✸✾✸✷✶✽ P♦st❛❧✿❘❡♥❞ ✻ ✻✹✷✳✻✼ ✼✵✷✳✻✼ ✹✳✷✸✵✻ ✵✳✻✹✺✹✾✼ P♦st❛❧✿❈r❡❞ ✷ ✻✹✻✳✺✺ ✻✾✽✳✺✺ ✵✳✸✹✹✶ ✵✳✽✹✶✾✷✺ P♦st❛❧✿❊♥tr ✷ ✻✹✻✳✸✵ ✻✾✽✳✸✵ ✵✳✺✾✸✹ ✵✳✼✹✸✷✽✵ P♦st❛❧✿Pt♦t❛❧ ✷ ✻✹✻✳✵✶ ✻✾✽✳✵✶ ✵✳✽✽✽✺ ✵✳✻✹✶✸✷✵ P♦st❛❧✿P❛❣❛s ✷ ✻✹✷✳✻✻ ✻✾✹✳✻✻ ✹✳✷✹✶✵ ✵✳✶✶✾✾✼✹ Pr♦❢✿❆♥t❣ ✸ ✻✹✺✳✵✻ ✻✾✾✳✵✻ ✶✳✽✸✺✷ ✵✳✻✵✼✷✾✺ Pr♦❢✿❘❡♥❞ ✾ ✻✹✹✳✹✻ ✼✶✵✳✹✻ ✷✳✹✹✵✷ ✵✳✾✽✷✹✹✾ Pr♦❢✿❈r❡❞ ✸ ✻✹✶✳✵✼ ✻✾✺✳✵✼ ✺✳✽✷✾✶ ✵✳✶✷✵✷✷✽ Pr♦❢✿❊♥tr ✸ ✻✹✻✳✵✹ ✼✵✵✳✵✹ ✵✳✽✺✶✼ ✵✳✽✸✼✵✻✼ Pr♦❢✿Pt♦t❛❧ ✸ ✻✹✺✳✻✻ ✻✾✾✳✻✻ ✶✳✷✸✷✶ ✵✳✼✹✺✸✶✻ Pr♦❢✿P❛❣❛s ✸ ✻✹✺✳✸✸ ✻✾✾✳✸✸ ✶✳✺✻✼✾ ✵✳✻✻✻✻✾✾ ❆♥t❣✿❘❡♥❞ ✸ ✻✹✺✳✽✸ ✻✾✾✳✽✸ ✶✳✵✻✺✶ ✵✳✼✽✺✺✵✶ ❆♥t❣✿❈r❡❞ ✶ ✻✹✻✳✶✸ ✻✾✻✳✶✸ ✵✳✼✻✽✻ ✵✳✸✽✵✻✻✺ ❆♥t❣✿❊♥tr ✶ ✻✹✻✳✹✽ ✻✾✻✳✹✽ ✵✳✹✶✻✶ ✵✳✺✶✽✾✵✽ ❆♥t❣✿Pt♦t❛❧ ✶ ✻✹✷✳✽✹ ✻✾✷✳✽✹ ✹✳✵✺✽✷ ✵✳✵✹✸✾✺✼ ✯ ❆♥t❣✿P❛❣❛s ✶ ✻✹✷✳✷✾ ✻✾✷✳✷✾ ✹✳✻✵✸✽ ✵✳✵✸✶✾✵✶ ✯ ❘❡♥❞✿❈r❡❞ ✸ ✻✹✸✳✻✶ ✻✾✼✳✻✶ ✸✳✷✽✷✽ ✵✳✸✺✵✵✹✺ ❘❡♥❞✿❊♥tr ✸ ✻✹✻✳✽✺ ✼✵✵✳✽✺ ✵✳✵✺✵✾ ✵✳✾✾✻✾✽✽ ❘❡♥❞✿Pt♦t❛❧ ✸ ✻✹✻✳✽✽ ✼✵✵✳✽✽ ✵✳✵✶✸✹ ✵✳✾✾✾✺✽✽ ❘❡♥❞✿P❛❣❛s ✸ ✻✹✺✳✵✵ ✻✾✾✳✵✵ ✶✳✽✾✹✽ ✵✳✺✾✹✺✸✸ ❈r❡❞✿❊♥tr ✶ ✻✹✻✳✻✼ ✻✾✻✳✻✼ ✵✳✷✷✹✽ ✵✳✻✸✺✸✾✻ ❈r❡❞✿Pt♦t❛❧ ✶ ✻✹✶✳✸✽ ✻✾✶✳✸✽ ✺✳✺✶✹✵ ✵✳✵✶✽✽✻✺ ✯ ❈r❡❞✿P❛❣❛s ✶ ✻✸✾✳✵✷ ✻✽✾✳✵✷ ✼✳✽✼✼✶ ✵✳✵✵✺✵✵✼ ✯✯ ❊♥tr✿Pt♦t❛❧ ✶ ✻✹✹✳✾✺ ✻✾✹✳✾✺ ✶✳✾✹✸✶ ✵✳✶✻✸✸✸✺ ❊♥tr✿P❛❣❛s ✶ ✻✹✺✳✼✷ ✻✾✺✳✼✷ ✶✳✶✼✻✸ ✵✳✷✼✽✶✶✵ Pt♦t❛❧✿P❛❣❛s ✶ ✻✸✵✳✻✻ ✻✽✵✳✻✻ ✶✻✳✷✹✵✾ ✺✳✺✼✽❡✲✵✺ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺

  ✬✳✬ ✵✳✶ ✬ ✬ ✶ ❆♥❛❧✐s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ♦ t❡st❡ ❡❢❡❝t✉❛❞♦ ✐♥❞✐❝❛ ❝♦♠♦ s❡♥❞♦ ❜❛s✲ t❛♥t❡ s✐❣♥✐✜❝❛t✐✈❛✱ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ❝♦✈❛r✐á✈❡✐s P❛❣❛s ❡ Pt♦t❛❧✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ ♦

  ♠♦❞❡❧♦ ❛ ❝♦♥s✐❞❡r❛r ❞❡✈❡r✐❛ s❡r✿ Def ault ∼ Idade + Dep + Civil + P ostal + Rend + Cred + Entr + P total × P agas.

  ❊st❛ s✉❣❡stã♦ ❞❡r✐✈❛ ❞❛ ✉t✐❧✐③❛çã♦ ❞❛ ✈❛r✐á✈❡❧ P❛❣❛s✴P❚♦t❛❧ ❛q✉❛♥❞♦ ❞❛ ❝♦♥str✉çã♦ ❞❛ ✈❛✲ r✐á✈❡❧ r❡s♣♦st❛ ❉❡❢❛✉❧t ❞✉r❛♥t❡ ❛ s✐♠✉❧❛çã♦ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✱ ✈❛r✐á✈❡❧ à q✉❛❧ ❢♦✐ ❛tr✐❜✉í❞♦ ✉♠ ♣❡s♦ ❡❧❡✈❛❞♦✳ ❈♦♠♦ t❛❧✱ ♦s r❡s✉❧t❛❞♦s ❞♦ ❚❡st❡ ❘❛③ã♦ ❞❡ ❱❡r♦s✐♠✐❧❤❛♥ç❛s ❞❡♠♦♥str❛♠ ✉♠❛ ♠❡❧❤♦r✐❛ ♥❛ q✉❛❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ q✉❛♥t♦ s❡ t❡♠ ❡♠ ❝♦♥t❛ ❡ss❡ ❢❛❝t♦r✳ ◆♦ ❡♥t❛♥t♦✱ ♦♣t♦✉✲s❡ ♣❡❧❛ ♥ã♦ ✐♥❝❧✉sã♦ ❞❡st❛ r❡❧❛çã♦ ❡♥tr❡ ❝♦✈❛r✐á✈❡✐s✱ ✉♠❛ ✈❡③ q✉❡ ❡st❛ ♣♦❞❡rá ♣r♦✈✐r ❞❛ s✐♠✉❧❛çã♦ ❜❡♠ ❝♦♠♦✱ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣rá❝t✐❝♦✱ ❡st❛ r❡❧❛çã♦ ♥ã♦ ❢❛rá ♠✉✐t♦ s❡♥t✐❞♦✳

  ❯♠❛ s❡❣✉♥❞❛ s✉❣❡stã♦✱ ❝♦♠ ❡❧❡✈❛❞♦ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛✱ q✉❡ r❡s✉❧t♦✉ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡st❛

  ✭✸✳✷✵✮ ❆ tít✉❧♦ ❞❡ ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠ ❝❧✐❡♥t❡ ❝♦♠ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❚❛❜❡❧❛ ♦ ✈❛❧♦r ❞❡ Y ✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❡q✉❛çã♦ ♦❜té♠✲s❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  3 DepP oucos +

  13 5.073

  Pt♦t❛❧ β

  14 3.595 × 10 −2

  P❛❣❛s

β

  15 −5.568 × 10 −02

  

❚❛❜❡❧❛ ✸✳✼✿ ❈♦❡✜❝✐❡♥t❡s ❞♦ ▼♦❞❡❧♦ ❆❥✉st❛❞♦

◆♦t❡✲s❡ q✉❡ ♦s ✈❛❧♦r❡s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❚❛❜❡❧❛ ❈♦♥✈é♠ ❛✐♥❞❛ r❡ss❛❧t❛r q✉❡

  

❛❧❣✉♠❛s ❝❛t❡❣♦r✐❛s ❞❛s ✈❛r✐á✈❡✐s ❝♦♥t❡♠♣❧❛❞❛s✱ ♥❡st❡ ♠♦❞❡❧♦✱ ❡stã♦ s✉❜❥❛❝❡♥t❡s ❛ β ✱ ❞❡s✐❣✲

♥❛❞❛♠❡♥t❡ ❛s ❝❛t❡❣♦r✐❛s✿ ■❞❛❞❡❆❞✉❧t♦s✱ ❉❡♣❆❧❣✉♥s✱ ❈✐✈✐❧❈❛s✱ P♦st❛❧❩♦♥❛✶ ✭q✉❡ ❝♦♥t❡♠♣❧❛ ❛

P♦st❛❧❩♦♥❛✷ ✐♥✐❝✐❛❧✮✱ ❘❡♥❞❆❧t♦ ❡ ❊♥tr▼❛✐s✶✵✳ ❊♥tã♦✱ ♣❡❧❛ ❡q✉❛çã♦ t❡♠✲s❡✿

  Y = β + β

  1 IdadeJovens + β

  2 IdadeIdosos + β

  4 DepM uitos + β

  12 −1.023 × 10 −5

  5 CivilDiv + β

  6 CivilSol +

  7 P ostalZona3 + β

  8 P ostalZona4 + β

  9 RendM inimo +

  10 RendBaixo + β

  11 RendM edio + β

  12 Cred +

  13 EntrM enos10 + β

  

14

P T otal + β

  15 P agas.

  ❊♥tr▼❡♥♦s✶✵

β

  ❈r❡❞ β

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✼✼

  ❈✐✈✐❧❉✐✈ β

  

❢✉♥çã♦ ❢♦✐ ♠✉❧t✐♣❧✐❝❛r ❛ ✈❛r✐á✈❡❧ ❈r❡❞ ♣❡❧❛ ✈❛r✐á✈❡❧ P❛❣❛s ♦ q✉❡✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡rá s❡r

❢r✉t♦ ❞❡ s✐♠✉❧❛çã♦ ❡ ♥ã♦ ❢❛③ ♠✉✐t♦ s❡♥t✐❞♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣rát✐❝♦✳ ◆♦ ❡♥t❛♥t♦✱ ♠❛✐s ✉♠❛

✈❡③✱ ❛ ✐♥tr♦❞✉çã♦ ❞❡st❛ r❡❧❛çã♦ tr❛❞✉③✲s❡ ♥✉♠❛ ♠❡❧❤♦r✐❛ ♥❛ q✉❛❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦✳ ✸✳✸✳✷ ❊st✐♠❛çã♦ ❞❛ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❉❡❢❛✉❧t

❆♣ós ♦ ❡st✉❞♦ ❡st❛tíst✐❝♦ ❞♦s ❞❛❞♦s ❡ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s ❞❡ s❡❧❡❝çã♦ ❞❡ ❝♦✈❛r✐á✈❡✐s ❡♥❝♦♥tr♦✉✲

s❡ ♦ ♠♦❞❡❧♦ ✜♥❛❧ ✭▼♦❞❙t♣ ♦✉ ▼♦❞❆■❈✮ q✉❡ ♠❡❧❤♦r ❡①♣❧✐❝❛ ❛ ✈❛r✐á✈❡❧ ❉❡❢❛✉❧t✳ ❊st❡ ♠♦❞❡❧♦

❝♦♥t❡♠♣❧❛ ❛s ✈❛r✐á✈❡✐s ■❞❛❞❡✱ ◆➸ ❞❡ ❉❡♣❡♥❞❡♥t❡s✱ ❊st❛❞♦ ❈✐✈✐❧✱ ❈ó❞✐❣♦ P♦st❛❧✱ ❘❡♥❞✐♠❡♥t♦

▲íq✉✐❞♦✱ ❈ré❞✐t♦✱ P❡r❝❡♥t❛❣❡♠ ❞❡ ❊♥tr❛❞❛✱ ◆➸ ❞❡ Pr❡st❛çõ❡s ❚♦t❛✐s ❡ ◆➸ ❞❡ Pr❡st❛çõ❡s P❛❣❛s✳

  ❉♦ ❛❥✉st❛♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ ✜♥❛❧ r❡s✉❧t❛♠ ♦s ❝♦❡✜❝✐❡♥t❡s β ′ i s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s✱ ♥❡st❛

s❡❝çã♦✱ ♣❛r❛ ❛ ❡st✐♠❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t✳ ❆ ❚❛❜❡❧❛ ❝♦♥té♠ ♦s ✈❛❧♦r❡s ❞♦s

❝♦❡✜❝✐❡♥t❡s✱ β

  ′ i s ✱ ❛ ❝♦♥s✐❞❡r❛r✳

  ❈❛r❛❝t❡ríst✐❝❛ ❈♦❡✜❝✐❡♥t❡s ✭■♥t❡r❝❡♣t✮ β −43.74 ■❞❛❞❡❏♦✈❡♥s β

  1 3.851

  ■❞❛❞❡■❞♦s♦s

β

  2 −2.146

  ❉❡♣P♦✉❝♦s β

  3 2.769

  ❉❡♣▼✉✐t♦s

β

  4 1.267

  5 3.219

  11 8.765

  ❈✐✈✐❧❙♦❧ β

  6 1.556

  P♦st❛❧❩♦♥❛✸ β

  7 3.450

  P♦st❛❧❩♦♥❛✹ β

  8 5.608

  ❘❡♥❞▼✐♥✐♠♦

β

  9

  23.500 ❘❡♥❞❇❛✐①♦ β

  10 17.060

  ❘❡♥❞▼❡❞✐♦

β

  • β
  • β
  • β
  • β

  

✼✽ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

b Y = β + β

  ✭✸✳✷✶✮ ◆❛ ❚❛❜❡❧❛ ❡stã♦ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ❡st✐♠❛❞❛s✱ ❝♦♥❢♦r♠❡ s❡ t❡♠ ✈✐♥❞♦

❛ ❞❡s❝r❡✈❡r✱ ♣❛r❛ ❝❧✐❡♥t❡s ❝♦♠ ❞✐❢❡r❡♥t❡s ❝❛r❛❝t❡ríst✐❝❛s✳ ◆❡st❛ ❚❛❜❡❧❛ ❝♦♥st❛ t❛♠❜é♠

♦ ✈❛❧♦r ❞♦ s♣r❡❛❞ ♠í♥✐♠♦ q✉❡ s❡ ❞❡✈❡ ❛tr✐❜✉✐r ❛ ❝❛❞❛ ❝❧✐❡♥t❡✱ ❡st✐♠❛❞♦ ❝♦♠ ❜❛s❡ ♥❛

♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t r❡s♣❡❝t✐✈❛ ❡ ♥✉♠❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ 0.85✱

s❡❣✉♥❞♦ ❛ ❊q✉❛çã♦

  

◆ ❆❞✉❧t♦s P♦✉❝♦s ❙♦❧ ❩♦♥❛✸ ▼✐♥✐♠♦ ✶✺✾✻✽✹ ▼❡♥♦s✶✵ ✹✽✵ ✾✹ ✸✳✵✵ ✾✺✳✷✹✪ ✶✹✳✷✾✪

❖ ❏♦✈❡♥s P♦✉❝♦s ❙♦❧ ❩♦♥❛✹ ▼✐♥✐♠♦ ✷✶✶✾✽✼ ▼❡♥♦s✶✵ ✸✻✵ ✶✵✽ ✸✳✸✽ ✾✻✳✼✵✪ ✶✹✳✺✵✪

❚❛❜❡❧❛ ✸✳✾✿ Pr♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❉❡❢❛✉❧t ❊st✐♠❛❞❛s ✲ ❊①❡♠♣❧♦s

  

❏ ❆❞✉❧t♦s ▼✉✐t♦s ❙♦❧ ❩♦♥❛✹ ▼✐♥✐♠♦ ✶✼✵✵✸✵ ▼❡♥♦s✶✵ ✹✽✵ ✶✼✶ ✲✵✳✼✹ ✸✷✳✷✾✪ ✹✳✽✹✪

❑ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s ❩♦♥❛✹ ▼✐♥✐♠♦ ✶✷✼✼✹✻ ▼❡♥♦s✶✵ ✹✽✵ ✶✻✶ ✵✳✶✾ ✺✹✳✽✺✪ ✽✳✷✸✪

▲ ❏♦✈❡♥s P♦✉❝♦s ❉✐✈ ❩♦♥❛✹ ▼❡❞✐♦ ✽✾✾✽✹ ▼❡♥♦s✶✵ ✺✹✵ ✻✵ ✵✳✼✵ ✻✻✳✼✹✪ ✶✵✳✵✶✪

▼ ■❞♦s♦s P♦✉❝♦s ❈❛s ❩♦♥❛✹ ▼✐♥✐♠♦ ✽✽✺✹✼ ▼❡♥♦s✶✵ ✹✷✵ ✼✵ ✶✳✸✻ ✼✾✳✺✼✪ ✶✶✳✾✹✪

  

❇ ■❞♦s♦s ❆❧❣✉♥s ❙♦❧ ❩♦♥❛✶ ▼✐♥✐♠♦ ✶✶✸✹✷✵ ▼❛✐s✶✵ ✷✹✵ ✶✾✾ ✲✷✹✳✹✹ ✵✳✵✵✪ ✵✳✵✵✪

❈ ❆❞✉❧t♦s ❆❧❣✉♥s ❈❛s ❩♦♥❛✹ ❇❛✐①♦ ✷✾✷✷✹✼ ▼❡♥♦s✶✵ ✷✹✵ ✶✺✽ ✲✶✾✳✶✻ ✵✳✵✵✪ ✵✳✵✵✪

❉ ❏♦✈❡♥s P♦✉❝♦s ❈❛s ❩♦♥❛✶ ▼❡❞✐♦ ✶✼✾✻✶✵ ▼❛✐s✶✵ ✹✽✵ ✼✸ ✲✶✼✳✵✵ ✵✳✵✵✪ ✵✳✵✵✪

❊ ❏♦✈❡♥s P♦✉❝♦s ❙♦❧ ❩♦♥❛✹ ❆❧t♦ ✷✻✷✵✸✾ ▼❡♥♦s✶✵ ✹✷✵ ✻✸ ✲✶✺✳✾✼ ✵✳✵✵✪ ✵✳✵✵✪

❋ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s ❩♦♥❛✸ ▼✐♥✐♠♦ ✼✾✺✸✼ ▼❡♥♦s✶✵ ✸✻✵ ✶✹✸ ✲✹✳✼✽ ✵✳✽✸✪ ✵✳✶✷✪

● ❏♦✈❡♥s P♦✉❝♦s ❈❛s ❩♦♥❛✹ ▼✐♥✐♠♦ ✶✵✼✼✶✵ ▼❛✐s✶✵ ✸✵✵ ✽✷ ✲✷✳✽✾ ✺✳✷✹✪ ✵✳✼✾✪

❍ ❆❞✉❧t♦s ❆❧❣✉♥s ❙♦❧ ❩♦♥❛✸ ▼✐♥✐♠♦ ✶✺✹✽✶✵ ▼❡♥♦s✶✵ ✹✷✵ ✾✶ ✲✶✳✼✶ ✶✺✳✷✽✪ ✷✳✷✾✪

■ ❏♦✈❡♥s P♦✉❝♦s ❉✐✈ ❩♦♥❛✹ ▼❡❞✐♦ ✶✼✽✹✶✹ ▼❡♥♦s✶✵ ✺✹✵ ✼✵ ✲✵✳✼✻ ✸✶✳✼✻✪ ✹✳✼✻✪

  b yi π b i (1 − b R)

❆ ❏♦✈❡♥s P♦✉❝♦s ❙♦❧ ❩♦♥❛✶ ▼❡❞✐♦ ✶✵✾✾✸✵ ▼❛✐s✶✵ ✷✹✵ ✾✽ ✲✷✹✳✼✺ ✵✳✵✵✪ ✵✳✵✵✪

  1+ e

  b yi

  e

  ❈❧✐❡♥t❡ ■❞❛❞❡ ❉❡♣ ❈✐✈✐❧ P♦st❛❧ ❘❡♥❞ ❈r❡❞ ❊♥tr Pt♦t❛❧ P❛❣❛s b y i

  i b s

  ❈❛r❛❝t❡ríst✐❝❛s b π

  Y i .

  3 DepP oucos + β

  Y i 1 + e

  ✱ ✉t✐❧✐③❛✲s❡ ❛ ❡q✉❛çã♦

♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

π i = e

  

❚❛❜❡❧❛ ✸✳✽✿ ❈❛r❛❝t❡ríst✐❝❛s ❞❡ ❯♠ ❈❧✐❡♥t❡ ✲ ❊①❡♠♣❧♦

❋✐♥❛❧♠❡♥t❡✱ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝❛❞❛ ❝❧✐❡♥t❡✱ π i

  15 P agas

■❞❛❞❡ ❉❡♣ ❈✐✈✐❧ P♦st❛❧ ❘❡♥❞ ❈r❡❞ ❊♥tr Pt♦t❛❧ P❛❣❛s

❆❞✉❧t♦s P♦✉❝♦s ❙♦❧ ❩♦♥❛✸ ▼✐♥✐♠♦ ✶✺✾✻✽✹ ▼❡♥♦s✶✵ ✹✽✵ ✾✹

  14 P total + β

  13 EntrM enos10 + β

  12 Cred + β

  9 RendM inimo + β

  

7

P ostalZona3 + β

  6 CivilSol + β

  ✸✳✸✳✸ ❆✈❛❧✐❛çã♦ ❞❛ ❈❛♣❛❝✐❞❛❞❡ Pr❡❞✐t✐✈❛ ❞♦ ▼♦❞❡❧♦

P❛r❛ ✈❛❧✐❞❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❞❡✈❡✲s❡✱ ❛♥t❡s ❞❛ ❛♣❧✐❝❛çã♦ ❞♦s ♠♦❞❡❧♦s✱ ❞✐✈✐❞✐r

❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❡♠ ❞✉❛s ❛♠♦str❛s✱ ❛ ❛♠♦str❛ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❛ ❛♠♦str❛

❞❡ ✈❛❧✐❞❛çã♦✳ ❖ ♦❜❥❡❝t✐✈♦ é ❛✈❛❧✐❛r s❡ ♦ ♠♦❞❡❧♦ ❝♦♥str✉í❞♦ ❛ ♣❛rt✐r ❞❛ ❛♠♦str❛ ❞❡

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✼✾

♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛çã♦✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❝❛rt❡✐r❛ ❞❡ ❝❧✐❡♥t❡s ✭✶✷✵✵✵ ❝❧✐❡♥t❡s✮ s✐♠✉❧❛❞❛

  

❢♦✐ ❞✐✈✐❞✐❞❛ ♥✉♠❛ ❛♠♦str❛ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ✭✶✵✵✵✵ ❝❧✐❡♥t❡s✮ ✉t✐❧✐③❛❞❛ ♥❛s s❡❝çõ❡s

❛♥t❡r✐♦r❡s ❡ ♥✉♠❛ ❛♠♦str❛ ❞❡ t❡st❡ ✭✷✵✵✵ ❝❧✐❡♥t❡s✮ q✉❡ é ✉t✐❧✐③❛❞❛ ♥❡st❛ s❡❝çã♦ ♣❛r❛

❛✈❛❧✐❛r ❛ ❝❛♣❛❝✐❞❛❞❡ ♣r❡❞✐t✐✈❛ ❞♦ ♠♦❞❡❧♦ ▼♦❞❆■❈ ❡st✐♠❛❞♦✳ ❆ ❛✈❛❧✐❛çã♦ ❞❛ ❝❛♣❛❝✐❞❛❞❡ ♣r❡❞✐t✐✈❛ ❞♦s ♠♦❞❡❧♦s ❢♦✐ r❡❛❧✐③❛❞❛ ❝♦♠ ❛ ❛❥✉❞❛ ❞❡ ♠❛tr✐✲

③❡s ❞❡ ❝❧❛ss✐✜❝❛çã♦✳ ❊st❛s ♠❛tr✐③❡s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❝♦♥s✐st❡♠ ♥✉♠❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡

❛ ❝❧❛ss✐✜❝❛çã♦ ✭❝✉♠♣r✐❞♦r❡s ♦✉ ✐♥❝✉♠♣r✐❞♦r❡s✮ ❞♦s ✷✵✵✵ ❝❧✐❡♥t❡s ❞❛ ❛♠♦str❛ ❞❡ t❡st❡ ❡

❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡st✐♠❛❞❛ ♣❡❧♦ ♠♦❞❡❧♦ ▲♦❣íst✐❝♦ ♣❛r❛ ♦s ♠❡s♠♦s ❝❧✐❡♥t❡s✳ ❆ ❡st✐♠❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❞♦s ❝❧✐❡♥t❡s ❞❛ ❛♠♦str❛ ❞❡ t❡st❡ ❢❡③✲s❡ ❞❡

❢♦r♠❛ ❛♥á❧♦❣❛ à ❡st✐♠❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❞♦s ❝❧✐❡♥t❡s ❞❛ ❝❛rt❡✐r❛ ✭✈❡r

❙❡❝çã♦

  ◆❛s t❛❜❡❧❛s q✉❡ s❡ s❡❣✉❡♠ ♦ ●r✉♣♦ ❈ ❞❡s✐❣♥❛ ♦ ❣r✉♣♦ ❞❡ ❝❧✐❡♥t❡s ❈✉♠♣r✐❞♦r❡s

❡ ♦ ●r✉♣♦ ■✱ ♦s ❝❧✐❡♥t❡s ■♥❝✉♠♣r✐❞♦r❡s✳ ❆s ♣❡r❝❡♥t❛❣❡♥s ❛ ♥❡❣r✐t♦ r❡♣r❡s❡♥t❛♠ ❛

♣❡r❝❡♥t❛❣❡♠ ❞❡ ❝❧✐❡♥t❡s ❜❡♠ ❝❧❛ss✐✜❝❛❞♦s✱ ♦✉ s❡❥❛✱ ♦s ❝❧✐❡♥t❡s q✉❡ ❛♣ós ❛ ❛♣❧✐❝❛çã♦

❞♦ ♠♦❞❡❧♦ ❧♦❣íst✐❝♦ s❡ ❝❧❛ss✐✜❝❛r❛♠ ❝♦rr❡❝t❛♠❡♥t❡ q✉❛♥t♦ ❛ s❡r❡♠ ❝✉♠♣r✐❞♦r❡s ♦✉

✐♥❝✉♠♣r✐❞♦r❡s✳ P❛r❛ ❛ ❝♦♥str✉çã♦ ❞❛s ♠❛tr✐③❡s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❝♦♥s✐❞❡r❛r❛♠✲s❡ ❞♦✐s ❝❛s♦s✿ ♥♦ ♣r✐✲

♠❡✐r♦✱ ❝♦♥s✐❞❡r♦✉✲s❡ q✉❡ ♦s ❝❧✐❡♥t❡s ❝♦♠ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡st✐♠❛❞❛ s✉♣❡✲

r✐♦r ❛ ✺✵✪ ❡r❛♠ ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s❀ ♥♦ s❡❣✉♥❞♦ ❝❛s♦✱ ❝♦♥s✐❞❡r♦✉✲s❡ q✉❡ ♦s ❝❧✐❡♥t❡s

❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡st✐♠❛❞❛ s✉♣❡r✐♦r ❛ ✷✺✪ ❡r❛♠ ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s✳ ❉❡st❛ ❢♦r♠❛✱ ♥✉♠❛ ♣r✐♠❡✐r❛ ❛♥á❧✐s❡✱ ❝♦♥t❛❜✐❧✐③❛r❛♠✲s❡ ♦s ❝❧✐❡♥t❡s ❞❛ ❛♠♦str❛ ❞❡

t❡st❡ q✉❡ ❡r❛♠ ✐♥❝✉♠♣r✐❞♦r❡s (1) ❡ ♦s ❝❧✐❡♥t❡s ❝✉♠♣r✐❞♦r❡s (0) ❡ ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡st❡s

❞♦✐s ❣r✉♣♦s✱ ♦s ❝❧✐❡♥t❡s ❝✉❥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡st✐♠❛❞❛ ❡r❛ s✉♣❡r✐♦r ♦✉ ✐❣✉❛❧ ❛

✺✵✪ ❡ ✐♥❢❡r✐♦r ❛ ✺✵✪✱ ❝♦♠♦ s❡ ♣♦❞❡ ♦❜s❡r✈❛r ♥❛ ❚❛❜❡❧❛

  ❖r✐❣✐♥❛❧ ❈❧❛ss✐✜❝❛çã♦ ❞♦ ▼♦❞❡❧♦ <

  50

  50 ≥ ✵ ✶✾✵✽ ✽ ✶✾✶✻

✶ ✷✾ ✺✺ ✽✹

  ✶✾✸✼ ✻✸ ❚♦t❛❧

❚❛❜❡❧❛ ✸✳✶✵✿ ❈♦♥t❛❜✐❧✐③❛çã♦ ❞❡ ❈❧✐❡♥t❡s ❈✉♠♣r✐❞♦r❡s ❡ ■♥❝✉♠♣r✐❞♦r❡s ✲ ✺✵✪

  ❈❧❛ss✐✜❝❛çã♦ ❞♦ ▼♦❞❡❧♦ ❈❧❛ss❡ ❖r✐❣✐♥❛❧ ●r✉♣♦ ❈ ●r✉♣♦ ■

●r✉♣♦ ❈ ✾✾✳✺✽✪ ✵✳✹✷✪

●r✉♣♦ ■ ✸✹✳✺✷✪ ✻✺✳✹✽✪

  

❚❛❜❡❧❛ ✸✳✶✶✿ ▼❛tr✐③ ❞❡ ❈❧❛ss✐✜❝❛çã♦ ❞♦ ▼♦❞❡❧♦ ❞❡ ❆♣r♦✈❛çã♦ ❞❡ ❈ré❞✐t♦ ✭✺✵✪✮

❖❜s❡r✈❛♥❞♦ ❛s ❚❛❜❡❧❛s ♣♦❞❡ ❞✐③❡r✲s❡ q✉❡ ♦ ♠♦❞❡❧♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ ❡s✲

t✐♠❛çã♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ❡stá ❛ ❝❧❛ss✐✜❝❛r ❝♦rr❡❝t❛♠❡♥t❡ ❝❡r❝❛ ❞❡ 99.58%

  

✽✵ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❛♣❡♥❛s s❡ ❝♦♥s✐❞❡r❛♠ ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s ♦s ❝❧✐❡♥t❡s ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t

❡st✐♠❛❞❛ s✉♣❡r✐♦r ♦✉ ✐❣✉❛❧ ❛ 50%✳ ❉♦♥❞❡✱ ❡♠ ♠é❞✐❛✱ ♦ ♠♦❞❡❧♦ ❝❧❛ss✐✜❝❛ ❝♦rr❡❝t❛♠❡♥t❡

82.53%

  ❞♦s ❝❧✐❡♥t❡s ❞❛ ❛♠♦str❛ ❞❡ t❡st❡✳ ◆✉♠❛ s❡❣✉♥❞❛ ❛♥á❧✐s❡✱ ❝♦♥s✐❞❡r♦✉✲s❡ q✉❡ ♦s ❝❧✐❡♥t❡s ❝✉❥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t

❡st✐♠❛❞❛ ❡r❛ s✉♣❡r✐♦r ♦✉ ✐❣✉❛❧ ❛ ✷✺✪ ❡r❛♠ ❝♦♥s✐❞❡r❛❞♦s ✐♥❝✉♠♣r✐❞♦r❡s ❡ ❝♦♥str✉ír❛♠✲s❡

❛s ❚❛❜❡❧❛s

  ❖r✐❣✐♥❛❧ ❈❧❛ss✐✜❝❛çã♦ ❞♦ ▼♦❞❡❧♦ <

25 ≥

  25 ✵ ✶✽✾✶ ✷✺ ✶✾✶✻

✶ ✶✽ ✻✻ ✽✹

  ✶✾✵✾ ✾✶ ❚♦t❛❧

❚❛❜❡❧❛ ✸✳✶✷✿ ❈♦♥t❛❜✐❧✐③❛çã♦ ❞❡ ❈❧✐❡♥t❡s ❈✉♠♣r✐❞♦r❡s ❡ ■♥❝✉♠♣r✐❞♦r❡s ✲ ✷✺✪

  ❈❧❛ss✐✜❝❛çã♦ ❞♦ ▼♦❞❡❧♦ ❈❧❛ss❡ ❖r✐❣✐♥❛❧ ●r✉♣♦ ❈ ●r✉♣♦ ■

●r✉♣♦ ❈ ✾✽✳✼✵✪ ✶✳✸✵✪

●r✉♣♦ ■ ✷✶✳✹✸✪ ✼✽✳✺✼✪

  

❚❛❜❡❧❛ ✸✳✶✸✿ ▼❛tr✐③ ❞❡ ❈❧❛ss✐✜❝❛çã♦ ❞♦ ▼♦❞❡❧♦ ❞❡ ❆♣r♦✈❛çã♦ ❞❡ ❈ré❞✐t♦ ✭✷✺✪✮

◆❡st❛ ú❧t✐♠❛ ❛♥á❧✐s❡✱ ♣♦❞❡ ❞✐③❡r✲s❡ q✉❡ ♦ ♠♦❞❡❧♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ ❡st✐♠❛çã♦ ❞❛s

♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ❡stá ❛ ❝❧❛ss✐✜❝❛r ❝♦rr❡❝t❛♠❡♥t❡ ❝❡r❝❛ ❞❡ 98.70% ❞♦s ❝❧✐❡♥✲

t❡s ❝✉♠♣r✐❞♦r❡s ❡ 78.57% ❞♦s ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s✱ ♣♦rt❛♥t♦✱ ❡♠ ♠é❞✐❛✱ ♦ ♠♦❞❡❧♦

❝❧❛ss✐✜❝❛ ❝♦rr❡❝t❛♠❡♥t❡ 88.63% ❞♦s ❝❧✐❡♥t❡s ❞❛ ❛♠♦str❛ ❞❡ t❡st❡✳ ❖❜s❡r✈❛✲s❡ t❛♠❜é♠✱ q✉❡ ♦ ♠♦❞❡❧♦ é t❛♠❜é♠ ♠❛✐s ❡✜❝✐❡♥t❡ ❛ ❝❧❛ss✐✜❝❛r ❝❧✐❡♥t❡s ❝✉♠✲

♣r✐❞♦r❡s✳ ❊st❡ r❡s✉❧t❛❞♦ ❞❡✈❡✲s❡✱ t❛❧✈❡③✱ ❛♦ ❢❛❝t♦ ❞❡ ♥❛ s✐♠✉❧❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❉❡❢❛✉❧t✱ s❡

t❡r ♦♣t❛❞♦ ♣♦r ❛♣❡♥❛s ❝♦♥s✐❞❡r❛r q✉❡ ♦s ❝❧✐❡♥t❡s ❝♦♠ ✉♠❛ ♣♦♥❞❡r❛çã♦ s✉♣❡r✐♦r ❛ ✽✺✳✺✪

✭✈❡❥❛✲s❡ ❊①♣r❡ssã♦ ❢♦ss❡♠ ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s✱ ♦✉ s❡❥❛✱ ❢♦r❛♠ ❡①❝❧✉í❞♦s ❝❧✐✲

❡♥t❡s q✉❡✱ ❡✈❡♥t✉❛❧♠❡♥t❡✱ ❡stã♦ ❛ s❡r ❝❧❛ss✐✜❝❛❞♦s ♥♦ ♠♦❞❡❧♦ ✜♥❛❧ ❝♦♠♦ ✐♥❝✉♠♣r✐❞♦r❡s✳

  ❱❡r✐✜❝❛✲s❡✱ ❞❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ q✉❡ ♦ ♠♦❞❡❧♦ ❞❡s❡♥✈♦❧✈✐❞♦ ✉t✐❧✐③❛♥❞♦ ❛ r❡❣r❡ssã♦

❧♦❣íst✐❝❛ ✭▼♦❞❆■❈✮ ♦❜t❡✈❡ ❜♦♥s r❡s✉❧t❛❞♦s ♥❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❝❧✐❡♥t❡s ❝♦♠♦ ❝✉♠♣r✐❞♦r❡s

❡ ✐♥❝✉♠♣r✐❞♦r❡s✳ ✸✳✸✳✹ ❆♥á❧✐s❡ ❞❡ ❘❡sí❞✉♦s

❆♣ós ♦ ❛❥✉st❛♠❡♥t♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ❞❛❞♦s✱ ❞❡✈❡✲s❡ ✐♥✈❡st✐❣❛r q✉ã♦ ❜❡♠ ♦ ♠♦❞❡❧♦

❝♦♥s❡❣✉❡ ❞❡s❝r❡✈❡r ♦s ❞❛❞♦s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❞❡✈❡✲s❡ ♣r♦❝✉r❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛❧❣✉♠❛

t❡♥❞ê♥❝✐❛ s✐st❡♠át✐❝❛ ♥❛ q✉❛❧✐❞❛❞❡ ❞♦ ❛❥✉st❛♠❡♥t♦✳ P♦r ❡①❡♠♣❧♦✱ s❡ ❛ q✉❛❧✐❞❛❞❡ ❞♦

❛❥✉st❛♠❡♥t♦ ❛✉♠❡♥t❛ ❝♦♠ ♦ ❛✉♠❡♥t♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♦❜s❡r✈❛çõ❡s✱ ♦✉ s❡ é ❢✉♥çã♦ ❞❡

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✽✶ ❙❡❣✉♥❞♦ ✱ ❛ ❛♥á❧✐s❡ ❞❡ r❡sí❞✉♦s é út✐❧✱ ♥ã♦ só ♣❛r❛ r❡❛❧✐③❛r

  

✉♠❛ ❛✈❛❧✐❛çã♦ ❧♦❝❛❧ ❞❛ q✉❛❧✐❞❛❞❡ ❞❡ ❛❥✉st❛♠❡♥t♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦

à ❡s❝♦❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦✱ ❞❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❡ ❞❡ t❡r♠♦s ❞♦ ♣r❡❞✐t♦r ❧✐♥❡❛r✱ ❝♦♠♦

t❛♠❜é♠ ♣❛r❛ ❛❥✉❞❛r ❛ ✐❞❡♥t✐✜❝❛r ♦❜s❡r✈❛çõ❡s ♠❛❧ ❛❥✉st❛❞❛s✱ ♦✉ s❡❥❛✱ q✉❡ ♥ã♦ sã♦ ❜❡♠

i

  

❡①♣❧✐❝❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦✳ ❯♠ r❡sí❞✉♦ R ❞❡✈❡ ❡①♣r✐♠✐r ❛ ❞✐s❝r❡♣â♥❝✐❛ ❡♥tr❡ ♦ ✈❛❧♦r

i i

  ♦❜s❡r✈❛❞♦ y ❡ ♦ ✈❛❧♦r by ❛❥✉st❛❞♦ ♣❡❧♦ ♠♦❞❡❧♦✳ ◆ã♦ t❡♥❞♦ ❝♦♠♦ ♦❜❥❡❝t✐✈♦✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ❢❛③❡r ✉♠ ❡st✉❞♦ ✐♥t❡♥s✐✈♦ s♦❜r❡ ❛ ❛♥á❧✐s❡

❞❡ r❡sí❞✉♦s s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛q✉✐ ❛❧❣✉♥s ❣rá✜❝♦s r❡s✉❧t❛♥t❡s ❞❛ ❛♥á❧✐s❡ ❞❡ r❡sí❞✉♦s

❞♦ ♠♦❞❡❧♦ ❡st✐♠❛❞♦ ❡ ❜r❡✈❡s ♦❜s❡r✈❛çõ❡s s♦❜r❡ ♦s ♠❡s♠♦s✳ ◆✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦

r❡❛❧ s❡r✐❛ ❡ss❡♥❝✐❛❧ ✉♠❛ ❛♥á❧✐s❡ ❞❡ r❡sí❞✉♦s ♠❛✐s ❛♣r♦❢✉♥❞❛❞❛✳ ❈♦♥✈é♠ r❡❢❡r✐r q✉❡ ❡①✐st❡ ❛❧❣✉♠❛ ❛♠❜✐❣✉✐❞❛❞❡ ❡♠ ❞❡✜♥✐r ♦s r❡sí❞✉♦s ♥❛ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ ✭✈❡r ◆❛ ❛♥á❧✐s❡ ❞❡ r❡sí❞✉♦s✱ ❡①✐st❡♠ ✈ár✐♦s t✐♣♦s ❞❡ r❡sí❞✉♦s ❞♦s q✉❛✐s s❡ ❞❡st❛❝❛♠ ♦s r❡sí❞✉♦s ❞❡ P❡❛rs♦♥ ❡ ♦s ❞❡s✈✐♦s r❡s✐❞✉❛✐s ✭♦✉ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✮✳

, m , x ) : 1 ≤ i ≤ n

i i i i

  ❈♦♥s✐❞❡r❡✲s❡ ♦s ❞❛❞♦s (Y ✱ ♦♥❞❡ ♦ Y sã♦ ✐♥❞❡♣❡♥❞❡♥t❡✲ , π ) ) = ♠❡♥t❡ ❞✐str✐❜✉í❞❛s ❝♦♠ ❞✐str✐❜✉✐çã♦ ❇✐♥♦♠✐❛❧(m i i ✱ ❡ ❛s π i s❛t✐s❢❛③❡♠ logit(π i

  T T T log [π /(1 − π )] = x β = (x , . . . , x ) i i i1 ik i ✱ ❝♦♠ x i ✈❡❝t♦r❡s ❝♦✈❛r✐á✈❡❧ ❝♦♥❤❡❝✐❞♦s ❡ k

  ′ β ∈ R

  ❞❡s❝♦♥❤❡❝✐❞♦✳ ❙❡❥❛ X ❛ ♠❛tr✐③ n × k ❞❡ ❞✐♠❡♥sã♦ k ❝♦♠ x ❛ i−és✐♠❛ ❧✐♥❤❛✱ i

  π (1 − π ) i i i

  

❡ D ❞❡♥♦t❛ ❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ n × n ❝♦♠ m ♦ i−és✐♠♦ ❡❧❡♠❡♥t♦ ❞✐❛❣♦♥❛❧✳

❉❡✜♥❡✲s❡ ❛✐♥❞❛

  1

  1 T −1 T

  2

  

X (X DX )

X D .

  2 H = D

  ✭✸✳✷✷✮ , p ) i i i i

  

❈♦♥s✐❞❡r❡✲s❡ ✉♠ r❡sí❞✉♦ R(y ✱ ❢✉♥çã♦ ❞❛ ♦❜s❡r✈❛çã♦ y ❡ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ p ✳

❆ ❡①♣r❡ssã♦ ♣❛r❛ ♦s r❡sí❞✉♦s ❞❡ P❡❛rs♦♥✱ s❡❣✉♥❞♦ ♣♦❞❡♠✲s❡ ❡s❝r❡✈❡r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ (y − m π ) i i i P

  

R (y , π ) = .

i i i

  ✭✸✳✷✸✮ 1/2

  [m i π i (1 − m i π i )] ❖ ❞❡s✈✐♦ r❡s✐❞✉❛❧ é ❛ ♠❡❞✐❞❛ ❞♦ ❞❡s✈✐♦ ❞♦ ❝♦♥tr✐❜✉t♦ ❞❡ ❝❛❞❛ ♦❜s❡r✈❛çã♦ ❡ ♣♦❞❡

s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ✈❡r✐✜❝❛r✱ ♥♦s ♠♦❞❡❧♦s ❧✐♥❡❛r❡s ❣❡♥❡r❛❧✐③❛❞♦s✱ ❛ ❛❞❡q✉❛çã♦ ❞♦ ♠♦❞❡❧♦

❛❥✉st❛❞♦ ♣❛r❛ ❝❛❞❛ ♦❜s❡r✈❛çã♦ ❡ ♣♦❞❡♠✲s❡ ❞❡t❡r♠✐♥❛r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  1/2 y m − y i i i D

  R (y i , π i ) = δ i 2 y i ln + (m i − y i ) ln , ✭✸✳✷✹✮ i m π m − m π i i i i i

  = sinal(y − m π ) ♦♥❞❡ δ i i i i ❡ s❡

  1

  ( D

  2

  

y i = 0 ⇒ R (y i , π i ) = − {2m i | log(1 − π i )|}

i

  1

  ✭✸✳✷✺✮ D

  i i i i i i i D

  2 y = m ⇒ R (y , π ) = {2m | log(π )|} .

  s✉❣❡r❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞♦s ❞❡s✈✐♦s r❡s✐❞✉❛✐s R ❡♠ ✈❡③ i

  

❞♦s r❡sí❞✉♦s ❞❡ P❡❛rs♦♥✱ ❥á q✉❡ ❡st❡s✱ ❡♠❜♦r❛ ❛♣r❡s❡♥t❡♠ ♣r♦♣r✐❡❞❛❞❡s ❞❡ s❡❣✉♥❞❛

  

✽✷ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

❙❡❣✉♥❞♦ ✉♠❛ ❞❛s r❛③õ❡s ♠❛✐s ❝♦♠✉♥s ♣❛r❛ ❛ ❢❛❧t❛ ❞❡ ❛❥✉st❡ é

❛tr❛✈és ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ♦✉t❧✐❡rs ♥♦s ❞❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱ é ✐♠♣♦rt❛♥t❡ ❡♥t❡♥❞❡r q✉❡ ✉♠

♣♦♥t♦ ♣♦❞❡ ♣❛r❡❝❡r s❡r ✉♠ ♦✉t❧✐❡r ❞❡✈✐❞♦ à ♠á ❡s♣❡❝✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦✱ ❡ ♥ã♦ ♣♦rq✉❡

❤❛❥❛ ❛❧❣♦ ❞❡ ❡rr❛❞♦ ❝♦♠ ♦s ❞❛❞♦s✳ P❛r❛ ❛♥❛❧✐s❛r ♦ ❣rá✜❝♦ ❞❛ ❋✐❣✉r❛ ❞❡✈❡ t❡r✲s❡ ❡♠ ❝♦♥t❛ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s✱ ❝♦♠♦

❛❧❛✈❛♥❝❛❣❡♠✱ ✐♥✢✉ê♥❝✐❛ ❡ ❞✐stâ♥❝✐❛ ❞❡ ❈♦♦❦✳ ❚❛✐s ❝♦♥❝❡✐t♦s s❡rã♦ ❜r❡✈❡♠❡♥t❡ ❡①♣❧✐❝❛✲

❞♦s ❛ s❡❣✉✐r✳

  P❛r❛ ❡♠ ❡st❛tíst✐❝❛✱ ❛❧❛✈❛♥❝❛❣❡♠ ✭❧❡✈❡r❛❣❡✮ é ✉♠ t❡r♠♦ ✉t✐❧✐③❛❞♦

❡♠ ❝♦♥❡①ã♦ ❝♦♠ ❛ ❛♥á❧✐s❡ ❞❡ r❡❣r❡ssã♦ ❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❡♠ ❛♥á❧✐s❡s q✉❡ ✈✐s❛♠ ✐❞❡♥t✐✜✲

❝❛r ❛s ♦❜s❡r✈❛çõ❡s q✉❡ tê♠ ✉♠ ❣r❛♥❞❡ ❡❢❡✐t♦ s♦❜r❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ♠❡❞✐çã♦ ❞❡ ♠♦❞❡❧♦s

❞❡ r❡❣r❡ssã♦✳ ❖s ♣♦♥t♦s ❞❡ ❛❧❛✈❛♥❝❛❣❡♠ sã♦ ❛s ♦❜s❡r✈❛çõ❡s ♦✉ ♦s ✈❛❧♦r❡s ❞❛s ✈❛r✐á✲

✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s t❛✐s q✉❡ ❛ ❢❛❧t❛ ❞❡ ♦❜s❡r✈❛çõ❡s ✈✐③✐♥❤❛s s✐❣♥✐✜❝❛ q✉❡ ♦ ♠♦❞❡❧♦ ❞❡

r❡❣r❡ssã♦ ♣❛ss❛rá ♣ró①✐♠♦ ❞❡ss❛ ♦❜s❡r✈❛çã♦ ♣❛rt✐❝✉❧❛r✳ ❆s ♦❜s❡r✈❛çõ❡s ❛❧❛✈❛♥❝❛❞❛s

❡♥❝♦♥tr❛♠✲s❡ ❞✐st❛♥❝✐❛❞❛s ❞♦ ✈❛❧♦r ♠é❞✐♦ ❞❡ y✳ ▼❡❞✐❞❛s ❞❡ ❛❧❛✈❛♥❝❛❣❡♠ ❞❡ ✉♠ ❞❡t❡r✲

  2

♠✐♥❛❞♦ ♣♦♥t♦ x sã♦ ♣r♦♣♦r❝✐♦♥❛✐s ❛ (y − y) ✳ ❆ ♠❡❞✐❞❛ ❞❡ ❛❧❛✈❛♥❝❛❣❡♠ ♠❛✐s ❝♦♠✉♠

é✿

  2 1 (y − y ) i

h + = ,

i

  P ✭✸✳✷✻✮

  2 n (y − y) j i

  ❝♦♠ h ♦ i−és✐♠♦ ❡❧❡♠❡♥t♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞❛ ♠❛tr✐③ H ❞❡✜♥✐❞❛ ❡♠ ➱ ❝♦♠✉♠ ❞✐③❡r✲s❡ q✉❡ ✉♠ ♣♦♥t♦ é ♠✉✐t♦ ✐♥✢✉❡♥t❡ s❡✿ 2p

h > ,

i

  ✭✸✳✷✼✮ n ♦♥❞❡ p é ♦ ♥ú♠❡r♦ ❞❡ ♣❛râ♠❡tr♦s ♥♦ ♠♦❞❡❧♦ ❡ n ♦ ♥ú♠❡r♦ ❞❡ ♦❜s❡r✈❛çõ❡s✳ ❆ ✐♥✢✉ê♥❝✐❛ ❞♦s ❞❛❞♦s ♥ã♦ ♣♦❞❡✱ ❞❡ t♦❞♦✱ s❡r ❛♥❛❧✐s❛❞❛ ❛♣❡♥❛s ❛tr❛✈és ❞♦s r❡sí❞✉♦s✱

♣♦✐s ✉♠ ♣♦♥t♦ ♠✉✐t♦ ✐♥✢✉❡♥t❡ ♣♦❞❡ ❢♦rç❛r ❛ ❧✐♥❤❛ ❞❡ r❡❣r❡ssã♦ ❛ ❛♣r♦①✐♠❛r✲s❡ ❞❡❧❡ ❡✱

♣♦rt❛♥t♦✱ ♦ ♣♦♥t♦ ✐♥✢✉❡♥t❡ ♣♦❞❡ t❡r ✉♠ r❡sí❞✉♦ ♣❡q✉❡♥♦✳ ❖❜s❡r✈❛çõ❡s ✐♥✢✉❡♥t❡s sã♦

♦❜s❡r✈❛çõ❡s q✉❡✱ s❡ r❡t✐r❛❞❛s ❞❛ ❛♥á❧✐s❡✱ ❣❡r❛r✐❛♠ ✈❛r✐❛çõ❡s ❛ss✐♥❛❧á✈❡✐s ♥♦ ❝♦♥❥✉♥t♦

❞♦s ✈❛❧♦r❡s ❛❥✉st❛❞♦s ❞❡ Y ❡ ❞♦s ♣❛râ♠❡tr♦s ❡st✐♠❛❞♦s✱ bβ✳

  ❯♠❛ ♠❡❞✐❞❛ ❢r❡q✉❡♥t❡ ♣❛r❛ ❛ ✐♥✢✉ê♥❝✐❛ ❞❛ ♦❜s❡r✈❛çã♦ i é ❛ ❞✐stâ♥❝✐❛ ❞❡ ❈♦♦❦✳ ❆

❞✐stâ♥❝✐❛ ❞❡ ❈♦♦❦ r❡♣r❡s❡♥t❛ ♦ ❡❢❡✐t♦ ❞❡ ❡①❝❧✉✐r ✉♠ ❞❛❞♦ ♦❜s❡r✈❛❞♦ ❡ é t❛♠❜é♠ ✉♠❛

t❡♥t❛t✐✈❛ ❞❡ ❝♦♠❜✐♥❛r ❛ ❛❧❛✈❛♥❝❛❣❡♠ ❡ ♦s r❡sí❞✉♦s ♥✉♠❛ ú♥✐❝❛ ♠❡❞✐❞❛✳ ❖s ♣♦♥t♦s ❝♦♠

r❡sí❞✉♦s ❡❧❡✈❛❞♦s ✭♦✉t❧✐❡rs✮ ❡✴♦✉ ❝♦♠ ❛❧t❛ ❛❧❛✈❛♥❝❛❣❡♠ ♣♦❞❡♠ ❞✐st♦r❝❡r ❛ ♣r❡❝✐sã♦ ❞❡

✉♠❛ r❡❣r❡ssã♦✳ ❆ ❞✐stâ♥❝✐❛ ❞❡ ❈♦♦❦✱ ❡♠ ✈❛❧♦r ❛❜s♦❧✉t♦✱ é ❞❛❞❛ ♣♦r✿

  1/2 n − p h i

  ∗

C i = |r | · .

  ✭✸✳✷✽✮ i p 1 − h i

  ❖❜s❡r✈❛çõ❡s ❛tí♣✐❝❛s✱ ✐♥✢✉❡♥t❡s ♦✉ ❛❧❛✈❛♥❝❛❞❛s✱ ❡♠❜♦r❛ ♣♦ss❛♠ ❡st❛r r❡❧❛❝✐♦♥❛❞❛s✱

♥ã♦ sã♦ ♦ ♠❡s♠♦ ❝♦♥❝❡✐t♦✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ ♦❜s❡r✈❛çã♦ ❝♦♠ r❡sí❞✉♦ st❛♥❞❛r❞✐③❛❞♦

i

  

❣r❛♥❞❡ ❡ h ❡❧❡✈❛❞♦✱ t❡♠ ❞❡ t❡r ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ ❈♦♦❦ ❣r❛♥❞❡✱ ♦✉ s❡❥❛ s❡r ✐♥✢✉❡♥t❡✳ ❙❡

i i

t✐✈❡r r ❣r❛♥❞❡ ❡ h ♣❡q✉❡♥♦✱ ♣♦❞❡✱ ♦✉ ♥ã♦✱ s❡r ✐♥✢✉❡♥t❡✱ ❝♦♥s♦❛♥t❡ ❛ ❣r❛♥❞❡③❛ r❡❧❛t✐✈❛

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✽✸ q✉❡ ♠❡r❡❝❡♠ ♠❛✐♦r ❛t❡♥çã♦ ❡ ❝♦♥s✐❞❡r❛çã♦✳

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  Cook's distance

  

❣❡♠ ❜❡♠ ❝♦♠♦ ❛ ❞✐stâ♥❝✐❛ ❞❡ ❈♦♦❦✱ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s ✈❛❧♦r❡s ♦❜s❡r✈❛❞♦s ❞❛ ✈❛r✐á✈❡❧

r❡s♣♦st❛✱ ❞❡st❛❝❛♥❞♦ ♦s ✈❛❧♦r❡s y q✉❡ tê♠ ♠❛✐♦r ❡❢❡✐t♦ s♦❜r❡ ❛s ❡st✐♠❛t✐✈❛s ❞♦ ♣❛râ✲

♠❡tr♦✳ P♦❞❡ ♦❜s❡r✈❛r✲s❡ q✉❡ ♦ ♣♦♥t♦ ♥ú♠❡r♦ ✹✶✶✶ t❡♠ ✉♠❛ ❛❧❛✈❛♥❝❛❣❡♠ ♣♦✉❝♦ s✉♣❡r✐♦r

❛♦ ♣♦♥t♦ ♥ú♠❡r♦ ✼✺✼✼ ❡ q✉❡ ❛♠❜❛s tê♠ ✉♠❛ ✐♥✢✉ê♥❝✐❛ q✉❛s❡ ✐❣✉❛❧ ♣♦✐s ❡stã♦ q✉❛s❡ à

♠❡s♠❛ ❞✐stâ♥❝✐❛ ❞♦ ❝♦♥t♦r♥♦ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ ❈♦♦❦✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ ♦ ♣♦♥t♦ ♥ú♠❡r♦

✽✼✶✽ ♦❜s❡r✈❛✲s❡ q✉❡✱ ❡♠ r❡❧❛çã♦ ❛♦s ♦✉tr♦s ❞♦✐s ♣♦♥t♦s ✐♥✢✉❡♥t❡s✱ tê♠ ✉♠❛ ❛❧❛✈❛♥❝❛✲

❣❡♠ ❡ ✉♠❛ ✐♥✢✉ê♥❝✐❛ ♠✉✐t♦ ✐♥❢❡r✐♦r✳

  8718

❋✐❣✉r❛ ✸✳✽✿ ❘❡s✐❞✉♦s ❞❡ P❡❛rs♦♥ P❛❞r♦♥✐③❛❞♦s × ❆❧❛✈❛♥❝❛❣❡♠

❖ ❣rá✜❝♦ ❞❛ ❋✐❣✉r❛ ✐❧✉str❛ ♦s r❡sí❞✉♦s st❛♥❞❛r❞✐③❛❞♦s ❡♠ ❢✉♥çã♦ ❞❛ ❛❧❛✈❛♥❝❛✲

  1 Residuals vs Leverage 4111 7577

  0.5

  0.5

  1

  ● glm(Default ~ Idade + Dep + Civil + Postal + Rend + Cred + Entr + Ptotal + ...

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  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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  0.0

  ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●

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  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ●

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  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  0.2

  0.4

  0.6

  −5

  5 Leverage Std. P earson resid.

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

  ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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  ❯♠❛ ❢♦r♠❛ ❞❡ ♦❜t❡r ♦ ❣rá✜❝♦ ❞❛ ❋✐❣✉r❛ é r❡❝♦rr❡r à ❢✉♥çã♦ rst❛♥❞❛r❞ ❞♦

❙♦❢t✇❛r❡ R q✉❡ ❡st✐♠❛ ♦s r❡sí❞✉♦s st❛♥❞❛r❞ q✉❛♥❞♦ s❡ t❡♠ ✉♠❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡

❛♣❡♥❛s ❝♦♠ ❞✉❛s ❝❛t❡❣♦r✐❛s✱ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❛ ✈❛r✐á✈❡❧ ❉❡❢❛✉❧t✳ ❖s ✈❛❧♦r❡s ❞♦s r❡sí❞✉♦s✱

♣❛r❛ ❝❛❞❛ ❝❛t❡❣♦r✐❛ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛✱ s❡rã♦ ✐❣✉❛✐s ❡♠ ♠❛❣♥✐t✉❞❡ ❡ ♦♣♦st♦s ❡♠ s✐♥❛❧

♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ✈❛❧♦r❡s ♥❛ ♠❡s♠❛ ❝❛t❡❣♦r✐❛✱ ❝♦♠♦ r❡✢❡①♦ ❞❛s ❊q✉❛çõ❡s

  

✽✹ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

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  1

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  3 Fitted values De viance Residuals

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❋✐❣✉r❛ ✸✳✾✿ ❉❡s✈✐♦s ❘❡s✐❞✉❛✐s × ❱❛❧♦r❡s ❆❥✉st❛❞♦s

♠❡tr♦s ❡ ♦s s❡✉s ❡rr♦s✱ ❞❡✈❡r✲s❡✲á r❡♣❡t✐r ♦ ♣r♦❝❡ss♦ ❞❡ ♠♦❞❡❧❛çã♦ ❡st❛tíst✐❝❛✱ ❡ ❡①❝❧✉✐r

♦s ♣♦♥t♦s ❡♠ q✉❡stã♦✳ ❆❧t❡r♥❛t✐✈❛♠❡♥t❡ ♣♦❞❡r✲s❡✲✐❛ ❢❛③❡r ✉♠ ✏❥❛❝❦❦♥✐❢❡✑ ❛♦s ❞❛❞♦s

✭✈❡❥❛✲s❡ ♦✉ s❡❥❛✱ ❡①❝❧✉✐r ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ✉♠ ❝❧✐❡♥t❡ ❞❡ ❝❛❞❛ ✈❡③

❡✱ ❡♠ ❝❛❞❛ ✉♠❛ ❞❡ss❛s ❡①❝❧✉sõ❡s✱ r❡❡st✐♠❛r ♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥t❡r❡ss❡✳ ❊♠❜♦r❛ ❡st❡ t✐♣♦

❞❡ ❛♥á❧✐s❡ ♥ã♦ s❡❥❛ ❢❡✐t❛ ♥❡st❡ tr❛❜❛❧❤♦✱ ♣♦❞❡ ❞✐③❡r✲s❡ q✉❡ ♦ ♠♦❞❡❧♦ ❡st✐♠❛❞♦ ♥ã♦ é ✉♠

❜♦♠ ♠♦❞❡❧♦ ♣❛r❛ ♦s três ♣♦♥t♦s ✐❞❡♥t✐✜❝❛❞♦s ♥❛ ❋✐❣✉r❛

  ✸✳✸ ❆♣❧✐❝❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✽✺

  ❈♦♥❝❧✉sã♦

❆♦ ❧♦♥❣♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ♦ ♦❜❥❡❝t✐✈♦ ❢♦✐ s❡♠♣r❡ ♦ ❞❛ ❡st✐♠❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡

❞❡❢❛✉❧t✱ ❝♦♠ ❜❛s❡ ♥❛s ❝❛r❛❝t❡ríst✐❝❛s ❢❛❝✉❧t❛❞❛s ♣❡❧♦ s♦❧✐❝✐t❛♥t❡ ❞❡ ❝ré❞✐t♦✳

  ❈♦♠❡ç♦✉✲s❡ ♣♦r ✉♠❛ ❛❜♦r❞❛❣❡♠ ❣❡r❛❧ ❞❡ ❝♦♥❝❡✐t♦s ❡ ♠ét♦❞♦s ❞❡ ❛✈❛❧✐❛çã♦ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✳ ◆♦ ❈❛♣ít✉❧♦ ♦❜t❡✈❡✲s❡ ✉♠ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❡st✐♠❛çã♦ ❞♦ s♣r❡❛❞ ♠í✲

♥✐♠♦ ❛ ❛tr✐❜✉✐r ❛ ❝❛❞❛ ❝❧✐❡♥t❡✳ ❊st❡ r❡s✉❧t❛❞♦ s✉❣❡r❡ q✉❡ ♦ s♣r❡❛❞ ♠í♥✐♠♦ ❛ ❛tr✐❜✉✐r

s❡❥❛ ❢✉♥çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ✉♠ ♠♦❞❡❧♦ ♣❛r❛ ❡st✐♠❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t

❡ ❞❡✐①♦✉✲s❡ ♣❛r❛ tr❛❜❛❧❤♦s ♣♦st❡r✐♦r❡s ❛ ❡st✐♠❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ r❡❝✉♣❡r❛çã♦✱

♥♦ ❡♥t❛♥t♦✱ ✈❛❧❡ ❛ ♣❡♥❛ r❡ss❛❧t❛r q✉❡✱ ✉t✐❧✐③❛♥❞♦ ❛s ✈❛r✐á✈❡✐s ❛❞❡q✉❛❞❛s✱ ❛ ❡st✐♠❛çã♦

❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ r❡❝✉♣❡r❛çã♦ ❢❛③✲s❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ❡st✐♠❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡

❞❡ ❞❡❢❛✉❧t✳

  ❆♣ós ♦ ❡st✉❞♦ ❞♦s ♠♦❞❡❧♦s ✉t✐❧✐③❛❞♦s ❝♦♠✉♠♠❡♥t❡ ♥❛ ❛♥á❧✐s❡ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✱

♦♣t♦✉✲s❡ ♣♦r ✉t✐❧✐③❛r ♦ ♠♦❞❡❧♦ ❞❡ ❈r❡❞✐t ❙❝♦r✐♥❣ ❞❡ ❛♣r♦✈❛çã♦ ❞❡ ❝ré❞✐t♦✳ P❛r❛ ❛♥❛❧✐s❛r

❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✱ ✉t✐❧✐③♦✉✲s❡ ❛✐♥❞❛ ❛ té❝♥✐❝❛ ❡st❛tíst✐❝❛ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r ♠ú❧t✐♣❧❛✱

❞❡s✐❣♥❛❞❛♠❡♥t❡ ♦ ♠♦❞❡❧♦ ▲♦❣✐t ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡ ❛s té❝♥✐❝❛s ❙t❡♣✇✐s❡✲

❇❛❝❦✇❛r❞ ❡ ❆■❈ ♣❛r❛ ❛ s❡❧❡❝çã♦ ❞❛s ✈❛r✐á✈❡✐s ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛s ♥♦ q✉❡ r❡s♣❡✐t❛ ❛

❡①♣❧✐❝❛r ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✳

  ❆♠❜❛s ❛s té❝♥✐❝❛s ❞❡ s❡❧❡❝çã♦ ❛♣❧✐❝❛❞❛s✱ ❙t❡♣✇✐s❡ ❡ ❆■❈✱ r❡✈❡❧❛r❛♠ q✉❡ ❛s ✈❛r✐á✈❡✐s

q✉❡ ♠❡❧❤♦r ❡①♣❧✐❝❛♠ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✱ ❞❡ ❡♥tr❡ t♦❞❛s ❛s q✉❡ ❢♦r❛♠ ❛♥❛❧✐s❛❞❛s✱

sã♦ ❛ ■❞❛❞❡✱ ♦ ◆➸ ❞❡ ❉❡♣❡♥❞❡♥t❡s✱ ♦ ❊st❛❞♦ ❈✐✈✐❧✱ ♦ ❈ó❞✐❣♦ P♦st❛❧✱ ♦ ❘❡♥❞✐♠❡♥t♦✱

♦ ❱❛❧♦r ❞♦ ❈ré❞✐t♦✱ ❛ P❡r❝❡♥t❛❣❡♠ ❞❡ ❊♥tr❛❞❛✱ ♦ ◆➸ ❞❡ Pr❡st❛çõ❡s ❚♦t❛✐s ❡ ♦ ◆➸ ❞❡

Pr❡st❛çõ❡s P❛❣❛s✳ ❘❡❧❡♠❜r❛♥❞♦ ♦s ✈❛❧♦r❡s ❛tr✐❜✉í❞♦s ❛♦s ♣❡s♦s ❞❛s ✈❛r✐á✈❡✐s ✭✈❡r ❚❛❜❡❧❛ ♥♦ ♣r♦✲

❝❡ss♦ ❞❡ s✐♠✉❧❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❉❡❢❛✉❧t ❝♦♥❝❧✉✐✉✲s❡ q✉❡✱ ❛s ✈❛r✐á✈❡✐s q✉❡ ♦s ♠ét♦❞♦s

❞❡ s❡❧❡❝çã♦ ❝♦♥s✐❞❡r❛♠ ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛s✱ ❝♦rr❡s♣♦♥❞❡♠ às ✈❛r✐á✈❡✐s ❝✉❥♦s ♣❡s♦s ❛tr✐✲

❜✉í❞♦s sã♦ ♠❛✐s ❡❧❡✈❛❞♦s✱ ✈❛r✐❛♥❞♦ ❡♥tr❡ ♦s ✺✪ ❡ ♦s ✷✹✪✱ ❡✈✐❞❡♥❝✐❛♥❞♦✲s❡ ❛ss✐♠✱ ❛

♣♦t❡♥❝✐❛❧✐❞❛❞❡ ❞❡st❡s ♠ét♦❞♦s ❞❡ s❡❧❡❝çã♦✳ ❆ ❝❛♣❛❝✐❞❛❞❡ ❞❡ s❡❧❡❝çã♦ ❞❡st❡s ♠ét♦❞♦s

é ❛✐♥❞❛ ✈❡r✐✜❝❛❞❛✱ q✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ P❛❣❛s é s❡❧❡❝❝✐♦♥❛❞❛ ❝♦♠♦ s✐❣♥✐✜❝❛t✐✈❛✱ ♠❡s♠♦

♥ã♦ t❡♥❞♦ s✐❞♦ ❛ss♦❝✐❛❞♦ ✉♠ ♣❡s♦ ❛ ❡st❛ ✈❛r✐á✈❡❧ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♠❛s s✐♠ à ✈❛r✐á✈❡❧

P❛❣❛s✴❚♦t❛❧✱ ♦✉ s❡❥❛✱ ♦ ♠ét♦❞♦ é s✉✜❝✐❡♥t❡♠❡♥t❡ r♦❜✉st♦ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♦ ♣❡s♦ q✉❡ ❛

✈❛r✐á✈❡❧ P❛❣❛s t❡♠ ♥♦ ♠♦❞❡❧♦ ♣❡❧♦ ❢❛❝t♦ ❞❡ ❡st❛r ❛ss♦❝✐❛❞❛ ❛ ♦✉tr❛ ✈❛r✐á✈❡❧✳

  ′ s ❉❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ♠♦❞❡❧♦ ▲♦❣✐t ♦❜t✐✈❡r❛♠✲s❡ ♦s ❝♦❡✜❝✐❡♥t❡s✱ β i ✱ q✉❡ r❡✢❡❝t❡♠

  

✽✻ ❈❛♣ít✉❧♦ ✸✳ ▼♦❞❡❧❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✲ ❆♣❧✐❝❛çã♦

♠♦❞♦✱ ♣♦❞❡ ❞✐③❡r✲s❡ q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❝♦♠ ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s ❝♦rr❡s♣♦♥❞❡♠ às ✈❛r✐á✈❡✐s

q✉❡ ♠❡♥♦s ❝♦♥tr✐❜✉❡♠ ♣❛r❛ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✱ ❛♦ ❝♦♥trár✐♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❝♦♠

✈❛❧♦r❡s ♣♦s✐t✐✈♦s✳ ❊st❡ ❢❛❝t♦ ❛❞✈é♠ ❞❛ ❡s❝♦❧❤❛ ❢❡✐t❛ ✐♥✐❝✐❛❧♠❡♥t❡✱ ♣❛r❛ ♦s ✈❛❧♦r❡s ❞❛

✈❛r✐á✈❡❧ ❉❡❢❛✉❧t✿ ✈❛❧♦r 0 ♣❛r❛ ❝❧✐❡♥t❡s ❝✉♠♣r✐❞♦r❡s ❡ ✈❛❧♦r 1 ♣❛r❛ ❝❧✐❡♥t❡s ✐♥❝✉♠♣r✐❞♦r❡s✳

P❛r❛ ❛✈❛❧✐❛r ❛ ❝❛♣❛❝✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ♣r❡✈❡r ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t ❡st✐♠❛r❛♠✲s❡

❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❞❡❢❛✉❧t ❞❡ t♦❞♦s ♦s ❝❧✐❡♥t❡s ❞❛ ❝❛rt❡✐r❛ ❛tr❛✈és ❞♦ ♠♦❞❡❧♦ ▲♦❣✐t ❡✱

♣❛r❛ ❝❛❞❛ ❝❧✐❡♥t❡✱ ❢♦r❛♠ ❝♦♠♣❛r❛❞♦s ♦s r❡s✉❧t❛❞♦s ❝♦♠ ♦s ✈❛❧♦r❡s ❞❛ ✈❛r✐á✈❡❧ ❉❡❢❛✉❧t

s✐♠✉❧❛❞❛ ♣r❡✈✐❛♠❡♥t❡✳ ❖s r❡s✉❧t❛❞♦s ♠♦str❛♠ q✉❡ ♦ ♠♦❞❡❧♦ ▲♦❣✐t ❝♦♥s❡❣✉❡ ♣r❡✈❡r✱ ❡♠ ♠é❞✐❛✱ ❛❝❡rt❛✲

❞❛♠❡♥t❡ ❝❡r❝❛ ❞❡ 82, 53% ❞♦s ❝❛s♦s✱ q✉❛♥❞♦ s❡ ❞❡✜♥❡✱ ♣❛r❛ ♦ ♠♦❞❡❧♦ ▲♦❣✐t✱ q✉❡ ❛s

♣r♦❜❛❜✐❧✐❞❛❞❡s ❛❝✐♠❛ ❞❡ 50% sã♦ ✐♥❝✉♠♣r✐♠❡♥t♦s ❡ ♣r❡✈ê✱ ❡♠ ♠é❞✐❛✱ ❝♦rr❡❝t❛♠❡♥t❡

88, 63% q✉❛♥❞♦ s❡ ❝♦♥s✐❞❡r❛ q✉❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡st✐♠❛❞❛s ❛❝✐♠❛ ❞♦s 25%✱ sã♦ ❝♦♥s✐✲

❞❡r❛❞❛s s✐t✉❛çõ❡s ❡♠ q✉❡ ♦❝♦rr❡ ❞❡❢❛✉❧t✳ ❉❡st❛ ❢♦r♠❛✱ t❡♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ❡❧❡✈❛❞❛

t❛①❛ ❞❡ ♣r❡✈✐s✐❜✐❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❛❥✉st❛❞♦ ❡ ♦s ✈❛❧♦r❡s ❡❧❡✈❛❞♦s ❞♦ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡✲

t❡r♠✐♥❛çã♦ ❆❥✉st❛❞♦✱ ❝♦♥s✐❞❡r❛✲s❡ q✉❡ ❡st❡ ♠♦❞❡❧♦ ♣♦❞❡rá s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡st✐♠❛r

❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ♣❛r❛ ♥♦✈♦s ❝❧✐❡♥t❡s s♦❧✐❝✐t❛♥t❡s ❞❡ ✉♠ ❝ré❞✐t♦ à ❍❛❜✐t❛çã♦✳

  ❋✐❝❛♠ ❡♠ ❛❜❡rt♦ ❛❧❣✉♠❛s q✉❡stõ❡s✱ q✉❡ ♥ã♦ ❢♦r❛♠ ❛♥❛❧✐s❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦✱ ♠❛s q✉❡ ♣♦❞❡♠ s❡r ♦❜❥❡❝t♦ ❞❡ ❡st✉❞♦s ❢✉t✉r♦s✱ ♥♦♠❡❛❞❛♠❡♥t❡✿ ❼ ❆ ❊st✐♠❛çã♦ ❞❛ t❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦✱ ♥❡❝❡ssár✐❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ s♣r❡❛❞ ♠í♥✐♠♦ ❛ ❛♣❧✐❝❛r✱ ♠❛s q✉❡ s❡ ❞❡❞✉③ s❡r ❡st✐♠❛❞❛ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡

  ❞❡❢❛✉❧t r❡❝♦rr❡♥❞♦✱ ♥♦ ❡♥t❛♥t♦✱ ❛ ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❞✐st✐♥t❛s✳ ❼ ❆♣❧✐❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❱órt✐❝❡s ❊st♦❝ást✐❝♦s✱ ✈❡r ❛ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❞✐♥â♠✐❝❛✱ ✐st♦ é✱ ❛❣r✉♣❛r ♦s ❝❧✐❡♥t❡s ❡♠ ✏❝❧❛ss❡s ❞❡ r✐s❝♦✑ ✭❝♦♠ ❜❛s❡ ♥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t ❡st✐♠❛❞❛✮✱ ❡st✐♠❛r ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❡♥tr❡ ❝❧❛ss❡s ❞❡ r✐s❝♦ ❡✱ ❛ss✉♠✐♥❞♦ q✉❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❛❜❡rt❛ ✭♣❡r♠✐t✐♥❞♦ ❛ s✉❜s❝r✐çã♦ ❞❡ ♥♦✈♦s ❝♦♥tr❛t♦s ❡ ❛ ❛♥✉❧❛çã♦ ❞❡ ❝♦♥tr❛t♦s ❡①✐st❡♥t❡s✮ ❛♥❛❧✐s❛r✱ ♥✉♠❛ ♣❡rs♣❡❝t✐✈❛ t❡♠♣♦r❛❧ ✱ ❛ ❡✈♦❧✉çã♦ ❞❛ ❞✐♠❡♥sã♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝❧❛ss❡s ❞❡ r✐s❝♦✱ ❡st✐♠❛♥❞♦ ♦ ♣❡s♦ r❡❧❛t✐✈♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝❧❛ss❡s ❞❡♥tr♦ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦✳

  ✽✼ ❆♣ê♥❞✐❝❡ ❆ ❚❛❜❡❧❛s

◆❡st❡ ❛♣ê♥❞✐❝❡ ❛♣r❡s❡♥t❛♠✲s❡ ❛❧❣✉♠❛s t❛❜❡❧❛s r❡❢❡r✐❞❛s ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❞❡✲

s✐❣♥❛❞❛♠❡♥t❡ ❛ ❚❛❜❡❧❛ ❞❛ ▼♦♦❞②✬s✿✏Pr♦❥❡çã♦ ●❧♦❜❛❧ ❞❡ Pr♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❚r❛♥s✐çã♦ ❛

✶✲❆♥♦✭P❡r❝❡♥t❛❣❡♠✮✑✱ ❛ ❚❛❜❡❧❛ ❞❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❡ r❡s♣❡❝t✐✈❛s ❝❛t❡❣♦r✐❛s ✉t✐❧✐✲

③❛❞❛s ♥❛ s✐♠✉❧❛çã♦ ❞❛ ❝❛rt❡✐r❛ ❞❡ ❝ré❞✐t♦ ❡ ❛ ❚❛❜❡❧❛ q✉❡ ✐❧✉str❛ ✉♠❛ ♣❛rt❡ ❞❛ ❝❛rt❡✐r❛

❞❡ ❝ré❞✐t♦ s✐♠✉❧❛❞❛✳

  ✽✽ ❆✳ ❚❛❜ ❡❧❛s

  ❆❛❛ ❆❛✶ ❆❛✷ ❆❛✸ ❆✶ ❆✷ ❆✸ ❇❛❛✶ ❇❛❛✷ ❇❛❛✸ ❇❛✶ ❇❛✷ ❇❛✸ ❇✶ ❇✷ ❇✸ ❈❛❛✶ ❈❛❛✷ ❈❛❛✸ ❈❛✲ ❈ ❉❊❋ ❆❛❛ ✾✷✳✹ ✷✳✷ ✶✳✸ ✵✳✶ ✵✳✵ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵

  ❆❛✶ ✹✳✸ ✽✹✳✹ ✹✳✻ ✷✳✵ ✵✳✸ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ❆❛✷ ✷✶✳✹ ✺✳✼ ✼✽✳✹ ✺✳✽ ✶✳✷ ✵✳✹ ✵✳✵ ✵✳✵ ✵✳✶ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ❆❛✸ ✸✵✳✶ ✶✳✵ ✹✳✶ ✽✸✳✻ ✸✳✺ ✶✳✸ ✵✳✹ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ❆✶ ✶✵✳✵ ✵✳✶ ✵✳✹ ✺✳✼ ✽✷✳✾ ✹✳✶ ✶✳✼ ✵✳✺ ✵✳✷ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ❆✷ ✷✵✳✵ ✵✳✵ ✵✳✷ ✶✳✵ ✺✳✵ ✽✷✳✾ ✹✳✶ ✶✳✹ ✵✳✹ ✵✳✷ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ❆✸ ✸✵✳✶ ✵✳✶ ✵✳✶ ✵✳✷ ✵✳✾ ✼✳✻ ✼✽✳✼ ✹✳✺ ✷✳✶ ✵✳✼ ✵✳✷ ✵✳✶ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ❇❛❛✶ ✶✵✳✶ ✵✳✶ ✵✳✶ ✵✳✶ ✵✳✶ ✵✳✼ ✺✳✾ ✼✾✳✽ ✺✳✵ ✶✳✼ ✵✳✹ ✵✳✶ ✵✳✶ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✶ ❇❛❛✷ ✷✵✳✶ ✵✳✶ ✵✳✶ ✵✳✶ ✵✳✶ ✵✳✼ ✶✳✻ ✺✳✺ ✼✽✳✻ ✺✳✶ ✶✳✷ ✵✳✼ ✵✳✹ ✵✳✷ ✵✳✶ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ❇❛❛✸ ✸✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✷ ✵✳✺ ✵✳✻ ✷✳✷ ✽✳✸ ✼✻✳✷ ✸✳✸ ✶✳✾ ✶✳✵ ✵✳✹ ✵✳✷ ✵✳✶ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✷ ❇❛✶ ✶✵✳✵ ✵✳✵ ✵✳✶ ✵✳✶ ✵✳✺ ✵✳✹ ✵✳✹ ✶✳✷ ✹✳✶ ✶✸✳✽ ✺✾✳✽ ✸✳✾ ✸✳✹ ✶✳✸ ✵✳✾ ✵✳✺ ✵✳✷ ✵✳✶ ✵✳✵ ✵✳✶ ✵✳✷ ❇❛✷ ✷✵✳✵ ✵✳✵ ✵✳✶ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✷ ✵✳✺ ✶✳✷ ✹✳✺ ✶✶✳✷ ✺✽✳✻ ✹✳✽ ✷✳✾ ✷✳✶ ✶✳✷ ✵✳✼ ✵✳✷ ✵✳✶ ✵✳✷ ✵✳✺ ❇❛✸ ✸✵✳✵ ✵✳✵ ✵✳✶ ✵✳✶ ✵✳✵ ✵✳✸ ✵✳✷ ✵✳✷ ✵✳✻ ✶✳✷ ✸✳✽ ✾✳✺ ✻✵✳✻ ✹✳✶ ✹✳✸ ✶✳✾ ✵✳✼ ✵✳✹ ✵✳✷ ✵✳✶ ✵✳✺ ❇✶ ✶✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✶ ✵✳✸ ✵✳✷ ✵✳✹ ✶✳✵ ✷✳✾ ✽✳✺ ✻✸✳✶ ✺✳✹ ✹✳✶ ✶✳✻ ✵✳✾ ✵✳✸ ✵✳✷ ✵✳✽ ❇✷ ✷✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✷ ✵✳✷ ✵✳✷ ✵✳✷ ✵✳✹ ✵✳✽ ✷✳✺ ✼✳✽ ✻✺✳✽ ✺✳✸ ✸✳✵ ✶✳✻ ✵✳✻ ✵✳✹ ✶✳✻ ❇✸ ✸✵✳✵ ✵✳✵ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✶ ✵✳✶ ✵✳✶ ✵✳✶ ✵✳✶ ✵✳✸ ✵✳✺ ✸✳✵ ✻✳✷ ✻✹✳✾ ✹✳✶ ✸✳✻ ✶✳✵ ✶✳✶ ✸✳✶ ❈❛❛✶ ✶✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✶ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✶ ✵✳✵ ✵✳✷ ✵✳✷ ✵✳✼ ✷✳✸ ✺✳✽ ✺✺✳✵ ✹✳✺ ✸✳✻ ✷✳✺ ✼✳✼ ❈❛❛✷ ✷✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✶ ✵✳✹ ✵✳✷ ✵✳✵ ✵✳✺ ✵✳✻ ✵✳✽ ✷✳✽ ✸✳✹ ✺✶✳✹ ✷✳✸ ✹✳✷ ✶✹✳✵ ❈❛❛✸ ✸✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✶ ✵✳✹ ✵✳✵ ✵✳✶ ✶✳✹ ✵✳✺ ✶✳✹ ✸✳✸ ✹✳✽ ✹✵✳✶ ✺✳✼ ✶✽✳✸ ❈❛✲❈ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✵ ✵✳✷ ✵✳✶ ✵✳✸ ✵✳✼ ✵✳✹ ✶✳✷ ✷✳✷ ✸✺✳✸ ✷✹✳✹

  ❚❛❜❡❧❛ ❆✳✶✿ ▼♦♦❞②✬s✿Pr♦❥❡çã♦ ●❧♦❜❛❧ ❞❡ Pr♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ❚r❛♥s✐çã♦ ❛ ✶✲❆♥♦✭✪✮

  ✽✾ ❱❛r✐á✈❡✐s ❈❛t❡❣♦r✐❛s ❉❡s❝r✐çã♦ ✪

  ❚✐♣♦ ✸ Pr♦❢❡ss♦r Pr✐♠ár✐♦✱ ❊♥❢❡r♠❡✐r❛✱ ❆ss✐st❡♥t❡ ❙♦✲ ❝✐❛❧✱ P❛rt❡✐r❛✱ ◗✉❛❞r♦ ▼é❞✐♦ ❋✉♥çã♦ Pú❜❧✐❝❛✱ ◗✉❛❞r♦ ▼é❞✐♦ ❙❡❝t♦r Pr✐✈❛❞♦✱ ❚é❝♥✐❝♦s✱ ❈❤❡❢❡ ❞❡ ❡q✉✐♣❛✱ ❈♦♥tr❛♠❡str❡ ✸✺✱✵✵✪

  ✷ ❃ ✷✺ ❡ ≤ ✺✵ ✲ ✸ ❃ ✺✵ ❡ ≤ ✼✺ ✲ ✹ ❃ ✼✺ ❡ ≤ ✶✵✵ ✲

  ✶✵ ✻✵✵ ✵✱✻✵✪

  ✷ ✶✷✵ ✶✱✵✻✪ ✸ ✶✽✵ ✹✱✷✺✪ ✹ ✷✹✵ ✶✶✱✶✺✪ ✺ ✸✵✵ ✷✵✱✵✼✪ ✻ ✸✻✵ ✷✺✱✵✽✪ ✼ ✹✷✵ ✷✶✱✺✵✪ ✽ ✹✽✵ ✶✷✱✵✾✪ ✾ ✺✹✵ ✹✱✵✸✪

  ▼❛✐s✶✵ ❃✶✵ ✲ ◆➸ ❚♦t❛❧ Pr❡st❛çõ❡s ✭♠❡s❡s✮ ✶ ✻✵ ✵✱✶✻✪

  ❆❧t♦ ❃ ✺✵✵✵ ✸✱✵✵✪ ❈ré❞✐t♦✭e✮

P❡r❝❡♥t❛❣❡♠ ❞❡ ❊♥tr❛❞❛ ▼❡♥♦s✶✵ ≤ 10 ✲

  ≤ ✼✺✵ ✹✵✱✵✪ ❇❛✐①♦ ❪✼✺✵✱✶✺✵✵❪ ✸✼✱✵✵✪ ▼é❞✐♦ ❪✶✺✵✵✱✺✵✵✵❪ ✷✵✱✵✵✪

  ❆♥t✐❣♦ ❃✶ ✾✷✱✺✵✪ ❘❡♥❞✐♠❡♥t♦ ▲íq✉✐❞♦ ✭e✮ ▼✐♥✐♠♦

  ✶✺✱✵✵✪ ❆♥t✐❣✉✐❞❛❞❡ Pr♦✜ssã♦ ✭❛♥♦s✮ ❘❡❝❡♥t❡ ≤ ✶ ✼✱✺✵✪

  ❚✐♣♦ ✹ ◗✉❛❞r♦ ❙✉♣❡r✐♦r ❞❛ ❋✉♥çã♦ Pú❜❧✐❝❛✱ Pr♦❢❡s✲ s♦r✱ ■♥✈❡st✐❣❛❞♦r✱ ▼é❞✐❝♦✱ ◗✉❛♥❞r♦ ❙✉♣❡r✐♦r ❙❡❝t♦r Pr✐✈❛❞♦✱ ❘❡❢♦r♠❛❞♦

  ♣r❡❣❛❞♦ ❊s❝r✐tór✐♦✱ ❊♠♣r❡❣❛❞♦ ❈♦♠ér❝✐♦✱ ❊♠♣r❡❣❛❞♦ ❙❡r✈✐ç♦s✱ ❖♣❡rár✐♦✱ ❖✉tr♦s ✸✵✱✵✵✪

  ■❞❛❞❡ ✭❛♥♦s✮ ❏♦✈❡♥s ❬✶✽✱✷✺❪ ✶✸✱✵✵✪ ❆❞✉❧t♦s ❬✷✻✱✺✵❪ ✽✺✱✵✵✪ ■❞♦s♦s ❃✺✵ ✷✱✵✵✪

  ✷✵✱✵✪ ❚✐♣♦ ✷ ❆❞♠✐♥✐str❛t✐✈♦✱ ❆✉①✐❧✐❛r ❋✉♥çã♦ Pú❜❧✐❝❛✱ ❊♠✲

  ❚✐♣♦ Pr♦✜ssã♦ ❚✐♣♦ ✶ ❆❣r✐❝✉❧t♦r✱ ❆◆■✴❊♠♣r❡sár✐♦✱ ❈♦♠❡r❝✐❛♥t❡✱ ●❡r❡♥t❡ ❞❡ ❙♦❝✐❡❞❛❞❡✱ Pr♦✜ssã♦ ▲✐❜❡r❛❧✱ ❆r✲ t✐st❛✱ ❏♦r♥❛❧✐st❛✱ P♦❧í❝✐❛✱ ▼✐❧✐t❛r✱ ●◆❘✱ ❈❛✲ ♠✐♦♥✐st❛✱ ❚❛①✐st❛✱ P❡s❝❛❞♦r✱ ❆ss❛❧❛r✐❛❞♦ ❆❣rí✲ ❝♦❧❛✱ ❉❡s❡♠♣r❡❣❛❞♦✱ ❱✐✈❡ ❞❡ ❘❡♥❞✐♠❡♥t♦s✱ ❊st✉❞❛♥t❡✱ ❉♦♠ést✐❝❛✱ ❚r❛❜❛❧❤❛❞♦r ❚❡♠♣♦rá✲ r✐♦

  ✺✵✱ ✺✶✱ ✺✷✱ ✺✹✱ ✻✶✱ ✻✸✱ ✻✹✱ ✼✶✱ ✼✺✱ ✼✻✱ ✼✽✱ ✽✵✱ ✽✷✱ ✽✻✱ ✽✼✱ ✽✽✱ ✽✾✱ ✾✶✱ ✾✸✱ ✾✹✱ ✾✺✱ ✾✾ ✺✱✵✵✪

  ❩♦♥❛ ✷ ✶✸✱ ✷✷✱ ✷✹✱ ✷✻✱ ✷✼✱ ✷✾✱ ✸✶✱ ✹✾✱ ✼✸✱ ✾✻ ✷✵✱✵✵✪ ❩♦♥❛ ✸ ✶✷✱ ✶✽✱ ✶✾✱ ✷✸✱ ✷✽✱ ✸✹✱ ✹✹✱ ✹✻✱ ✹✼✱ ✼✵✱ ✼✹✱ ✾✵ ✶✵✱✵✵✪ ❩♦♥❛ ✹ ✶✵✱ ✶✹✱ ✶✺✱ ✶✻✱ ✸✵✱ ✸✷✱ ✸✺✱ ✸✻✱ ✸✼✱ ✹✵✱ ✹✺✱ ✹✽✱

  ❈ó❞✐❣♦s P♦st❛✐s ❩♦♥❛ ✶ ✶✶✱ ✶✼✱ ✷✵✱ ✷✶✱ ✷✺✱ ✸✸✱ ✸✽✱ ✹✶✱ ✹✷✱ ✹✸✱ ✺✸✱ ✻✵✱ ✻✷✱ ✼✷✱ ✼✼✱ ✼✾✱ ✽✱ ✽✸✱ ✽✺✱ ✾✷✱ ✾✼✱ ✾✽ ✻✺✱✵✪

  ❊st❛❞♦ ❈✐✈✐❧ ❉✐✈ ❉✐✈♦r❝✐❛❞♦✱ ❙❡♣❛r❛❞♦ ✷✱✻✵✪ ❙♦❧ ❙♦❧t❡✐r♦✱ ❯♥✐ã♦ ▲✐✈r❡✱ ❱✐ú✈♦ ✹✼✱✽✵✪ ❈❛s ❈❛s❛❞♦ ✹✾✱✻✵✪

  ■❞❛❞❡ ❞❛ ❈❛s❛ ◆♦✈❛ ≤ ✺ ❛♥♦s ✼✺✱✵✵✪ ❯s❛❞❛ ❬✻✱✶✺❪ ❛♥♦s ✷✵✱✵✵✪ ❱❡❧❤❛ ❃ ✶✺ ✺✱✵✵✪

  ❚✐♣♦ ❍❛❜✐t❛çã♦ ❈❛s❛ ▼♦r❛❞✐❛✱ ❆♣❛rt❛♠❡♥t♦ ✽✾✱✵✵✪ ❊♠♣r❡s❛ Pré❞✐♦✱ ❊♠♣r❡s❛ ✸✱✵✵✪ ❈♦♠❡r❝✐♦ ❈♦♠ér❝✐♦✱ ❚❡rr❡♥♦✱ ❖✉tr♦ ✽✱✵✵✪

  

◆➸ ❉❡♣❡♥❞❡♥t❡s P♦✉❝♦s ✵ ♦✉ ✶ ✽✵✱✵✵✪

❆❧❣✉♥s ✷ ♦✉ ✸ ✶✺✱✵✵✪ ▼✉✐t♦s ≥ ✹ ✺✱✵✵✪

P❛❣❛s ◆➸ ❆❧❡❛tór✐♦ ❡♥tr❡ ✻✵ ❡ ◆➸ ❚♦t❛❧ Pr❡st❛çõ❡s P❛❣❛s ✴ ❚♦t❛❧ ✭✪✮ ✶ ≤ ✷✺ ✲

  ✾✵ ❆✳ ❚❛❜ ❡❧❛s

  ❈❧✐❡♥t❡ ■❞❛❞❡ ❉❡♣ ❍❛❜ ■❞❝❛s❛ ❈✐✈✐❧ P♦st❛❧ Pr♦❢ ❆♥t❣ ❘❡♥❞ ❈r❡❞ ❊♥tr Pt♦t❛❧ P❛❣❛s ❉❡❝♣❡r❝♥t ❉❡❢❛✉❧t ✶ ❆❞✉❧t♦s P♦✉❝♦s ❈♦♠❡r❝✐♦ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✷ ❘❡❝❡♥t❡ ▼❡❞✐♦ ✶✺✾✷✶✻ ▼❛✐s✶✵ ✹✷✵ ✾✹ ▼❡♥♦s❊♥tr❡✷✺❡✺✵✺ ✵ ✷ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✾✻✽✼✽ ▼❡♥♦s✶✵ ✸✻✵ ✶✻✵ ❊♥tr❡✷✺❡✺✵ ✶ ✸ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✶✷✶✻✼✶ ▼❡♥♦s✶✵ ✷✹✵ ✶✻✹ ❊♥tr❡✺✵❡✼✺ ✵ ✹ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ❱❡❧❤❛ ❙♦❧ ❩♦♥❛✸ ❚✐♣♦✹ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✷✶✶✸✽✻ ▼❛✐s✶✵ ✹✷✵ ✸✵✵ ❊♥tr❡✺✵❡✼✺ ✵ ✺ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ❱❡❧❤❛ ❙♦❧ ❩♦♥❛✷ ❚✐♣♦✹ ❆♥t✐❣♦ ❇❛✐①♦ ✷✽✶✼✵✷ ▼❡♥♦s✶✵ ✸✵✵ ✾✼ ❊♥tr❡✷✺❡✺✵ ✵ ✻ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✸ ❚✐♣♦✷ ❆♥t✐❣♦ ❇❛✐①♦ ✷✷✾✺✼✽ ▼❡♥♦s✶✵ ✷✹✵ ✶✼✺ ❊♥tr❡✺✵❡✼✺ ✵ ✼ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✷ ❘❡❝❡♥t❡ ▼✐♥✐♠♦ ✾✷✺✵✵ ▼❡♥♦s✶✵ ✸✻✵ ✷✵✵ ❊♥tr❡✺✵❡✼✺ ✵ ✽ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✸ ❚✐♣♦✷ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✷✷✽✾✺✺ ▼❛✐s✶✵ ✹✷✵ ✷✽✼ ❊♥tr❡✺✵❡✼✺ ✵ ✾ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✷ ❚✐♣♦✷ ❆♥t✐❣♦ ▼❡❞✐♦ ✶✹✶✺✹✸ ▼❡♥♦s✶✵ ✹✷✵ ✶✾✷ ❊♥tr❡✷✺❡✺✵ ✵

  ✶✵ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ❯s❛❞❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✷ ❘❡❝❡♥t❡ ▼✐♥✐♠♦ ✶✾✹✷✷✽ ▼❛✐s✶✵ ✸✵✵ ✽✽ ❊♥tr❡✷✺❡✺✵ ✵ ✶✶ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ❇❛✐①♦ ✶✺✽✽✵✽ ▼❡♥♦s✶✵ ✹✽✵ ✶✺✺ ❊♥tr❡✷✺❡✺✵ ✵ ✶✷ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✹ ❚✐♣♦✷ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✶✻✻✸✸✺ ▼❡♥♦s✶✵ ✸✵✵ ✶✶✵ ❊♥tr❡✷✺❡✺✵ ✵ ✶✸ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✶ ❚✐♣♦✸ ❆♥t✐❣♦ ❆❧t♦ ✾✽✷✼✼ ▼❛✐s✶✵ ✸✻✵ ✸✸✶ ▼❛✐s✼✺ ✵ ✶✹ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✶✾✽✷✶✾ ▼❡♥♦s✶✵ ✹✷✵ ✷✼✻ ❊♥tr❡✺✵❡✼✺ ✵ ✶✺ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✹ ❚✐♣♦✹ ❆♥t✐❣♦ ▼❡❞✐♦ ✶✷✶✹✵✾ ▼❡♥♦s✶✵ ✸✵✵ ✶✼✼ ❊♥tr❡✺✵❡✼✺ ✵ ✶✻ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✸ ❚✐♣♦✶ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✸✾✼✺✵✻ ▼❡♥♦s✶✵ ✸✻✵ ✷✸✽ ❊♥tr❡✺✵❡✼✺ ✵ ✶✼ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✶ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✶✵✶✹✵✷ ▼❛✐s✶✵ ✸✻✵ ✶✼✶ ❊♥tr❡✷✺❡✺✵ ✵ ✶✽ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✶ ❆♥t✐❣♦ ❇❛✐①♦ ✶✷✶✾✾✾ ▼❛✐s✶✵ ✸✵✵ ✷✽✼ ▼❛✐s✼✺ ✵ ✶✾ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✶✶✶✷✺✷ ▼❡♥♦s✶✵ ✹✷✵ ✷✽✻ ❊♥tr❡✺✵❡✼✺ ✵ ✷✵ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✸ ❚✐♣♦✹ ❆♥t✐❣♦ ▼❡❞✐♦ ✶✻✼✶✺✶ ▼❛✐s✶✵ ✹✽✵ ✹✵✸ ▼❛✐s✼✺ ✵ ✷✶ ❆❞✉❧t♦s P♦✉❝♦s ❈♦♠❡r❝✐♦ ❯s❛❞❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✶✻✵✶✵✸ ▼❛✐s✶✵ ✷✹✵ ✶✹✽ ❊♥tr❡✺✵❡✼✺ ✵ ✷✷ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✶ ❚✐♣♦✸ ❆♥t✐❣♦ ❇❛✐①♦ ✸✽✾✷✵✾ ▼❛✐s✶✵ ✹✷✵ ✸✵✺ ❊♥tr❡✺✵❡✼✺ ✵ ✷✸ ❆❞✉❧t♦s ❆❧❣✉♥s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✷ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✸✷✷✵✵✸ ▼❡♥♦s✶✵ ✸✻✵ ✽✻ ▼❡♥♦s❊♥tr❡✷✺❡✺✵✺ ✵ ✷✹ ❆❞✉❧t♦s P♦✉❝♦s ❊♠♣r❡s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ❇❛✐①♦ ✶✸✹✽✽✷ ▼❡♥♦s✶✵ ✹✽✵ ✸✷✹ ❊♥tr❡✺✵❡✼✺ ✵ ✷✺ ❆❞✉❧t♦s P♦✉❝♦s ❈♦♠❡r❝✐♦ ◆♦✈❛ ❙♦❧ ❩♦♥❛✹ ❚✐♣♦✹ ❆♥t✐❣♦ ▼❡❞✐♦ ✷✹✾✷✹✸ ▼❡♥♦s✶✵ ✹✽✵ ✶✽✷ ❊♥tr❡✷✺❡✺✵ ✵ ✷✻ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ▼✐♥✐♠♦ ✶✼✺✽✸✻ ▼❛✐s✶✵ ✹✽✵ ✷✹✶ ❊♥tr❡✺✵❡✼✺ ✵ ✷✼ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✹ ❚✐♣♦✶ ❆♥t✐❣♦ ▼❡❞✐♦ ✾✵✾✼✻ ▼❛✐s✶✵ ✸✻✵ ✶✾✻ ❊♥tr❡✺✵❡✼✺ ✵ ✷✽ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❈❛s ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ▼❡❞✐♦ ✾✻✼✵✶ ▼❛✐s✶✵ ✸✻✵ ✽✻ ▼❡♥♦s❊♥tr❡✷✺❡✺✵✺ ✵ ✷✾ ❆❞✉❧t♦s ❆❧❣✉♥s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✹ ❚✐♣♦✶ ❆♥t✐❣♦ ❇❛✐①♦ ✶✵✾✺✶✸ ▼❛✐s✶✵ ✸✻✵ ✻✶ ▼❡♥♦s❊♥tr❡✷✺❡✺✵✺ ✵ ✸✵ ❆❞✉❧t♦s P♦✉❝♦s ❈❛s❛ ◆♦✈❛ ❙♦❧ ❩♦♥❛✹ ❚✐♣♦✸ ❆♥t✐❣♦ ❇❛✐①♦ ✶✾✾✾✻✽ ▼❛✐s✶✵ ✹✷✵ ✷✵✻ ❊♥tr❡✷✺❡✺✵ ✵

  ❚❛❜❡❧❛ ❆✳✸✿ ❈❛rt❡✐r❛ ❞❡ ❈ré❞✐t♦ ✲ ❇❛s❡ ❞❡ ❉❛❞♦s

  ✾✶ ❆♣ê♥❞✐❝❡ ❇ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

◆❡st❡ ❛♣ê♥❞✐❝❡ ❛♣r❡s❡♥t❛♠✲s❡ ❛❧❣✉♥s ✐♥♣✉ts ✉t✐❧✐③❛❞♦s ❡ ♦s ♦✉t♣✉ts ❣❡r❛❞♦s✭❞❡✈❡ t❡r✲s❡

❡♠ ❝♦♥s✐❞❡r❛çã♦ q✉❡ ♦s ✐♥♣✉ts ❛♣r❡s❡♥t❛❞♦s sã♦ ❛♣❡♥❛s ❡①❡♠♣❧✐✜❝❛t✐✈♦s✱ ❡ ♦ ❛✉t♦r ♥ã♦

s❡ r❡s♣♦♥s❛❜✐❧✐③❛ ♣♦r q✉❛❧q✉❡r ✉t✐❧✐③❛çã♦ ✐♥❞❡✈✐❞❛ ❞♦s ♠❡s♠♦s✮✳ P❛r❛ ✹ ❩♦♥❛s P♦st❛✐s ❃ ❞❛t❛ ❁✲ r❡❛❞✳t❛❜❧❡✭✧❉❆❉❖❙❈❘③✶③✷s❡♠❉❡❝✳t①t✧✱ ❤❡❛❞❡r ❂ ❚❘❯❊✮ ❃ ❛tt❛❝❤✭❞❛t❛✮ ❃ ♥❛♠❡s✭❞❛t❛✮

  ❬✶❪ ✧■❞❛❞❡✧ ✧❉❡♣✧ ✧❍❛❜✧ ✧■❞❝❛s❛✧ ✧❈✐✈✐❧✧ ❬✻❪ ✧P♦st❛❧✧ ✧Pr♦❢✧ ✧❆♥t❣✧ ✧❘❡♥❞✧ ✧❈r❡❞✧ ❬✶✶❪ ✧❊♥tr✧ ✧Pt♦t❛❧✧ ✧P❛❣❛s✧ ✧P❛❣♣❡r❝♥t✧ ✧❉❡❢❛✉❧t✧

  ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔ ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛

  ▼♦❞❡❧♦ ❈♦♠♣❧❡t♦ ➔ ❃ ▼♦❞❢✉❧❧ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✾✷❡✰✵✵ ✲✼✳✶✸❡✲✵✸ ✲✸✳✼✾❡✲✵✹ ✲✶✳✷✾❡✲✵✺ ✷✳✽✵❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✺✾❡✰✵✶ ✻✳✵✼❡✰✵✷ ✲✵✳✵✽ ✵✳✾✹✵ ■❞❛❞❡■❞♦s♦s ✲✷✳✵✾❡✰✵✵ ✽✳✼✹❡✲✵✶ ✲✷✳✹✵ ✵✳✵✶✼ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✾✺❡✰✵✵ ✸✳✷✷❡✲✵✶ ✶✷✳✷✾ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✸✵❡✰✵✵ ✻✳✽✾❡✲✵✶ ✶✳✽✾ ✵✳✵✺✾ ✳ ❉❡♣P♦✉❝♦s ✷✳✽✺❡✰✵✵ ✸✳✺✸❡✲✵✶ ✽✳✵✺ ✽✳✶❡✲✶✻ ✯✯✯ ❍❛❜❈♦♠❡r❝✐♦ ✶✳✷✻❡✲✵✷ ✸✳✻✽❡✲✵✶ ✵✳✵✸ ✵✳✾✼✸ ❍❛❜❊♠♣r❡s❛ ✲✷✳✼✵❡✲✵✶ ✻✳✵✾❡✲✵✶ ✲✵✳✹✹ ✵✳✻✺✼ ■❞❝❛s❛❯s❛❞❛ ✲✻✳✼✸❡✲✵✷ ✷✳✻✶❡✲✵✶ ✲✵✳✷✻ ✵✳✼✾✻ ■❞❝❛s❛❱❡❧❤❛ ✺✳✹✺❡✲✵✶ ✹✳✽✷❡✲✵✶ ✶✳✶✸ ✵✳✷✺✾ ❈✐✈✐❧❉✐✈ ✸✳✸✽❡✰✵✵ ✻✳✵✹❡✲✵✶ ✺✳✻✵ ✷✳✶❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✻✵❡✰✵✵ ✷✳✷✽❡✲✵✶ ✻✳✾✾ ✷✳✼❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✷ ✷✳✸✺❡✰✵✵ ✶✳✹✾❡✰✵✵ ✶✳✺✽ ✵✳✶✶✺

  ✾✷ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  P♦st❛❧❩♦♥❛✸ ✺✳✹✽❡✰✵✵ ✶✳✹✶❡✰✵✵ ✸✳✾✵ ✾✳✽❡✲✵✺ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✼✳✻✾❡✰✵✵ ✶✳✹✷❡✰✵✵ ✺✳✹✶ ✻✳✸❡✲✵✽ ✯✯✯ Pr♦❢❚✐♣♦✷ ✲✸✳✸✷❡✲✵✶ ✷✳✾✽❡✲✵✶ ✲✶✳✶✷ ✵✳✷✻✺ Pr♦❢❚✐♣♦✸ ✲✺✳✶✶❡✲✵✶ ✷✳✽✺❡✲✵✶ ✲✶✳✼✾ ✵✳✵✼✸ ✳ Pr♦❢❚✐♣♦✹ ✲✻✳✸✻❡✲✵✶ ✸✳✻✸❡✲✵✶ ✲✶✳✼✺ ✵✳✵✽✵ ✳ ❆♥t❣❘❡❝❡♥t❡ ✲✺✳✷✽❡✲✵✶ ✸✳✽✹❡✲✵✶ ✲✶✳✸✽ ✵✳✶✻✾ ❘❡♥❞❇❛✐①♦ ✶✳✼✷❡✰✵✶ ✻✳✵✼❡✰✵✷ ✵✳✵✸ ✵✳✾✼✼ ❘❡♥❞▼❡❞✐♦ ✽✳✼✹❡✰✵✵ ✻✳✵✼❡✰✵✷ ✵✳✵✶ ✵✳✾✽✾ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✽❡✰✵✶ ✻✳✵✼❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✻❡✲✵✺ ✶✳✻✼❡✲✵✻ ✲✻✳✸✼ ✶✳✾❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✶✹❡✰✵✵ ✸✳✼✺❡✲✵✶ ✶✸✳✼✵ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✻❡✲✵✷ ✷✳✷✽❡✲✵✸ ✶✻✳✵✻ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✼✵❡✲✵✷ ✸✳✻✺❡✲✵✸ ✲✶✺✳✻✷ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮

  ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✹✹✳✷✽ ♦♥ ✾✾✼✺ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✾✹✳✸ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ➔

  ▼♦❞❡❧♦ ◆✉❧♦ ❃ ▼♦❞♥✉❧❧ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✷✳✻✶✺ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✸✳✸✽✼✶ ✵✳✵✺✻✷ ✲✻✵✳✷ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✷✽✽✷ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✻ ➔ ❈♦♠♣❛r❛çã♦ ▼♦❞❡❧♦ ◆✉❧♦ ❱s ▼♦❞❡❧♦ ❈♦♠♣❧❡t♦ ❃ ❛♥♦✈❛✭▼♦❞♥✉❧❧✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ✶ ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s

  ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮ ✶ ✾✾✾✾ ✷✽✽✵ ✷ ✾✾✼✺ ✻✹✹ ✷✹ ✷✷✸✻ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔

  ✾✸ P❛r❛ ✸ ❩♦♥❛s P♦st❛✐s ❃ ❞❛t❛ ❁✲ r❡❛❞✳t❛❜❧❡✭✧❉❆❉❖❙❈❘③✶③✸s❡♠❉❡❝✳t①t✧✱ ❤❡❛❞❡r ❂ ❚❘❯❊✮ ❃ ❛tt❛❝❤✭❞❛t❛✮ ❃ ♥❛♠❡s✭❞❛t❛✮

  ❬✶❪ ✧■❞❛❞❡✧ ✧❉❡♣✧ ✧❍❛❜✧ ✧■❞❝❛s❛✧ ✧❈✐✈✐❧✧ ❬✻❪ ✧P♦st❛❧✧ ✧Pr♦❢✧ ✧❆♥t❣✧ ✧❘❡♥❞✧ ✧❈r❡❞✧ ❬✶✶❪ ✧❊♥tr✧ ✧Pt♦t❛❧✧ ✧P❛❣❛s✧ ✧P❛❣♣❡r❝♥t✧ ✧❉❡❢❛✉❧t✧

  ❘❡❣r❡ssã♦ ▲♦❣íst✐❝❛ ▼♦❞❡❧♦ ❈♦♠♣❧❡t♦

  ➔ ❃ ▼♦❞❢✉❧❧ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✾✶❡✰✵✵ ✲✼✳✷✾❡✲✵✸ ✲✹✳✵✷❡✲✵✹ ✲✶✳✹✵❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✾❡✰✵✶ ✻✳✵✼❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✷ ■❞❛❞❡■❞♦s♦s ✲✷✳✵✼❡✰✵✵ ✽✳✼✸❡✲✵✶ ✲✷✳✸✼ ✵✳✵✶✽ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✾✷❡✰✵✵ ✸✳✶✽❡✲✵✶ ✶✷✳✸✶ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✸✵❡✰✵✵ ✻✳✽✽❡✲✵✶ ✶✳✽✾ ✵✳✵✺✽ ✳ ❉❡♣P♦✉❝♦s ✷✳✽✹❡✰✵✵ ✸✳✺✸❡✲✵✶ ✽✳✵✻ ✼✳✾❡✲✶✻ ✯✯✯ ❍❛❜❈♦♠❡r❝✐♦ ✶✳✷✻❡✲✵✷ ✸✳✻✽❡✲✵✶ ✵✳✵✸ ✵✳✾✼✸ ❍❛❜❊♠♣r❡s❛ ✲✸✳✷✸❡✲✵✶ ✺✳✾✾❡✲✵✶ ✲✵✳✺✹ ✵✳✺✾✵ ■❞❝❛s❛❯s❛❞❛ ✲✻✳✾✷❡✲✵✷ ✷✳✻✵❡✲✵✶ ✲✵✳✷✼ ✵✳✼✾✵ ■❞❝❛s❛❱❡❧❤❛ ✹✳✼✾❡✲✵✶ ✹✳✾✵❡✲✵✶ ✵✳✾✽ ✵✳✸✷✽ ❈✐✈✐❧❉✐✈ ✸✳✸✽❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✻✸ ✶✳✽❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✻✵❡✰✵✵ ✷✳✷✽❡✲✵✶ ✼✳✵✷ ✷✳✸❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✺✽❡✰✵✵ ✻✳✺✼❡✲✵✶ ✺✳✹✺ ✺✳✶❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✼✽❡✰✵✵ ✻✳✻✺❡✲✵✶ ✽✳✻✾ ❁ ✷❡✲✶✻ ✯✯✯ Pr♦❢❚✐♣♦✷ ✲✸✳✼✷❡✲✵✶ ✷✳✾✻❡✲✵✶ ✲✶✳✷✻ ✵✳✷✵✾ Pr♦❢❚✐♣♦✸ ✲✺✳✹✾❡✲✵✶ ✷✳✽✹❡✲✵✶ ✲✶✳✾✹ ✵✳✵✺✸ ✳ Pr♦❢❚✐♣♦✹ ✲✻✳✻✷❡✲✵✶ ✸✳✻✷❡✲✵✶ ✲✶✳✽✸ ✵✳✵✻✽ ✳ ❆♥t❣❘❡❝❡♥t❡ ✲✺✳✷✶❡✲✵✶ ✸✳✽✶❡✲✵✶ ✲✶✳✸✼ ✵✳✶✼✷ ❘❡♥❞❇❛✐①♦ ✶✳✼✷❡✰✵✶ ✻✳✵✼❡✰✵✷ ✵✳✵✸ ✵✳✾✼✼ ❘❡♥❞▼❡❞✐♦ ✽✳✼✾❡✰✵✵ ✻✳✵✼❡✰✵✷ ✵✳✵✶ ✵✳✾✽✽ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✼❡✰✵✶ ✻✳✵✼❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✺❡✲✵✺ ✶✳✻✻❡✲✵✻ ✲✻✳✸✹ ✷✳✸❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✶✶❡✰✵✵ ✸✳✼✸❡✲✵✶ ✶✸✳✼✶ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✺❡✲✵✷ ✷✳✷✼❡✲✵✸ ✶✻✳✶✶ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✻✼❡✲✵✷ ✸✳✻✷❡✲✵✸ ✲✶✺✳✻✼ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮

  ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✹✻✳✾ ♦♥ ✾✾✼✻ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✾✹✳✾ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾

  ➔ ▼♦❞❡❧♦ ◆✉❧♦

  ✾✹ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  ❃ ▼♦❞♥✉❧❧ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✲✵✳✷✺✽ ✷✳✻✶✺ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✸✳✸✽✼✶ ✵✳✵✺✻✷ ✲✻✵✳✷ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✷✽✽✷ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✻ ➔ ❈♦♠♣❛r❛çã♦ ▼♦❞❡❧♦ ◆✉❧♦ ❱s ▼♦❞❡❧♦ ❈♦♠♣❧❡t♦ ❃ ❛♥♦✈❛✭▼♦❞♥✉❧❧✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ✶ ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s

  ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮ ✶ ✾✾✾✾ ✷✽✽✵ ✷ ✾✾✼✻ ✻✹✼ ✷✸ ✷✷✸✸ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ❃ ❘✷❴❢✉❧❧ ❬✶❪ ✵✳✼✼✺ ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔

  ❙t❡♣✇✐s❡✲❇❛❝❦✇❛r❞ ❃ ❛❴✶ ❁✲ ✭▼♦❞❢✉❧❧ ✲ ❘❡♥❞✮ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✸✳✶✹✽✷ ✲✵✳✶✶✾✹ ✲✵✳✵✸✾✺ ✲✵✳✵✶✷✸ ✷✳✽✶✺✺ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✶✳✶✼❡✰✵✶ ✼✳✵✸❡✲✵✶ ✲✶✻✳✼✵ ❁ ✷❡✲✶✻ ✯✯✯ ■❞❛❞❡■❞♦s♦s ✲✼✳✺✼❡✲✵✶ ✻✳✺✽❡✲✵✶ ✲✶✳✶✺ ✵✳✷✺ ■❞❛❞❡❏♦✈❡♥s ✶✳✼✾❡✰✵✵ ✶✳✼✺❡✲✵✶ ✶✵✳✷✵ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✻✳✾✷❡✲✵✶ ✹✳✾✻❡✲✵✶ ✶✳✸✾ ✵✳✶✻ ❉❡♣P♦✉❝♦s ✶✳✷✾❡✰✵✵ ✷✳✸✽❡✲✵✶ ✺✳✹✶ ✻✳✷❡✲✵✽ ✯✯✯ ❍❛❜❈♦♠❡r❝✐♦ ✶✳✶✼❡✲✵✶ ✷✳✸✽❡✲✵✶ ✵✳✹✾ ✵✳✻✷ ❍❛❜❊♠♣r❡s❛ ✶✳✸✻❡✲✵✷ ✸✳✽✶❡✲✵✶ ✵✳✵✹ ✵✳✾✼ ■❞❝❛s❛❯s❛❞❛ ✶✳✶✺❡✲✵✶ ✶✳✻✼❡✲✵✶ ✵✳✻✾ ✵✳✹✾ ■❞❝❛s❛❱❡❧❤❛ ✶✳✷✻❡✲✵✶ ✸✳✸✺❡✲✵✶ ✵✳✸✽ ✵✳✼✶

  ✾✺ ❈✐✈✐❧❉✐✈ ✶✳✺✶❡✰✵✵ ✸✳✼✸❡✲✵✶ ✹✳✵✻ ✹✳✾❡✲✵✺ ✯✯✯ ❈✐✈✐❧❙♦❧ ✼✳✼✷❡✲✵✶ ✶✳✹✻❡✲✵✶ ✺✳✷✽ ✶✳✸❡✲✵✼ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✶✳✾✷❡✰✵✵ ✸✳✾✽❡✲✵✶ ✹✳✽✸ ✶✳✹❡✲✵✻ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✷✳✼✺❡✰✵✵ ✸✳✼✹❡✲✵✶ ✼✳✸✺ ✷✳✵❡✲✶✸ ✯✯✯ Pr♦❢❚✐♣♦✷ ✲✷✳✷✼❡✲✵✶ ✶✳✾✼❡✲✵✶ ✲✶✳✶✺ ✵✳✷✺ Pr♦❢❚✐♣♦✸ ✲✶✳✽✹❡✲✵✶ ✶✳✾✷❡✲✵✶ ✲✵✳✾✻ ✵✳✸✹ Pr♦❢❚✐♣♦✹ ✲✷✳✵✽❡✲✵✶ ✷✳✸✼❡✲✵✶ ✲✵✳✽✽ ✵✳✸✽ ❆♥t❣❘❡❝❡♥t❡ ✲✸✳✻✵❡✲✵✶ ✷✳✽✶❡✲✵✶ ✲✶✳✷✽ ✵✳✷✵ ❈r❡❞ ✲✹✳✾✹❡✲✵✻ ✶✳✵✺❡✲✵✻ ✲✹✳✼✶ ✷✳✺❡✲✵✻ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✷✳✻✻❡✰✵✵ ✷✳✶✸❡✲✵✶ ✶✷✳✺✶ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✶✳✼✽❡✲✵✷ ✾✳✼✹❡✲✵✹ ✶✽✳✸✷ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✷✳✽✺❡✲✵✷ ✶✳✻✷❡✲✵✸ ✲✶✼✳✺✽ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✶✹✽✸✳✺ ♦♥ ✾✾✼✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✶✺✷✻ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✾ ❃ ❛♥♦✈❛✭❛❴✶✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰

  ❆♥t❣ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✼✾ ✶✹✽✹ ✷ ✾✾✼✻ ✻✹✼ ✸ ✽✸✼ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ❃ ❛❴✷ ❁✲ ✭▼♦❞❢✉❧❧ ✲ ❍❛❜✮ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✾✵✽✸✽✹ ✲✵✳✵✵✼✸✷✵ ✲✵✳✵✵✵✹✵✵ ✲✵✳✵✵✵✵✶✹ ✷✳✼✾✻✷✸✾ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✾❡✰✵✶ ✻✳✵✼❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✷ ■❞❛❞❡■❞♦s♦s ✲✷✳✵✾❡✰✵✵ ✽✳✼✵❡✲✵✶ ✲✷✳✹✵ ✵✳✵✶✻ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✾✸❡✰✵✵ ✸✳✶✽❡✲✵✶ ✶✷✳✸✺ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✸✷❡✰✵✵ ✻✳✽✻❡✲✵✶ ✶✳✾✷ ✵✳✵✺✺ ✳ ❉❡♣P♦✉❝♦s ✷✳✽✹❡✰✵✵ ✸✳✺✸❡✲✵✶ ✽✳✵✻ ✼✳✺❡✲✶✻ ✯✯✯ ■❞❝❛s❛❯s❛❞❛ ✲✼✳✼✹❡✲✵✷ ✷✳✺✾❡✲✵✶ ✲✵✳✸✵ ✵✳✼✻✺ ■❞❝❛s❛❱❡❧❤❛ ✹✳✺✻❡✲✵✶ ✹✳✽✻❡✲✵✶ ✵✳✾✹ ✵✳✸✹✽ ❈✐✈✐❧❉✐✈ ✸✳✸✽❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✻✹ ✶✳✼❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✻✵❡✰✵✵ ✷✳✷✽❡✲✵✶ ✼✳✵✶ ✷✳✹❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✺✾❡✰✵✵ ✻✳✺✺❡✲✵✶ ✺✳✹✾ ✹✳✵❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✼✾❡✰✵✵ ✻✳✻✷❡✲✵✶ ✽✳✼✺ ❁ ✷❡✲✶✻ ✯✯✯ Pr♦❢❚✐♣♦✷ ✲✸✳✼✶❡✲✵✶ ✷✳✾✻❡✲✵✶ ✲✶✳✷✺ ✵✳✷✶✵ Pr♦❢❚✐♣♦✸ ✲✺✳✺✹❡✲✵✶ ✷✳✽✸❡✲✵✶ ✲✶✳✾✻ ✵✳✵✺✵ ✳ Pr♦❢❚✐♣♦✹ ✲✻✳✻✺❡✲✵✶ ✸✳✻✷❡✲✵✶ ✲✶✳✽✹ ✵✳✵✻✻ ✳ ❆♥t❣❘❡❝❡♥t❡ ✲✺✳✶✾❡✲✵✶ ✸✳✽✶❡✲✵✶ ✲✶✳✸✻ ✵✳✶✼✸ ❘❡♥❞❇❛✐①♦ ✶✳✼✷❡✰✵✶ ✻✳✵✼❡✰✵✷ ✵✳✵✸ ✵✳✾✼✼ ❘❡♥❞▼❡❞✐♦ ✽✳✼✽❡✰✵✵ ✻✳✵✼❡✰✵✷ ✵✳✵✶ ✵✳✾✽✽ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✼❡✰✵✶ ✻✳✵✼❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾

  ✾✻ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  ❈r❡❞ ✲✶✳✵✺❡✲✵✺ ✶✳✻✺❡✲✵✻ ✲✻✳✸✼ ✶✳✾❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✶✶❡✰✵✵ ✸✳✼✸❡✲✵✶ ✶✸✳✻✾ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✺❡✲✵✷ ✷✳✷✻❡✲✵✸ ✶✻✳✶✶ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✻✼❡✲✵✷ ✸✳✻✶❡✲✵✸ ✲✶✺✳✻✾ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✹✼✳✷ ♦♥ ✾✾✼✽ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✾✶✳✷ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ❃ ❛♥♦✈❛✭❛❴✷✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰

  ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✼✽ ✻✹✼ ✷ ✾✾✼✻ ✻✹✼ ✷ ✵✳✸✵✷ ✵✳✽✻ ❃ ❛❴✸ ❁✲ ✭❛❴✷✲ ■❞❝❛s❛✮ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✾✸❡✰✵✵ ✲✼✳✹✾❡✲✵✸ ✲✹✳✶✺❡✲✵✹ ✲✶✳✹✺❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✽❡✰✵✶ ✻✳✵✾❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✸ ■❞❛❞❡■❞♦s♦s ✲✷✳✵✺❡✰✵✵ ✽✳✼✷❡✲✵✶ ✲✷✳✸✺ ✵✳✵✶✾ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✾✷❡✰✵✵ ✸✳✶✼❡✲✵✶ ✶✷✳✸✺ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✸✻❡✰✵✵ ✻✳✽✼❡✲✵✶ ✶✳✾✼ ✵✳✵✹✽ ✯ ❉❡♣P♦✉❝♦s ✷✳✽✺❡✰✵✵ ✸✳✺✷❡✲✵✶ ✽✳✵✽ ✻✳✸❡✲✶✻ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✸✻❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✻✶ ✷✳✶❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✺✽❡✰✵✵ ✷✳✷✼❡✲✵✶ ✻✳✾✼ ✸✳✶❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✺✻❡✰✵✵ ✻✳✺✶❡✲✵✶ ✺✳✹✼ ✹✳✹❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✼✹❡✰✵✵ ✻✳✺✽❡✲✵✶ ✽✳✼✹ ❁ ✷❡✲✶✻ ✯✯✯ Pr♦❢❚✐♣♦✷ ✲✸✳✻✹❡✲✵✶ ✷✳✾✺❡✲✵✶ ✲✶✳✷✸ ✵✳✷✶✼ Pr♦❢❚✐♣♦✸ ✲✺✳✻✷❡✲✵✶ ✷✳✽✸❡✲✵✶ ✲✶✳✾✾ ✵✳✵✹✼ ✯ Pr♦❢❚✐♣♦✹ ✲✻✳✼✻❡✲✵✶ ✸✳✻✷❡✲✵✶ ✲✶✳✽✼ ✵✳✵✻✷ ✳ ❆♥t❣❘❡❝❡♥t❡ ✲✺✳✶✶❡✲✵✶ ✸✳✽✶❡✲✵✶ ✲✶✳✸✹ ✵✳✶✽✵ ❘❡♥❞❇❛✐①♦ ✶✳✼✷❡✰✵✶ ✻✳✵✾❡✰✵✷ ✵✳✵✸ ✵✳✾✼✼ ❘❡♥❞▼❡❞✐♦ ✽✳✼✽❡✰✵✵ ✻✳✵✾❡✰✵✷ ✵✳✵✶ ✵✳✾✽✾ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✼❡✰✵✶ ✻✳✵✾❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✺❡✲✵✺ ✶✳✻✺❡✲✵✻ ✲✻✳✸✻ ✷✳✶❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✶✵❡✰✵✵ ✸✳✼✷❡✲✵✶ ✶✸✳✼✶ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✹❡✲✵✷ ✷✳✷✺❡✲✵✸ ✶✻✳✶✹ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✻✹❡✲✵✷ ✸✳✺✾❡✲✵✸ ✲✶✺✳✼✶ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮

  ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✹✽✳✷ ♦♥ ✾✾✽✵ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✽✽✳✷ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾

  ✾✼ ❃ ❛♥♦✈❛✭❛❴✸✱ ❛❴✷✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰

  ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✵ ✻✹✽ ✷ ✾✾✼✽ ✻✹✼ ✷ ✶✳✵✶ ✵✳✻ ❃ ❛❴✹ ❁✲ ✭❛❴✸✲ Pr♦❢✮ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✾✶❡✰✵✵ ✲✼✳✾✶❡✲✵✸ ✲✹✳✹✼❡✲✵✹ ✲✶✳✻✸❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✽❡✰✵✶ ✻✳✶✹❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✸ ■❞❛❞❡■❞♦s♦s ✲✷✳✶✾❡✰✵✵ ✽✳✼✷❡✲✵✶ ✲✷✳✺✶ ✵✳✵✶✷ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✽✻❡✰✵✵ ✸✳✶✷❡✲✵✶ ✶✷✳✸✼ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✷✻❡✰✵✵ ✻✳✾✶❡✲✵✶ ✶✳✽✷ ✵✳✵✻✽ ✳ ❉❡♣P♦✉❝♦s ✷✳✼✾❡✰✵✵ ✸✳✹✾❡✲✵✶ ✼✳✾✾ ✶✳✸❡✲✶✺ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✷✻❡✰✵✵ ✺✳✾✸❡✲✵✶ ✺✳✺✵ ✸✳✽❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✺✺❡✰✵✵ ✷✳✷✺❡✲✵✶ ✻✳✽✽ ✺✳✾❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✹✷❡✰✵✵ ✻✳✸✶❡✲✵✶ ✺✳✹✷ ✻✳✵❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✺✾❡✰✵✵ ✻✳✸✻❡✲✵✶ ✽✳✼✾ ❁ ✷❡✲✶✻ ✯✯✯ ❆♥t❣❘❡❝❡♥t❡ ✲✹✳✽✹❡✲✵✶ ✸✳✽✷❡✲✵✶ ✲✶✳✷✼ ✵✳✷✵✺ ❘❡♥❞❇❛✐①♦ ✶✳✼✶❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✸ ✵✳✾✼✽ ❘❡♥❞▼❡❞✐♦ ✽✳✼✶❡✰✵✵ ✻✳✶✹❡✰✵✷ ✵✳✵✶ ✵✳✾✽✾ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✺❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✶❡✲✵✺ ✶✳✻✸❡✲✵✻ ✲✻✳✷✵ ✺✳✻❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✶✵❡✰✵✵ ✸✳✼✷❡✲✵✶ ✶✸✳✼✶ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✺✾❡✲✵✷ ✷✳✷✶❡✲✵✸ ✶✻✳✷✹ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✺✽❡✲✵✷ ✸✳✺✸❡✲✵✸ ✲✶✺✳✽✶ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✺✸✳✶✾ ♦♥ ✾✾✽✸ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✽✼✳✷ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ❃ ❛♥♦✈❛✭❛❴✹✱ ❛❴✸✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰

  ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✸ ✻✺✸ ✷ ✾✾✽✵ ✻✹✽ ✸ ✹✳✾✽ ✵✳✶✼ ❃ ❛❴✺ ❁✲ ✭❛❴✹✲ ❆♥t❣✮

  ✾✽ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✽✾❡✰✵✵ ✲✼✳✾✷❡✲✵✸ ✲✹✳✺✶❡✲✵✹ ✲✶✳✻✼❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✼❡✰✵✶ ✻✳✶✹❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✸ ■❞❛❞❡■❞♦s♦s ✲✷✳✶✺❡✰✵✵ ✽✳✼✶❡✲✵✶ ✲✷✳✹✻ ✵✳✵✶✹ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✽✺❡✰✵✵ ✸✳✶✷❡✲✵✶ ✶✷✳✸✺ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✷✼❡✰✵✵ ✻✳✽✻❡✲✵✶ ✶✳✽✺ ✵✳✵✻✺ ✳ ❉❡♣P♦✉❝♦s ✷✳✼✼❡✰✵✵ ✸✳✹✽❡✲✵✶ ✼✳✾✼ ✶✳✻❡✲✶✺ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✷✷❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✸✻ ✽✳✶❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✺✻❡✰✵✵ ✷✳✷✺❡✲✵✶ ✻✳✾✸ ✹✳✸❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✹✺❡✰✵✵ ✻✳✸✵❡✲✵✶ ✺✳✹✽ ✹✳✸❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✻✶❡✰✵✵ ✻✳✸✻❡✲✵✶ ✽✳✽✷ ❁ ✷❡✲✶✻ ✯✯✯ ❘❡♥❞❇❛✐①♦ ✶✳✼✶❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✸ ✵✳✾✼✽ ❘❡♥❞▼❡❞✐♦ ✽✳✼✻❡✰✵✵ ✻✳✶✹❡✰✵✷ ✵✳✵✶ ✵✳✾✽✾ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✺❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✷❡✲✵✺ ✶✳✻✸❡✲✵✻ ✲✻✳✷✽ ✸✳✸❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✵✼❡✰✵✵ ✸✳✻✾❡✲✵✶ ✶✸✳✼✸ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✵❡✲✵✷ ✷✳✷✷❡✲✵✸ ✶✻✳✷✸ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✺✼❡✲✵✷ ✸✳✺✷❡✲✵✸ ✲✶✺✳✽✶ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮

  ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✺✹✳✽✻ ♦♥ ✾✾✽✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✽✻✳✾ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ❃ ❛♥♦✈❛✭❛❴✺✱ ❛❴✹✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰

  Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✹ ✻✺✺ ✷ ✾✾✽✸ ✻✺✸ ✶ ✶✳✻✼ ✵✳✷✵ ➔ ▼❡❧❤♦r ▼♦❞❡❧♦ ❊st✐♠❛❞♦ ❝♦♠ ❙t❡♣✇✐s❡

  ❃ ▼♠♦❞❴❙t♣ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✽✾❡✰✵✵ ✲✼✳✾✷❡✲✵✸ ✲✹✳✺✶❡✲✵✹ ✲✶✳✻✼❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✼❡✰✵✶ ✻✳✶✹❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✸ ■❞❛❞❡■❞♦s♦s ✲✷✳✶✺❡✰✵✵ ✽✳✼✶❡✲✵✶ ✲✷✳✹✻ ✵✳✵✶✹ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✽✺❡✰✵✵ ✸✳✶✷❡✲✵✶ ✶✷✳✸✺ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✷✼❡✰✵✵ ✻✳✽✻❡✲✵✶ ✶✳✽✺ ✵✳✵✻✺ ✳ ❉❡♣P♦✉❝♦s ✷✳✼✼❡✰✵✵ ✸✳✹✽❡✲✵✶ ✼✳✾✼ ✶✳✻❡✲✶✺ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✷✷❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✸✻ ✽✳✶❡✲✵✽ ✯✯✯

  ✾✾ ❈✐✈✐❧❙♦❧ ✶✳✺✻❡✰✵✵ ✷✳✷✺❡✲✵✶ ✻✳✾✸ ✹✳✸❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✹✺❡✰✵✵ ✻✳✸✵❡✲✵✶ ✺✳✹✽ ✹✳✸❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✻✶❡✰✵✵ ✻✳✸✻❡✲✵✶ ✽✳✽✷ ❁ ✷❡✲✶✻ ✯✯✯ ❈r❡❞ ✲✶✳✵✷❡✲✵✺ ✶✳✻✸❡✲✵✻ ✲✻✳✷✽ ✸✳✸❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✵✼❡✰✵✵ ✸✳✻✾❡✲✵✶ ✶✸✳✼✸ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✵❡✲✵✷ ✷✳✷✷❡✲✵✸ ✶✻✳✷✸ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✺✼❡✲✵✷ ✸✳✺✷❡✲✵✸ ✲✶✺✳✽✶ ❁ ✷❡✲✶✻ ✯✯✯ ❘❡♥❞❇❛✐①♦ ✶✳✼✶❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✸ ✵✳✾✼✽ ❘❡♥❞▼❡❞✐♦ ✽✳✼✻❡✰✵✵ ✻✳✶✹❡✰✵✷ ✵✳✵✶ ✵✳✾✽✾ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✺❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✺✹✳✽✻ ♦♥ ✾✾✽✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✽✻✳✾ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔

  ❙t❡♣ ❆■❈ ❙t❛rt✿ ❆■❈❂✻✾✺ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰

  ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ✲ ❍❛❜ ✷ ✻✹✼ ✻✾✶ ✲ ■❞❝❛s❛ ✷ ✻✹✽ ✻✾✷ ✲ Pr♦❢ ✸ ✻✺✷ ✻✾✹ ✲ ❆♥t❣ ✶ ✻✹✾ ✻✾✺ ❁♥♦♥❡❃ ✻✹✼ ✻✾✺ ✲ ❈r❡❞ ✶ ✻✾✹ ✼✹✵ ✲ ❈✐✈✐❧ ✷ ✼✶✾ ✼✻✸ ✲ ❉❡♣ ✷ ✼✸✼ ✼✽✶ ✲ P♦st❛❧ ✷ ✽✹✼ ✽✾✶ ✲ ■❞❛❞❡ ✷ ✽✻✶ ✾✵✺ ✲ ❊♥tr ✶ ✶✵✹✾ ✶✵✾✺ ✲ Pt♦t❛❧ ✶ ✶✹✶✼ ✶✹✻✸ ✲ ❘❡♥❞ ✸ ✶✹✽✹ ✶✺✷✻ ✲ P❛❣❛s ✶ ✶✼✾✶ ✶✽✸✼ ❙t❡♣✿ ❆■❈❂✻✾✶ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰

  ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ✲ ■❞❝❛s❛ ✷ ✻✹✽ ✻✽✽ ✲ Pr♦❢ ✸ ✻✺✷ ✻✾✵ ✲ ❆♥t❣ ✶ ✻✹✾ ✻✾✶ ❁♥♦♥❡❃ ✻✹✼ ✻✾✶ ✰ ❍❛❜ ✷ ✻✹✼ ✻✾✺ ✲ ❈r❡❞ ✶ ✻✾✺ ✼✸✼ ✲ ❈✐✈✐❧ ✷ ✼✶✾ ✼✺✾ ✲ ❉❡♣ ✷ ✼✸✼ ✼✼✼ ✲ P♦st❛❧ ✷ ✽✹✽ ✽✽✽ ✲ ■❞❛❞❡ ✷ ✽✻✸ ✾✵✸ ✲ ❊♥tr ✶ ✶✵✹✾ ✶✵✾✶ ✲ Pt♦t❛❧ ✶ ✶✹✶✾ ✶✹✻✶ ✲ ❘❡♥❞ ✸ ✶✹✽✹ ✶✺✷✷ ✲ P❛❣❛s ✶ ✶✼✾✸ ✶✽✸✺

  ✶✵✵ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  ❙t❡♣✿ ❆■❈❂✻✽✽ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈

  ✲ Pr♦❢ ✸ ✻✺✸ ✻✽✼ ✲ ❆♥t❣ ✶ ✻✺✵ ✻✽✽ ❁♥♦♥❡❃ ✻✹✽ ✻✽✽ ✰ ■❞❝❛s❛ ✷ ✻✹✼ ✻✾✶ ✰ ❍❛❜ ✷ ✻✹✽ ✻✾✷ ✲ ❈r❡❞ ✶ ✻✾✻ ✼✸✹ ✲ ❈✐✈✐❧ ✷ ✼✷✵ ✼✺✻ ✲ ❉❡♣ ✷ ✼✸✾ ✼✼✺ ✲ P♦st❛❧ ✷ ✽✹✽ ✽✽✹ ✲ ■❞❛❞❡ ✷ ✽✻✹ ✾✵✵ ✲ ❊♥tr ✶ ✶✵✺✵ ✶✵✽✽ ✲ Pt♦t❛❧ ✶ ✶✹✶✾ ✶✹✺✼ ✲ ❘❡♥❞ ✸ ✶✹✽✹ ✶✺✶✽ ✲ P❛❣❛s ✶ ✶✼✾✻ ✶✽✸✹ ❙t❡♣✿ ❆■❈❂✻✽✼ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰

  ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ✲ ❆♥t❣ ✶ ✻✺✺ ✻✽✼ ❁♥♦♥❡❃ ✻✺✸ ✻✽✼ ✰ Pr♦❢ ✸ ✻✹✽ ✻✽✽ ✰ ■❞❝❛s❛ ✷ ✻✺✷ ✻✾✵ ✰ ❍❛❜ ✷ ✻✺✸ ✻✾✶ ✲ ❈r❡❞ ✶ ✻✾✽ ✼✸✵ ✲ ❈✐✈✐❧ ✷ ✼✷✷ ✼✺✷ ✲ ❉❡♣ ✷ ✼✹✷ ✼✼✷ ✲ P♦st❛❧ ✷ ✽✺✵ ✽✽✵ ✲ ■❞❛❞❡ ✷ ✽✻✼ ✽✾✼ ✲ ❊♥tr ✶ ✶✵✺✻ ✶✵✽✽ ✲ Pt♦t❛❧ ✶ ✶✹✷✵ ✶✹✺✷ ✲ ❘❡♥❞ ✸ ✶✹✽✻ ✶✺✶✹ ✲ P❛❣❛s ✶ ✶✼✾✽ ✶✽✸✵ ❙t❡♣✿ ❆■❈❂✻✽✼ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰

  Pt♦t❛❧ ✰ P❛❣❛s ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ❁♥♦♥❡❃ ✻✺✺ ✻✽✼ ✰ ❆♥t❣ ✶ ✻✺✸ ✻✽✼ ✰ Pr♦❢ ✸ ✻✺✵ ✻✽✽ ✰ ■❞❝❛s❛ ✷ ✻✺✹ ✻✾✵ ✰ ❍❛❜ ✷ ✻✺✺ ✻✾✶ ✲ ❈r❡❞ ✶ ✼✵✶ ✼✸✶ ✲ ❈✐✈✐❧ ✷ ✼✷✹ ✼✺✷ ✲ ❉❡♣ ✷ ✼✹✷ ✼✼✵ ✲ P♦st❛❧ ✷ ✽✺✷ ✽✽✵ ✲ ■❞❛❞❡ ✷ ✽✻✽ ✽✾✻ ✲ ❊♥tr ✶ ✶✵✺✼ ✶✵✽✼ ✲ Pt♦t❛❧ ✶ ✶✹✷✷ ✶✹✺✷ ✲ ❘❡♥❞ ✸ ✶✹✽✼ ✶✺✶✸ ✲ P❛❣❛s ✶ ✶✼✾✾ ✶✽✷✾ ❈♦❡❢❢✐❝✐❡♥ts✿ ✭■♥t❡r❝❡♣t✮ ■❞❛❞❡■❞♦s♦s ■❞❛❞❡❏♦✈❡♥s ❉❡♣▼✉✐t♦s ❉❡♣P♦✉❝♦s ❈✐✈✐❧❉✐✈

  ✲✹✳✸✼❡✰✵✶ ✲✷✳✶✺❡✰✵✵ ✸✳✽✺❡✰✵✵ ✶✳✷✼❡✰✵✵ ✷✳✼✼❡✰✵✵ ✸✳✷✷❡✰✵✵ ❈✐✈✐❧❙♦❧ P♦st❛❧❩♦♥❛✸ P♦st❛❧❩♦♥❛✹ ❘❡♥❞❇❛✐①♦ ❘❡♥❞▼❡❞✐♦

  ✶✵✶ ✶✳✺✻❡✰✵✵ ✸✳✹✺❡✰✵✵ ✺✳✻✶❡✰✵✵ ✶✳✼✶❡✰✵✶ ✽✳✼✻❡✰✵✵ ❘❡♥❞▼✐♥✐♠♦ ❈r❡❞ ❊♥tr▼❡♥♦s✶✵ Pt♦t❛❧ P❛❣❛s

  ✷✳✸✺❡✰✵✶ ✲✶✳✵✷❡✲✵✺ ✺✳✵✼❡✰✵✵ ✸✳✻✵❡✲✵✷ ✲✺✳✺✼❡✲✵✷ ❉❡❣r❡❡s ♦❢ ❋r❡❡❞♦♠✿ ✾✾✾✾ ❚♦t❛❧ ✭✐✳❡✳ ◆✉❧❧✮❀ ✾✾✽✹ ❘❡s✐❞✉❛❧ ◆✉❧❧ ❉❡✈✐❛♥❝❡✿ ✷✽✽✵ ❘❡s✐❞✉❛❧ ❉❡✈✐❛♥❝❡✿ ✻✺✺ ❆■❈✿ ✻✽✼

  ➔ ▼❡❧❤♦r ▼♦❞❡❧♦ ❊st✐♠❛❞♦ ❝♦♠ ❙t❡♣ ❆■❈ ❃ ▼♦❞❆■❈

  ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✽✾❡✰✵✵ ✲✼✳✾✷❡✲✵✸ ✲✹✳✺✶❡✲✵✹ ✲✶✳✻✼❡✲✵✺ ✷✳✼✾❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✸✼❡✰✵✶ ✻✳✶✹❡✰✵✷ ✲✵✳✵✼ ✵✳✾✹✸ ■❞❛❞❡■❞♦s♦s ✲✷✳✶✺❡✰✵✵ ✽✳✼✶❡✲✵✶ ✲✷✳✹✻ ✵✳✵✶✹ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✽✺❡✰✵✵ ✸✳✶✷❡✲✵✶ ✶✷✳✸✺ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✷✼❡✰✵✵ ✻✳✽✻❡✲✵✶ ✶✳✽✺ ✵✳✵✻✺ ✳ ❉❡♣P♦✉❝♦s ✷✳✼✼❡✰✵✵ ✸✳✹✽❡✲✵✶ ✼✳✾✼ ✶✳✻❡✲✶✺ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✷✷❡✰✵✵ ✻✳✵✵❡✲✵✶ ✺✳✸✻ ✽✳✶❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✺✻❡✰✵✵ ✷✳✷✺❡✲✵✶ ✻✳✾✸ ✹✳✸❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✹✺❡✰✵✵ ✻✳✸✵❡✲✵✶ ✺✳✹✽ ✹✳✸❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✻✶❡✰✵✵ ✻✳✸✻❡✲✵✶ ✽✳✽✷ ❁ ✷❡✲✶✻ ✯✯✯ ❘❡♥❞❇❛✐①♦ ✶✳✼✶❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✸ ✵✳✾✼✽ ❘❡♥❞▼❡❞✐♦ ✽✳✼✻❡✰✵✵ ✻✳✶✹❡✰✵✷ ✵✳✵✶ ✵✳✾✽✾ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✺❡✰✵✶ ✻✳✶✹❡✰✵✷ ✵✳✵✹ ✵✳✾✻✾ ❈r❡❞ ✲✶✳✵✷❡✲✵✺ ✶✳✻✸❡✲✵✻ ✲✻✳✷✽ ✸✳✸❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✵✼❡✰✵✵ ✸✳✻✾❡✲✵✶ ✶✸✳✼✸ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✸✳✻✵❡✲✵✷ ✷✳✷✷❡✲✵✸ ✶✻✳✷✸ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✺✳✺✼❡✲✵✷ ✸✳✺✷❡✲✵✸ ✲✶✺✳✽✶ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮

  ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✺✹✳✽✻ ♦♥ ✾✾✽✹ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✽✻✳✾ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ❃ ❘✷❴❆■❈ ❬✶❪ ✵✳✼✼✷ ❃ ❛♥♦✈❛✭▼♦❞❆■❈✱ ▼♦❞❢✉❧❧✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰

  Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✹ ✻✺✺ ✷ ✾✾✼✻ ✻✹✼ ✽ ✼✳✾✻ ✵✳✹✹ ❃ ▼♦❞❆■❈✷ ❁✲ ✭▼♦❞❆■❈ ✲ ❘❡♥❞✮

  ✶✵✷ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✸✳✶✶✶✼ ✲✵✳✶✷✶✵ ✲✵✳✵✸✾✼ ✲✵✳✵✶✷✹ ✷✳✼✽✷✹ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✶✳✶✽❡✰✵✶ ✻✳✾✵❡✲✵✶ ✲✶✼✳✶✻ ❁ ✷❡✲✶✻ ✯✯✯ ■❞❛❞❡■❞♦s♦s ✲✼✳✻✵❡✲✵✶ ✻✳✺✽❡✲✵✶ ✲✶✳✶✺ ✵✳✷✺ ■❞❛❞❡❏♦✈❡♥s ✶✳✼✽❡✰✵✵ ✶✳✼✺❡✲✵✶ ✶✵✳✶✻ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✻✳✽✷❡✲✵✶ ✹✳✾✸❡✲✵✶ ✶✳✸✾ ✵✳✶✼ ❉❡♣P♦✉❝♦s ✶✳✷✼❡✰✵✵ ✷✳✸✼❡✲✵✶ ✺✳✸✺ ✽✳✽❡✲✵✽ ✯✯✯ ❈✐✈✐❧❉✐✈ ✶✳✹✽❡✰✵✵ ✸✳✼✷❡✲✵✶ ✸✳✾✾ ✻✳✻❡✲✵✺ ✯✯✯ ❈✐✈✐❧❙♦❧ ✼✳✼✵❡✲✵✶ ✶✳✹✻❡✲✵✶ ✺✳✷✽ ✶✳✸❡✲✵✼ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✶✳✾✶❡✰✵✵ ✸✳✾✽❡✲✵✶ ✹✳✼✾ ✶✳✼❡✲✵✻ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✷✳✼✸❡✰✵✵ ✸✳✼✸❡✲✵✶ ✼✳✸✶ ✷✳✽❡✲✶✸ ✯✯✯ ❈r❡❞ ✲✹✳✾✷❡✲✵✻ ✶✳✵✺❡✲✵✻ ✲✹✳✻✾ ✷✳✼❡✲✵✻ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✷✳✻✼❡✰✵✵ ✷✳✶✷❡✲✵✶ ✶✷✳✺✼ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✶✳✼✽❡✲✵✷ ✾✳✼✷❡✲✵✹ ✶✽✳✸✸ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✲✷✳✽✺❡✲✵✷ ✶✳✻✷❡✲✵✸ ✲✶✼✳✻✵ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✶✹✽✼✳✹ ♦♥ ✾✾✽✼ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✶✺✶✸ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✾ ❃ ❛♥♦✈❛✭▼♦❞❆■❈✷✱ ▼♦❞❆■❈✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰

  P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✼ ✶✹✽✼ ✷ ✾✾✽✹ ✻✺✺ ✸ ✽✸✸ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ❃ ❘✷❴❆■❈✷ ❬✶❪ ✵✳✹✽✸ ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔

  ❈♦♥✜♠❛çã♦ ❙❡❧❡❝çã♦ ✈❛r✐á✈❡✐s ✶➟ ❢✉♥çã♦

  ➔ ❃ ▼♦❞❢✉❧❧ ❙✐♥❣❧❡ t❡r♠ ❞❡❧❡t✐♦♥s ▼♦❞❡❧✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰

  ✶✵✸ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ▲❘❚ Pr✭❈❤✐✮

  ❁♥♦♥❡❃ ✻✹✼ ✻✾✺ ■❞❛❞❡ ✷ ✽✻✶ ✾✵✺ ✷✶✺ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣ ✷ ✼✸✼ ✼✽✶ ✾✵ ❁ ✷❡✲✶✻ ✯✯✯ ❍❛❜ ✷ ✻✹✼ ✻✾✶ ✵ ✵✳✽✻ ■❞❝❛s❛ ✷ ✻✹✽ ✻✾✷ ✶ ✵✳✺✾ ❈✐✈✐❧ ✷ ✼✶✾ ✼✻✸ ✼✷ ❁ ✷❡✲✶✻ ✯✯✯ P♦st❛❧ ✷ ✽✹✼ ✽✾✶ ✷✵✵ ❁ ✷❡✲✶✻ ✯✯✯ Pr♦❢ ✸ ✻✺✷ ✻✾✹ ✺ ✵✳✶✾ ❆♥t❣ ✶ ✻✹✾ ✻✾✺ ✷ ✵✳✶✻ ❘❡♥❞ ✸ ✶✹✽✹ ✶✺✷✻ ✽✸✼ ❁ ✷❡✲✶✻ ✯✯✯ ❈r❡❞ ✶ ✻✾✹ ✼✹✵ ✹✽ ✺✳✺❡✲✶✷ ✯✯✯ ❊♥tr ✶ ✶✵✹✾ ✶✵✾✺ ✹✵✷ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✶ ✶✹✶✼ ✶✹✻✸ ✼✼✵ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✶ ✶✼✾✶ ✶✽✸✼ ✶✶✹✹ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ❃▼♦❞❆■❈ ❙✐♥❣❧❡ t❡r♠ ❞❡❧❡t✐♦♥s ▼♦❞❡❧✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s

  ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ▲❘❚ Pr✭❈❤✐✮ ❁♥♦♥❡❃ ✻✺✺ ✻✽✼ ■❞❛❞❡ ✷ ✽✻✽ ✽✾✻ ✷✶✸ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣ ✷ ✼✹✷ ✼✼✵ ✽✽ ❁ ✷❡✲✶✻ ✯✯✯ ❈✐✈✐❧ ✷ ✼✷✹ ✼✺✷ ✻✾ ✶✳✶❡✲✶✺ ✯✯✯ P♦st❛❧ ✷ ✽✺✷ ✽✽✵ ✶✾✼ ❁ ✷❡✲✶✻ ✯✯✯ ❘❡♥❞ ✸ ✶✹✽✼ ✶✺✶✸ ✽✸✸ ❁ ✷❡✲✶✻ ✯✯✯ ❈r❡❞ ✶ ✼✵✶ ✼✸✶ ✹✻ ✾✳✽❡✲✶✷ ✯✯✯ ❊♥tr ✶ ✶✵✺✼ ✶✵✽✼ ✹✵✷ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✶ ✶✹✷✷ ✶✹✺✷ ✼✻✼ ❁ ✷❡✲✶✻ ✯✯✯ P❛❣❛s ✶ ✶✼✾✾ ✶✽✷✾ ✶✶✹✹ ❁ ✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶

  ➔✷➟ ❢✉♥çã♦ ❃ ▼♦❞❢✉❧❧ ❙✐♥❣❧❡ t❡r♠ ❛❞❞✐t✐♦♥s ▼♦❞❡❧✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❍❛❜ ✰ ■❞❝❛s❛ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ Pr♦❢ ✰ ❆♥t❣ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰ P❛❣❛s

  ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ▲❘❚ Pr✭❈❤✐✮ ❁♥♦♥❡❃ ✻✹✼ ✻✾✺ ■❞❛❞❡✿❉❡♣ ✸ ✻✹✻ ✼✵✵ ✶✳✷✷ ✵✳✼✹✾ ■❞❛❞❡✿❍❛❜ ✹ ✻✹✷ ✻✾✽ ✹✳✻✽ ✵✳✸✷✷ ■❞❛❞❡✿■❞❝❛s❛ ✹ ✻✹✺ ✼✵✶ ✶✳✽✷ ✵✳✼✻✾ ■❞❛❞❡✿❈✐✈✐❧ ✹ ✻✹✻ ✼✵✷ ✶✳✶✸ ✵✳✽✽✾ ■❞❛❞❡✿P♦st❛❧ ✹ ✻✹✹ ✼✵✵ ✸✳✵✼ ✵✳✺✹✻ ■❞❛❞❡✿Pr♦❢ ✻ ✻✹✺ ✼✵✺ ✷✳✶✺ ✵✳✾✵✻ ■❞❛❞❡✿❆♥t❣ ✷ ✻✹✻ ✻✾✽ ✵✳✹✵ ✵✳✽✶✼ ■❞❛❞❡✿❘❡♥❞ ✻ ✻✹✼ ✼✵✼ ✵✳✷✾ ✶✳✵✵✵ ■❞❛❞❡✿❈r❡❞ ✷ ✻✹✵ ✻✾✷ ✻✳✽✽ ✵✳✵✸✷ ✯ ■❞❛❞❡✿❊♥tr ✷ ✻✹✻ ✻✾✽ ✵✳✺✽ ✵✳✼✹✻ ■❞❛❞❡✿Pt♦t❛❧ ✷ ✻✹✻ ✻✾✽ ✵✳✽✻ ✵✳✻✺✵ ■❞❛❞❡✿P❛❣❛s ✷ ✻✹✺ ✻✾✼ ✶✳✼✽ ✵✳✹✶✶

  ✶✵✹ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  ❉❡♣✿❍❛❜ ✹ ✻✸✼ ✻✾✸ ✾✳✼✽ ✵✳✵✹✹ ✯ ❉❡♣✿■❞❝❛s❛ ✹ ✻✹✹ ✼✵✵ ✷✳✹✹ ✵✳✻✺✻ ❉❡♣✿❈✐✈✐❧ ✹ ✻✹✹ ✼✵✵ ✸✳✶✸ ✵✳✺✸✻ ❉❡♣✿P♦st❛❧ ✹ ✻✹✺ ✼✵✶ ✷✳✸✾ ✵✳✻✻✺ ❉❡♣✿Pr♦❢ ✻ ✻✹✸ ✼✵✸ ✹✳✶✸ ✵✳✻✺✽ ❉❡♣✿❆♥t❣ ✷ ✻✹✼ ✻✾✾ ✵✳✸✸ ✵✳✽✹✼ ❉❡♣✿❘❡♥❞ ✻ ✻✹✻ ✼✵✻ ✶✳✵✷ ✵✳✾✽✺ ❉❡♣✿❈r❡❞ ✷ ✻✹✹ ✻✾✻ ✷✳✺✺ ✵✳✷✽✵ ❉❡♣✿❊♥tr ✷ ✻✹✸ ✻✾✺ ✸✳✻✷ ✵✳✶✻✸ ❉❡♣✿Pt♦t❛❧ ✷ ✻✹✷ ✻✾✹ ✺✳✸✼ ✵✳✵✻✽ ✳ ❉❡♣✿P❛❣❛s ✷ ✻✹✶ ✻✾✸ ✻✳✷✻ ✵✳✵✹✹ ✯ ❍❛❜✿■❞❝❛s❛ ✹ ✻✹✸ ✻✾✾ ✸✳✾✵ ✵✳✹✶✾ ❍❛❜✿❈✐✈✐❧ ✹ ✻✸✸ ✻✽✾ ✶✸✳✼✾ ✵✳✵✵✽ ✯✯ ❍❛❜✿P♦st❛❧ ✹ ✻✹✸ ✻✾✾ ✸✳✾✵ ✵✳✹✶✾ ❍❛❜✿Pr♦❢ ✻ ✻✸✽ ✻✾✽ ✽✳✽✽ ✵✳✶✽✶ ❍❛❜✿❆♥t❣ ✷ ✻✹✺ ✻✾✼ ✶✳✾✽ ✵✳✸✼✶ ❍❛❜✿❘❡♥❞ ✻ ✻✹✹ ✼✵✹ ✸✳✵✺ ✵✳✽✵✷ ❍❛❜✿❈r❡❞ ✷ ✻✹✻ ✻✾✽ ✵✳✻✾ ✵✳✼✵✾ ❍❛❜✿❊♥tr ✷ ✻✹✺ ✻✾✼ ✷✳✸✹ ✵✳✸✶✵ ❍❛❜✿Pt♦t❛❧ ✷ ✻✹✺ ✻✾✼ ✶✳✻✷ ✵✳✹✹✹ ❍❛❜✿P❛❣❛s ✷ ✻✹✸ ✻✾✺ ✸✳✻✻ ✵✳✶✻✵ ■❞❝❛s❛✿❈✐✈✐❧ ✹ ✻✹✸ ✻✾✾ ✹✳✶✹ ✵✳✸✽✼ ■❞❝❛s❛✿P♦st❛❧ ✹ ✻✹✸ ✻✾✾ ✸✳✺✺ ✵✳✹✼✵ ■❞❝❛s❛✿Pr♦❢ ✻ ✻✹✵ ✼✵✵ ✻✳✹✼ ✵✳✸✼✷ ■❞❝❛s❛✿❆♥t❣ ✷ ✻✹✻ ✻✾✽ ✵✳✾✸ ✵✳✻✸✵ ■❞❝❛s❛✿❘❡♥❞ ✻ ✻✹✹ ✼✵✹ ✸✳✵✵ ✵✳✽✵✾ ■❞❝❛s❛✿❈r❡❞ ✷ ✻✹✻ ✻✾✽ ✵✳✽✼ ✵✳✻✹✼ ■❞❝❛s❛✿❊♥tr ✷ ✻✹✺ ✻✾✼ ✶✳✺✼ ✵✳✹✺✻ ■❞❝❛s❛✿Pt♦t❛❧ ✷ ✻✹✼ ✻✾✾ ✵✳✷✾ ✵✳✽✻✻ ■❞❝❛s❛✿P❛❣❛s ✷ ✻✹✻ ✻✾✽ ✵✳✽✺ ✵✳✻✺✹ ❈✐✈✐❧✿P♦st❛❧ ✹ ✻✸✻ ✻✾✷ ✶✵✳✼✷ ✵✳✵✸✵ ✯ ❈✐✈✐❧✿Pr♦❢ ✻ ✻✹✶ ✼✵✶ ✺✳✻✻ ✵✳✹✻✸ ❈✐✈✐❧✿❆♥t❣ ✷ ✻✹✻ ✻✾✽ ✵✳✽✷ ✵✳✻✻✹ ❈✐✈✐❧✿❘❡♥❞ ✻ ✻✹✺ ✼✵✺ ✷✳✷✾ ✵✳✽✾✶ ❈✐✈✐❧✿❈r❡❞ ✷ ✻✹✺ ✻✾✼ ✶✳✹✵ ✵✳✹✾✼ ❈✐✈✐❧✿❊♥tr ✷ ✻✹✹ ✻✾✻ ✸✳✶✵ ✵✳✷✶✷ ❈✐✈✐❧✿Pt♦t❛❧ ✷ ✻✹✹ ✻✾✻ ✸✳✵✽ ✵✳✷✶✹ ❈✐✈✐❧✿P❛❣❛s ✷ ✻✹✻ ✻✾✽ ✵✳✾✺ ✵✳✻✷✷ P♦st❛❧✿Pr♦❢ ✻ ✻✹✺ ✼✵✺ ✶✳✼✺ ✵✳✾✹✷ P♦st❛❧✿❆♥t❣ ✷ ✻✹✺ ✻✾✼ ✶✳✽✼ ✵✳✸✾✸ P♦st❛❧✿❘❡♥❞ ✻ ✻✹✸ ✼✵✸ ✹✳✷✸ ✵✳✻✹✺ P♦st❛❧✿❈r❡❞ ✷ ✻✹✼ ✻✾✾ ✵✳✸✹ ✵✳✽✹✷ P♦st❛❧✿❊♥tr ✷ ✻✹✻ ✻✾✽ ✵✳✺✾ ✵✳✼✹✸ P♦st❛❧✿Pt♦t❛❧ ✷ ✻✹✻ ✻✾✽ ✵✳✽✾ ✵✳✻✹✶ P♦st❛❧✿P❛❣❛s ✷ ✻✹✸ ✻✾✺ ✹✳✷✹ ✵✳✶✷✵ Pr♦❢✿❆♥t❣ ✸ ✻✹✺ ✻✾✾ ✶✳✽✹ ✵✳✻✵✼ Pr♦❢✿❘❡♥❞ ✾ ✻✹✹ ✼✶✵ ✷✳✹✹ ✵✳✾✽✷ Pr♦❢✿❈r❡❞ ✸ ✻✹✶ ✻✾✺ ✺✳✽✸ ✵✳✶✷✵ Pr♦❢✿❊♥tr ✸ ✻✹✻ ✼✵✵ ✵✳✽✺ ✵✳✽✸✼ Pr♦❢✿Pt♦t❛❧ ✸ ✻✹✻ ✼✵✵ ✶✳✷✸ ✵✳✼✹✺ Pr♦❢✿P❛❣❛s ✸ ✻✹✺ ✻✾✾ ✶✳✺✼ ✵✳✻✻✼ ❆♥t❣✿❘❡♥❞ ✸ ✻✹✻ ✼✵✵ ✶✳✵✼ ✵✳✼✽✻ ❆♥t❣✿❈r❡❞ ✶ ✻✹✻ ✻✾✻ ✵✳✼✼ ✵✳✸✽✶ ❆♥t❣✿❊♥tr ✶ ✻✹✻ ✻✾✻ ✵✳✹✷ ✵✳✺✶✾ ❆♥t❣✿Pt♦t❛❧ ✶ ✻✹✸ ✻✾✸ ✹✳✵✻ ✵✳✵✹✹ ✯ ❆♥t❣✿P❛❣❛s ✶ ✻✹✷ ✻✾✷ ✹✳✻✵ ✵✳✵✸✷ ✯ ❘❡♥❞✿❈r❡❞ ✸ ✻✹✹ ✻✾✽ ✸✳✷✽ ✵✳✸✺✵ ❘❡♥❞✿❊♥tr ✸ ✻✹✼ ✼✵✶ ✵✳✵✺ ✵✳✾✾✼ ❘❡♥❞✿Pt♦t❛❧ ✸ ✻✹✼ ✼✵✶ ✵✳✵✶ ✶✳✵✵✵ ❘❡♥❞✿P❛❣❛s ✸ ✻✹✺ ✻✾✾ ✶✳✽✾ ✵✳✺✾✺ ❈r❡❞✿❊♥tr ✶ ✻✹✼ ✻✾✼ ✵✳✷✷ ✵✳✻✸✺ ❈r❡❞✿Pt♦t❛❧ ✶ ✻✹✶ ✻✾✶ ✺✳✺✶ ✵✳✵✶✾ ✯ ❈r❡❞✿P❛❣❛s ✶ ✻✸✾ ✻✽✾ ✼✳✽✽ ✵✳✵✵✺ ✯✯ ❊♥tr✿Pt♦t❛❧ ✶ ✻✹✺ ✻✾✺ ✶✳✾✹ ✵✳✶✻✸ ❊♥tr✿P❛❣❛s ✶ ✻✹✻ ✻✾✻ ✶✳✶✽ ✵✳✷✼✽

  ✶✵✺ Pt♦t❛❧✿P❛❣❛s ✶ ✻✸✶ ✻✽✶ ✶✻✳✷✹ ✺✳✻❡✲✵✺ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ➔

  ❚❡st❛r s✉❣❡stã♦ ❞♦ ♦✉t♣✉t ❛♥t❡r✐♦r ❃ ▼♦❞❆■❈✸ ✭Pt♦t❛❧ ✯ P❛❣❛s✮ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✽✷❡✰✵✵ ✲✻✳✸✵❡✲✵✸ ✲✸✳✷✶❡✲✵✹ ✲✶✳✵✾❡✲✵✺ ✷✳✺✼❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✸✳✽✹❡✰✵✶ ✻✳✸✷❡✰✵✷ ✲✵✳✵✻ ✵✳✾✺✶✺ ■❞❛❞❡■❞♦s♦s ✲✷✳✷✼❡✰✵✵ ✽✳✻✹❡✲✵✶ ✲✷✳✻✷ ✵✳✵✵✽✼ ✯✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✽✽❡✰✵✵ ✸✳✶✼❡✲✵✶ ✶✷✳✷✸ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✷✹❡✰✵✵ ✻✳✾✸❡✲✵✶ ✶✳✼✾ ✵✳✵✼✹✶ ✳ ❉❡♣P♦✉❝♦s ✷✳✼✽❡✰✵✵ ✸✳✹✾❡✲✵✶ ✼✳✾✻ ✶✳✼❡✲✶✺ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✷✹❡✰✵✵ ✺✳✾✽❡✲✵✶ ✺✳✹✸ ✺✳✽❡✲✵✽ ✯✯✯ ❈✐✈✐❧❙♦❧ ✶✳✺✼❡✰✵✵ ✷✳✷✽❡✲✵✶ ✻✳✾✵ ✺✳✵❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✸✾❡✰✵✵ ✻✳✷✻❡✲✵✶ ✺✳✹✶ ✻✳✷❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✺✺❡✰✵✵ ✻✳✸✶❡✲✵✶ ✽✳✼✾ ❁ ✷❡✲✶✻ ✯✯✯ ❘❡♥❞❇❛✐①♦ ✶✳✻✽❡✰✵✶ ✻✳✸✷❡✰✵✷ ✵✳✵✸ ✵✳✾✼✽✽ ❘❡♥❞▼❡❞✐♦ ✽✳✾✸❡✰✵✵ ✻✳✸✷❡✰✵✷ ✵✳✵✶ ✵✳✾✽✽✼ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✷❡✰✵✶ ✻✳✸✷❡✰✵✷ ✵✳✵✹ ✵✳✾✼✵✼ ❈r❡❞ ✲✶✳✵✷❡✲✵✺ ✶✳✻✹❡✲✵✻ ✲✻✳✷✶ ✺✳✺❡✲✶✵ ✯✯✯ ❊♥tr▼❡♥♦s✶✵ ✺✳✵✼❡✰✵✵ ✸✳✻✽❡✲✵✶ ✶✸✳✼✼ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✷✳✹✻❡✲✵✷ ✸✳✸✾❡✲✵✸ ✼✳✷✹ ✹✳✺❡✲✶✸ ✯✯✯ P❛❣❛s ✲✶✳✵✹❡✲✵✶ ✶✳✸✹❡✲✵✷ ✲✼✳✽✵ ✻✳✵❡✲✶✺ ✯✯✯ Pt♦t❛❧✿P❛❣❛s ✶✳✵✼❡✲✵✹ ✷✳✼✵❡✲✵✺ ✸✳✾✼ ✼✳✶❡✲✵✺ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✸✾✳✻✾ ♦♥ ✾✾✽✸ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✼✸✳✼ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ❃ ❛♥♦✈❛✭▼♦❞❆■❈✷✱ ▼♦❞❆■❈✸✱ t❡st ❂ ✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✰

  P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✯ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✼ ✶✹✽✼ ✷ ✾✾✽✸ ✻✹✵ ✹ ✽✹✽ ❁✷❡✲✶✻ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ❃ ❘✷❴❆■❈✸ ❬✶❪ ✵✳✼✼✽ ➔ ✷➟ ❢✉♥çã♦

  ✶✵✻ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  ❃ ▼♦❞❆■❈✸ ❙✐♥❣❧❡ t❡r♠ ❛❞❞✐t✐♦♥s ▼♦❞❡❧✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✯ P❛❣❛s

  ❉❢ ❉❡✈✐❛♥❝❡ ❆■❈ ▲❘❚ Pr✭❈❤✐✮ ❁♥♦♥❡❃ ✻✹✵ ✻✼✹ ■❞❛❞❡✿❉❡♣ ✸ ✻✸✾ ✻✼✾ ✶✳✶✶ ✵✳✼✼✹✸ ■❞❛❞❡✿❈✐✈✐❧ ✹ ✻✸✾ ✻✽✶ ✵✳✽✹ ✵✳✾✸✸✹ ■❞❛❞❡✿P♦st❛❧ ✹ ✻✸✼ ✻✼✾ ✷✳✻✶ ✵✳✻✷✹✽ ■❞❛❞❡✿❘❡♥❞ ✻ ✻✹✵ ✻✽✻ ✵✳✶✷ ✶✳✵✵✵✵ ■❞❛❞❡✿❈r❡❞ ✷ ✻✸✹ ✻✼✷ ✺✳✼✽ ✵✳✵✺✺✼ ✳ ■❞❛❞❡✿❊♥tr ✷ ✻✸✾ ✻✼✼ ✵✳✹✹ ✵✳✽✵✹✶ ■❞❛❞❡✿Pt♦t❛❧ ✷ ✻✸✾ ✻✼✼ ✵✳✸✶ ✵✳✽✺✹✸ ■❞❛❞❡✿P❛❣❛s ✷ ✻✸✾ ✻✼✼ ✵✳✽✵ ✵✳✻✼✶✹ ❉❡♣✿❈✐✈✐❧ ✹ ✻✸✻ ✻✼✽ ✸✳✷✶ ✵✳✺✷✸✵ ❉❡♣✿P♦st❛❧ ✹ ✻✸✽ ✻✽✵ ✷✳✵✸ ✵✳✼✷✾✼ ❉❡♣✿❘❡♥❞ ✻ ✻✸✾ ✻✽✺ ✵✳✼✶ ✵✳✾✾✹✸ ❉❡♣✿❈r❡❞ ✷ ✻✸✽ ✻✼✻ ✷✳✵✺ ✵✳✸✺✽✹ ❉❡♣✿❊♥tr ✷ ✻✸✼ ✻✼✺ ✷✳✾✷ ✵✳✷✸✷✵ ❉❡♣✿Pt♦t❛❧ ✷ ✻✸✺ ✻✼✸ ✹✳✼✵ ✵✳✵✾✺✺ ✳ ❉❡♣✿P❛❣❛s ✷ ✻✸✺ ✻✼✸ ✹✳✾✵ ✵✳✵✽✻✷ ✳ ❈✐✈✐❧✿P♦st❛❧ ✹ ✻✸✵ ✻✼✷ ✾✳✼✹ ✵✳✵✹✺✶ ✯ ❈✐✈✐❧✿❘❡♥❞ ✻ ✻✸✻ ✻✽✷ ✸✳✷✻ ✵✳✼✼✺✺ ❈✐✈✐❧✿❈r❡❞ ✷ ✻✸✽ ✻✼✻ ✶✳✻✺ ✵✳✹✸✼✷ ❈✐✈✐❧✿❊♥tr ✷ ✻✸✽ ✻✼✻ ✶✳✹✹ ✵✳✹✽✻✻ ❈✐✈✐❧✿Pt♦t❛❧ ✷ ✻✸✺ ✻✼✸ ✹✳✺✷ ✵✳✶✵✹✺ ❈✐✈✐❧✿P❛❣❛s ✷ ✻✸✽ ✻✼✻ ✷✳✶✶ ✵✳✸✹✽✸ P♦st❛❧✿❘❡♥❞ ✻ ✻✸✻ ✻✽✷ ✸✳✻✶ ✵✳✼✷✾✼ P♦st❛❧✿❈r❡❞ ✷ ✻✸✾ ✻✼✼ ✵✳✸✷ ✵✳✽✺✶✸ P♦st❛❧✿❊♥tr ✷ ✻✸✾ ✻✼✼ ✵✳✹✻ ✵✳✼✾✹✺ P♦st❛❧✿Pt♦t❛❧ ✷ ✻✸✾ ✻✼✼ ✵✳✼✼ ✵✳✻✽✵✶ P♦st❛❧✿P❛❣❛s ✷ ✻✸✻ ✻✼✹ ✸✳✹✹ ✵✳✶✼✽✾ ❘❡♥❞✿❈r❡❞ ✸ ✻✸✻ ✻✼✻ ✸✳✻✺ ✵✳✸✵✶✻ ❘❡♥❞✿❊♥tr ✸ ✻✹✵ ✻✽✵ ✵✳✵✸ ✵✳✾✾✽✻ ❘❡♥❞✿Pt♦t❛❧ ✸ ✻✹✵ ✻✽✵ ✵✳✵✷ ✵✳✾✾✾✵ ❘❡♥❞✿P❛❣❛s ✸ ✻✸✽ ✻✼✽ ✶✳✼✶ ✵✳✻✸✹✾ ❈r❡❞✿❊♥tr ✶ ✻✸✾ ✻✼✺ ✵✳✻✺ ✵✳✹✷✵✷ ❈r❡❞✿Pt♦t❛❧ ✶ ✻✸✻ ✻✼✷ ✸✳✺✻ ✵✳✵✺✾✶ ✳ ❈r❡❞✿P❛❣❛s ✶ ✻✸✷ ✻✻✽ ✼✳✺✶ ✵✳✵✵✻✶ ✯✯ ❊♥tr✿Pt♦t❛❧ ✶ ✻✸✽ ✻✼✹ ✶✳✸✶ ✵✳✷✺✷✻ ❊♥tr✿P❛❣❛s ✶ ✻✸✼ ✻✼✸ ✷✳✹✶ ✵✳✶✷✵✹ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶

  ➔❚❡st❛r s✉❣❡stã♦ ❞♦ ♦✉t♣✉t ❛♥t❡r✐♦r ❃ ▼♦❞❆■❈✹ ❁✲✭Pt♦t❛❧ ✯ P❛❣❛s ✰ ❈r❡❞ ✯ P❛❣❛s✮ ❉❡✈✐❛♥❝❡ ❘❡s✐❞✉❛❧s✿ ▼✐♥ ✶◗ ▼❡❞✐❛♥ ✸◗ ▼❛① ✲✷✳✽✻❡✰✵✵ ✲✺✳✾✵❡✲✵✸ ✲✷✳✼✹❡✲✵✹ ✲✽✳✵✽❡✲✵✻ ✷✳✻✻❡✰✵✵ ❈♦❡❢❢✐❝✐❡♥ts✿

  ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ③ ✈❛❧✉❡ Pr✭❃⑤③⑤✮ ✭■♥t❡r❝❡♣t✮ ✲✹✳✵✾❡✰✵✶ ✻✳✷✾❡✰✵✷ ✲✵✳✵✻ ✵✳✾✹✽✷ ■❞❛❞❡■❞♦s♦s ✲✷✳✶✽❡✰✵✵ ✽✳✺✸❡✲✵✶ ✲✷✳✺✻ ✵✳✵✶✵✺ ✯ ■❞❛❞❡❏♦✈❡♥s ✸✳✾✻❡✰✵✵ ✸✳✷✸❡✲✵✶ ✶✷✳✷✹ ❁ ✷❡✲✶✻ ✯✯✯ ❉❡♣▼✉✐t♦s ✶✳✷✻❡✰✵✵ ✻✳✾✺❡✲✵✶ ✶✳✽✶ ✵✳✵✻✾✼ ✳ ❉❡♣P♦✉❝♦s ✷✳✽✹❡✰✵✵ ✸✳✺✺❡✲✵✶ ✽✳✵✵ ✶✳✷❡✲✶✺ ✯✯✯ ❈✐✈✐❧❉✐✈ ✸✳✸✶❡✰✵✵ ✺✳✾✻❡✲✵✶ ✺✳✺✻ ✷✳✼❡✲✵✽ ✯✯✯

  ✶✵✼ ❈✐✈✐❧❙♦❧ ✶✳✻✵❡✰✵✵ ✷✳✸✵❡✲✵✶ ✻✳✾✼ ✸✳✸❡✲✶✷ ✯✯✯ P♦st❛❧❩♦♥❛✸ ✸✳✹✸❡✰✵✵ ✻✳✷✾❡✲✵✶ ✺✳✹✺ ✺✳✵❡✲✵✽ ✯✯✯ P♦st❛❧❩♦♥❛✹ ✺✳✻✷❡✰✵✵ ✻✳✸✺❡✲✵✶ ✽✳✽✺ ❁ ✷❡✲✶✻ ✯✯✯ ❘❡♥❞❇❛✐①♦ ✶✳✻✾❡✰✵✶ ✻✳✷✾❡✰✵✷ ✵✳✵✸ ✵✳✾✼✽✺ ❘❡♥❞▼❡❞✐♦ ✽✳✾✽❡✰✵✵ ✻✳✷✾❡✰✵✷ ✵✳✵✶ ✵✳✾✽✽✻ ❘❡♥❞▼✐♥✐♠♦ ✷✳✸✹❡✰✵✶ ✻✳✷✾❡✰✵✷ ✵✳✵✹ ✵✳✾✼✵✸ ❈r❡❞ ✼✳✻✼❡✲✵✼ ✹✳✸✵❡✲✵✻ ✵✳✶✽ ✵✳✽✺✽✸ ❊♥tr▼❡♥♦s✶✵ ✺✳✶✺❡✰✵✵ ✸✳✼✹❡✲✵✶ ✶✸✳✼✼ ❁ ✷❡✲✶✻ ✯✯✯ Pt♦t❛❧ ✷✳✺✼❡✲✵✷ ✸✳✸✻❡✲✵✸ ✼✳✻✻ ✶✳✽❡✲✶✹ ✯✯✯ P❛❣❛s ✲✽✳✻✻❡✲✵✷ ✶✳✹✹❡✲✵✷ ✲✻✳✵✸ ✶✳✼❡✲✵✾ ✯✯✯ Pt♦t❛❧✿P❛❣❛s ✶✳✵✶❡✲✵✹ ✷✳✺✾❡✲✵✺ ✸✳✾✵ ✾✳✽❡✲✵✺ ✯✯✯ ❈r❡❞✿P❛❣❛s ✲✶✳✵✻❡✲✵✼ ✸✳✾✸❡✲✵✽ ✲✷✳✻✾ ✵✳✵✵✼✷ ✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ✭❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r ❢♦r ❜✐♥♦♠✐❛❧ ❢❛♠✐❧② t❛❦❡♥ t♦ ❜❡ ✶✮ ◆✉❧❧ ❞❡✈✐❛♥❝❡✿ ✷✽✽✵✳✶✷ ♦♥ ✾✾✾✾ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❘❡s✐❞✉❛❧ ❞❡✈✐❛♥❝❡✿ ✻✸✷✳✶✽ ♦♥ ✾✾✽✷ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❆■❈✿ ✻✻✽✳✷ ◆✉♠❜❡r ♦❢ ❋✐s❤❡r ❙❝♦r✐♥❣ ✐t❡r❛t✐♦♥s✿ ✶✾ ❃❛♥♦✈❛✭▼♦❞❆■❈✱▼♦❞❆■❈✹✱t❡st❂✧❈❤✐sq✧✮ ❆♥❛❧②s✐s ♦❢ ❉❡✈✐❛♥❝❡ ❚❛❜❧❡ ▼♦❞❡❧ ✶✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰

  Pt♦t❛❧ ✰ P❛❣❛s ▼♦❞❡❧ ✷✿ ❉❡❢❛✉❧t ⑦ ■❞❛❞❡ ✰ ❉❡♣ ✰ ❈✐✈✐❧ ✰ P♦st❛❧ ✰ ❘❡♥❞ ✰ ❈r❡❞ ✰ ❊♥tr ✰ Pt♦t❛❧ ✯ P❛❣❛s ✰ ❈r❡❞ ✯ P❛❣❛s ❘❡s✐❞✳ ❉❢ ❘❡s✐❞✳ ❉❡✈ ❉❢ ❉❡✈✐❛♥❝❡ P✭❃⑤❈❤✐⑤✮

  ✶ ✾✾✽✹ ✻✺✺ ✷ ✾✾✽✷ ✻✸✷ ✷ ✷✷✳✼ ✶✳✷❡✲✵✺ ✯✯✯ ✲✲✲ ❙✐❣♥✐❢✳ ❝♦❞❡s✿ ✵ ✬✯✯✯✬ ✵✳✵✵✶ ✬✯✯✬ ✵✳✵✶ ✬✯✬ ✵✳✵✺ ✬✳✬ ✵✳✶ ✬ ✬ ✶ ❃❘✷❴❆■❈✹ ❬✶❪ ✵✳✼✽ ❃ ▼♦❞❆■❈ ❈♦❡❢❢✐❝✐❡♥ts✿ ✭■♥t❡r❝❡♣t✮ ■❞❛❞❡■❞♦s♦s ■❞❛❞❡❏♦✈❡♥s ❉❡♣▼✉✐t♦s ❉❡♣P♦✉❝♦s ❈✐✈✐❧❉✐✈

  ✲✹✳✸✼❡✰✵✶ ✲✷✳✶✺❡✰✵✵ ✸✳✽✺❡✰✵✵ ✶✳✷✼❡✰✵✵ ✷✳✼✼❡✰✵✵ ✸✳✷✷❡✰✵✵ ❈✐✈✐❧❙♦❧ P♦st❛❧❩♦♥❛✸ P♦st❛❧❩♦♥❛✹ ❘❡♥❞❇❛✐①♦ ❘❡♥❞▼❡❞✐♦ ❘❡♥❞▼✐♥✐♠♦ ✶✳✺✻❡✰✵✵ ✸✳✹✺❡✰✵✵ ✺✳✻✶❡✰✵✵ ✶✳✼✶❡✰✵✶ ✽✳✼✻❡✰✵✵ ✷✳✸✺❡✰✵✶ ❈r❡❞ ❊♥tr▼❡♥♦s✶✵ Pt♦t❛❧ P❛❣❛s

  ✲✶✳✵✷❡✲✵✺ ✺✳✵✼❡✰✵✵ ✸✳✻✵❡✲✵✷ ✲✺✳✺✼❡✲✵✷ ❉❡❣r❡❡s ♦❢ ❋r❡❡❞♦♠✿ ✾✾✾✾ ❚♦t❛❧ ✭✐✳❡✳ ◆✉❧❧✮❀ ✾✾✽✹ ❘❡s✐❞✉❛❧ ◆✉❧❧ ❉❡✈✐❛♥❝❡✿ ✷✽✽✵ ❘❡s✐❞✉❛❧ ❉❡✈✐❛♥❝❡✿ ✻✺✺ ❆■❈✿ ✻✽✼ ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔

  ✶✵✾ ●❧♦ssár✐♦

  ❆ ❆❧❛✈❛♥❝❛❣❡♠ ✭▲❡✈❡r❛❣❡✮ s✐❣♥✐✜❝❛ ✐♥✲ ✈❡st✐r ♠❛✐s ❞♦ q✉❡ ❛ ❝❛♣❛❝✐❞❛❞❡ ✜♥❛♥❝❡✐r❛

  ♣❡r♠✐t❡✳ ❉❡s✐❣♥❛ ♦ ✉s♦ ❞❡ ✈ár✐♦s ✐♥str✉✲ ♠❡♥t♦s ✜♥❛♥❝❡✐r♦s✱ t❛✐s ❝♦♠♦ ❡♠♣rést✐♠♦s ❞❡ ❝❛♣✐t❛❧✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❛✉♠❡♥t❛r ♦ r❡t♦r♥♦ ♣♦t❡♥❝✐❛❧ ❞❛s ♦♣❡r❛çõ❡s ✜♥❛♥❝❡✐✲ r❛s✱ ❛✉♠❡♥t❛❞♦ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ t❛♠❜é♠ ♦ s❡✉ r✐s❝♦✳ ❆ ❛❧❛✈❛♥❝❛❣❡♠ t❡♠✱ ♣♦rt❛♥t♦✱ ✉♠ ❡❢❡✐t♦ ❛♠♣❧✐❛❞♦r t❛♥t♦ ❞♦s ❣❛♥❤♦s ❝♦♠♦ ❞❛s ♣❡r❞❛s✳

  ❆r❜✐tr❛❣❡♠ ♥♦ ♠❡r❝❛❞♦ ✜♥❛♥❝❡✐r♦ ❡ ❡♠ ❡❝♦♥♦♠✐❛✱ ❡♥t❡♥❞❡✲s❡ ♣♦r ✉♠❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♠♣r❛ ❡ ✈❡♥❞❛ ❞❡ ✈❛❧♦r❡s ♥❡❣♦❝✐á✈❡✐s✱ r❡❛❧✐③❛❞❛ ❝♦♠ ♦ ♦❜❥❡❝t✐✈♦ ❞❡ ❣❛♥❤♦s ❡❝♦♥ó✲ ♠✐❝♦s s♦❜r❡ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ♣r❡ç♦s ❡①✐st❡♥t❡✱ ♣❛r❛ ✉♠ ♠❡s♠♦ ❛❝t✐✈♦✱ ❡♥tr❡ ❞♦✐s ♠❡r❝❛✲ ❞♦s✳ ❚r❛t❛✲s❡ ❞❡ ✉♠❛ ♦♣❡r❛çã♦ s❡♠ r✐s❝♦ ✭♦✉ ❞❡ r✐s❝♦ r❡❞✉③✐❞♦✮ ❡♠ q✉❡ ❛q✉❡❧❡ q✉❡ r❡❛❧✐③❛ ❛ ❛r❜✐tr❛❣❡♠ ❛♣r♦✈❡✐t❛ ♦ ❡s♣❛ç♦ ❞❡ t❡♠♣♦ ❡①✐st❡♥t❡ ❡♥tr❡ ❛ ❝♦♠♣r❛ ❡ ❛ ✈❡♥❞❛✱ ❡♠ ♠❡r❝❛❞♦s ❞✐❢❡r❡♥t❡s ✭❡♠ q✉❡ ♦ ♣r❡ç♦ ❞♦ ❛❝t✐✈♦ ❛✐♥❞❛ ♥ã♦ s❡ ❛❥✉st♦✉✮ ♣❛r❛ ❛✉❢❡r✐r ❧✉✲ ❝r♦✳

  ❇ ❇♦♥❞s ✈❡r ❖❜r✐❣❛çã♦✳

  ❈ ❈❛♣✐t❛❧ ❊❝♦♥ó♠✐❝♦ ❝❛♣✐t❛❧ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦ ❜❛♥❝♦ ♣❡r♠❛♥❡❝❡r s♦❧✈❡♥t❡ ❡♠ ❝❛s♦

  ❞❡ ♣❡r❞❛ ❡①tr❡♠❛✳ ❖ ❝❛♣✐t❛❧ ❡❝♦♥ó♠✐❝♦ é ♦ ♥ú♠❡r♦ ❞❡ ❞❡s✈✐♦s✲♣❛❞rã♦ à ❞✐stâ♥❝✐❛ ❞❛ ♣❡r❞❛ ❡s♣❡r❛❞❛ q✉❡ é ♥❡❝❡ssár✐♦ ♣❛r❛ ♣r♦✲ t❡❣❡r ♦ ❜❛♥❝♦ ❞❛ ✐♥s♦❧✈ê♥❝✐❛✱ ❡♠ ❝❛s♦ ❞❡ ♣❡r❞❛s ❡①tr❡♠❛s ♥❛ ❝❛rt❡✐r❛ ❞♦ ❜❛♥❝♦✱ ❞❡✲ ✈✐❞♦ ❛♦ r✐s❝♦ ❞❡ ✐♥❝✉♠♣r✐♠❡♥t♦✳ ❚❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✏❝❛♣✐t❛❧ ❞❡ r✐s❝♦✑✳ ❈❛rt❡✐r❛ ❞❡ ❊♠♣rést✐♠♦s ❝♦❧❡❝çã♦ ❞❡ ❛❝t✐✈♦s s❡♠ ❧✐q✉✐❞❡③✱ ❡♠♣rést✐♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♠❛♥t✐❞❛ ♣♦r ✉♠❛ ✐♥st✐t✉✐çã♦ ❜❛♥✲ ❝ár✐❛✳ ❈❛s❤✲✢♦✇ ✭❋❧✉①♦ ❞❡ ❈❛✐①❛✮ r❡❢❡r❡✲ s❡ ❛♦ ♠♦♥t❛♥t❡ ❞❡ ❝❛✐①❛ r❡❝❡❜✐❞♦ ❡ ❣❛st♦ ♣♦r ✉♠❛ ❡♠♣r❡s❛ ❞✉r❛♥t❡ ✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ❞❡✜♥✐❞♦✳ ❘❡❢❡r❡✲s❡ à ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ✜♥❛♥❝❡✐r❛ ❞❛ ❡♠♣r❡s❛✳ ➱ ❛ s♦♠❛ ❞♦s ❧✉❝r♦s ❛❝r❡s❝✐❞❛ ❞❛ ❞❡♣r❡❝✐❛çã♦✱ ♣♦r ♦✉tr❛s ♣❛❧❛✲ ✈r❛s✱ é ♦ ✢✉①♦ ❞❡ ❞✐♥❤❡✐r♦ ❞✐s♣♦♥í✈❡❧✳ ❖ s❡✉ ❛✉♠❡♥t♦ ♦✉ ❞✐♠✐♥✉✐çã♦ ♠♦str❛ ❛ ❝❛♣❛✲ ❝✐❞❛❞❡ ♣❛r❛ ❣❡r❛r r✐q✉❡③❛✱ ❡ ❝❛❧❝✉❧❛✲s❡ ❝♦♠ ❜❛s❡ ♥♦ ❧✉❝r♦ ❝♦♥tá❜✐❧ ❞❡ ✉♠❛ ❡♠♣r❡s❛✳ ❈❧✐❡♥t❡ ❈✉♠♣r✐❞♦r t♦❞♦ ♦ ❝❧✐❡♥t❡ ❝❛✲ ♣❛③ ❞❡ ❝✉♠♣r✐r ❛s ❝♦♥❞✐çõ❡s ❞❡ ✉♠ ❝♦♥✲ tr❛❝t♦ ❞❡ ❞í✈✐❞❛ ❞♦ q✉❛❧ é t✐t✉❧❛r✳

  ❈❧✐❡♥t❡ ■♥❝✉♠♣r✐❞♦r t♦❞♦ ♦ ❝❧✐❡♥t❡ q✉❡ ✱ ❛♥t❡s ❞♦ ✜♥❛❧ ❞♦ ❝♦♥tr❛❝t♦✱ ❢❛❧❤❛ ❛❧✲ ❣✉♠ ♣❛❣❛♠❡♥t♦✳ ❈♦♥tr❛♣❛rt❡ ✉♠❛ ❝♦♥tr❛♣❛rt❡ é ✉♠ ♣❛rt✐❝✐♣❛♥t❡ ♥✉♠ ♥❡❣ó❝✐♦✳ ❯♠ ♥❡❣ó❝✐♦ t❡♠ ✈ár✐❛s ♣❛rt❡s ✭♣❛rt✐❝✐♣❛♥t❡s✮ q✉❡ ❡❢❡❝t✉❛♠ ♦ ♥❡❣ó❝✐♦ ❡♥tr❡ s✐✱ s❡♥❞♦ ❝❛❞❛ ✉♠ ❞❡❧❡s ❝♦♥✲

  ✶✶✵ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘ tr❛♣❛rt❡ ❞❡ t♦❞♦s ♦s ♦✉tr♦s✳ P♦r ❡①❡♠♣❧♦✱ ♥✉♠❛ tr❛♥s❛❝çã♦ ❞❡ ❛❝çõ❡s✱ ♦ ✈❡♥❞❡❞♦r é ❝♦♥tr❛♣❛rt❡ ❞♦ ❝♦♠♣r❛❞♦r✱ ❡ ♦ ❝♦♠♣r❛❞♦r✱ ❝♦♥tr❛♣❛rt❡ ❞♦ ✈❡♥❞❡❞♦r✳

  ❈♦r♣♦r❛t❡ ❇♦♥❞s tít✉❧♦s ❞❡ ❞í✈✐❞❛ ❡♠✐t✐❞♦s ♣♦r ✉♠❛ ❡♠♣r❡s❛ ❡ ✈❡♥❞✐❞♦s ❛ ✐♥✈❡st✐❞♦r❡s✱ ♣❛r❛ ✜♥❛♥❝✐❛r ✐♥✈❡st✐♠❡♥t♦s ❞❡ ❝❛♣✐t❛❧ ❡ ❣❡r✐r ♦ ❝❛s❤✲✢♦✇✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱ ♦s ❛❝t✐✈♦s ❞❛ ❡♠♣r❡s❛ ♣♦❞❡♠ s❡r ✉s❛✲ ❞♦s ❝♦♠♦ ❣❛r❛♥t✐❛ ♣❛r❛ ❛s ♦❜r✐❣❛çõ❡s✳ ❙ã♦ ❝♦♥s✐❞❡r❛❞♦s tít✉❧♦s ❞❡ ♠❛✐♦r r✐s❝♦ ❞♦ q✉❡ ♦s tít✉❧♦s ❞♦ ❣♦✈❡r♥♦✳ ❈♦♠♦ r❡s✉❧t❛❞♦✱ ❛s t❛①❛s ❞❡ ❥✉r♦s sã♦ q✉❛s❡ s❡♠♣r❡ ♠❛✐s ❡❧❡✈❛✲ ❞❛s✱ ♠❡s♠♦ ♣❛r❛ ❛s ❡♠♣r❡s❛s ❝♦♠ ❛ ♠❡❧❤♦r q✉❛❧✐❞❛❞❡ ❞❡ ❝ré❞✐t♦✳ ❆ ♠❛✐♦r✐❛ ❞❛s ❝♦r♣♦✲ r❛t❡ ❜♦♥❞s sã♦ tr✐❜✉tá✈❡✐s ❝♦♠ ♦s t❡r♠♦s ❛ ♠❛✐s ❞❡ ✉♠ ❛♥♦✳ ❆ ❞í✈✐❞❛ ❞❡ ❝♦r♣♦r❛t❡ q✉❡ ✈❡♥❝❡ ❡♠ ♠❡♥♦s ❞❡ ✉♠ ❛♥♦ é ❣❡r❛❧♠❡♥t❡ ❞❡s✐❣♥❛❞❛ ♣♦r ✏♣❛♣❡❧ ❝♦♠❡r❝✐❛❧✑✳ ❈ré❞✐t♦ s✐❣♥✐✜❝❛ ❝♦♥✜❛♥ç❛✴s❡❣✉r❛♥ç❛✳ ➱ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛ss✉♠✐r ❝♦♠♣r♦♠✐ss♦s✱ q✉❡r ♣❛r❛ ✜♥❛♥❝✐❛♠❡♥t♦s✱ q✉❡r ♣❛r❛ ❡♠✲ ♣rést✐♠♦s✱ ❥✉♥t♦ ❞♦ s✐st❡♠❛ ✜♥❛♥❝❡✐r♦✳ ❆ ❝❡❞ê♥❝✐❛ ❞❡ ❝ré❞✐t♦ é ✉♠❛ r❡❧❛çã♦ ❝♦♥tr❛t✉❛❧ ♠❡❞✐❛♥t❡ ❛ q✉❛❧ ✉♠❛ ❞❛s ♣❛rt❡s✱ ❛ ❝r❡❞♦r❛✱ ❡♥tr❡❣❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠♦♥t❛♥t❡ ❛ ♦✉tr❛ ♣❛rt❡✱ ❛ ❞❡✈❡❞♦r❛✱ s♦❜ ❛ ♣r♦♠❡ss❛ ❞❡ ♣❛❣❛✲ ♠❡♥t♦ ❞♦ ✈❛❧♦r ❡♠ ❞í✈✐❞❛✳ ❖ ❝r❡❞♦r é t♦❞❛ ❛ ♣❡ss♦❛ t✐t✉❧❛r ❞❡ ✉♠ ❝ré❞✐t♦✱ ♦✉✱ q✉❡ t❡♠ ❛ r❡❝❡❜❡r ❞❡ ♦✉tr❡♠ ✉♠❛ ❝❡rt❛ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ❞✐♥❤❡✐r♦✳ ❈♦♥❝❡♥tr❛çã♦ ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ✭❈♦♥❝❡♥tr❛t✐♦♥ ❘✐s❦✮ s❡ ✉♠❛ ❝❛rt❡✐r❛ ❞❡ ❡♠♣rést✐♠♦s é ❛❧t❛♠❡♥t❡ ❝♦♥❝❡♥tr❛❞♦✱ ❡♠ t❡r♠♦s ❞❡ ❡①♣♦s✐çã♦ ❛ ✉♠ s❡❝t♦r ❡s♣❡❝í✜❝♦ ♦✉ ❛ ✉♠ ♣❛ís✱ é ❛❧t❛♠❡♥t❡ s✉s❝❡♣tí✈❡❧ à ❝♦r✲ r❡❧❛çã♦ ❡♥tr❡ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t ❡ ❡✈❡♥t♦s ❞❡ ♠✐❣r❛çã♦ ❞❡ ❝ré❞✐t♦✳ ❚❛♠❜é♠ s❡ ❞❡s✐❣♥❛ ❘✐s❝♦ ❞❡ ❈♦♥❝❡♥tr❛çã♦✳ ❈r❡❞✐t▼❡tr✐❝s ♠♦❞❡❧♦ ✉t✐❧✐③❛❞♦ ♣❡❧♦ ❏P ▼♦r❣❛♥ ♣❛r❛ ❛ q✉❛♥t✐✜❝❛çã♦ ❞♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❡♠ ❝❛rt❡✐r❛s ❞❡ ♣r♦❞✉t♦s ❞❡ ❝ré❞✐t♦✱

  ✐♥str✉♠❡♥t♦s ❞❡ r❡♥❞❛ ✜①❛ ❡ ✐♥str✉♠❡♥t♦s ✜♥❛♥❝❡✐r♦s s✉❥❡✐t♦s ❛♦ ✐♥❝✉♠♣r✐♠❡♥t♦ ❞❛ ❝♦♥tr❛♣❛rt❡✳ ❈✉r✈❛ ❞❡ ❏✉r♦s ✭❨✐❡❧❞ ❈✉r✈❡✮ r❡✲ ♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❛ ❡str✉t✉r❛ ❛ t❡r♠♦ ✭t❡r♠ str✉❝t✉r❡✮ ❞❛s t❛①❛s ❞❡ ❥✉r♦✳ ■❧✉s✲ tr❛✱ ❣❡r❛❧♠❡♥t❡✱ ♦ r❡♥❞✐♠❡♥t♦ ❧♦❝❛❧ ❞♦s tít✉✲ ❧♦s ❝♦♠ ✈❡♥❝✐♠❡♥t♦s ❞✐❢❡r❡♥t❡s ♠❛s ❝♦♠ ♦s ♠❡s♠♦s ❢❛❝t♦r❡s ❞❡ r✐s❝♦ ✭t❛✐s ❝♦♠♦ ❛ s♦❧✈❛✲ ❜✐❧✐❞❛❞❡ ❞♦ ❡♠✐t❡♥t❡✮✱ r❡❧❛t✐✈❛♠❡♥t❡ à ♠❛✲ t✉r✐❞❛❞❡✳

  ❉ ❉❡❢❛✉❧t ❢❛❧ê♥❝✐❛ ♦✉ ✐♥❝✉♠♣r✐♠❡♥t♦❀ ❢❛✲ ❧❤❛ ♥♦ ♣❛❣❛♠❡♥t♦ ❞❡ ✉♠ ❝✉♣ã♦ ♦✉ ❞♦ r❡❡♠✲

  ❜♦❧s♦ ❞❛ ❞í✈✐❞❛✳ ❉❡❢❛✉❧t ▲♦ss ♣❛r❛ ❝❛❞❛ ❞❡✈❡❞♦r✱ é ♦ ♠♦♥t❛♥t❡ ♣❡r❞✐❞♦ ♣❡❧♦ ❝r❡❞♦r ❞❡✈✐❞♦ à ♦❝♦r✲ rê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t✳

  ❉❡r✐✈❛❞♦ ✉♠ ✐♥str✉♠❡♥t♦ ❞❡r✐✈❛❞♦ é ✉♠ ♣r♦❞✉t♦ ❝✉❥♦ ✈❛❧♦r ✈❛r✐❛ ❝♦♠ ❛s ♠✉❞❛♥✲ ç❛s q✉❡ ♦❝♦rr❡♠ ♥✉♠❛ ♦✉ ♠❛✐s ✈❛r✐á✈❡✐s ❞♦ ♠❡r❝❛❞♦ s✉❜❥❛❝❡♥t❡✱ ❝♦♠♦ ❝❛♣✐t❛❧ ♣ró♣r✐♦ ♦✉ ♣r❡ç♦s ❞❡ ❝♦♠♠♦❞✐t✐❡s✱ t❛①❛s ❞❡ ❥✉r♦s ♦✉ t❛①❛s ❞❡ ❝â♠❜✐♦✳ ❉❡r✐✈❛❞♦s ❜ás✐❝♦s ✐♥❝❧✉❡♠ ❢♦r✇❛r❞s✱ ❢✉t✉r♦s✱ s✇❛♣s✱ ♦♣çõ❡s✱ ✇❛rr❛♥ts ❡ ♦❜r✐❣❛çõ❡s ❝♦♥✈❡rtí✈❡✐s✳

  ❊ ❊♠♣rést✐♠♦ ❈♦♥tr❛t♦ ♥♦ q✉❛❧ ✉♠❛ ♣❛rt❡ ❢♦r♥❡❝❡ ❞✐♥❤❡✐r♦✱ ♣♦r ✉♠ ❞❡t❡r♠✐♥❛❞♦

  ♣❡rí♦❞♦✱ ❛ ♦✉tr❛ ♣❛rt❡ ❡♠ tr♦❝❛ ❞❡ ♣❛❣❛✲ ♠❡♥t♦ r❡❣✉❧❛r ❞❡ ❥✉r♦s ❡ ❞♦ ♥♦♠✐♥❛❧✳ ❊str✉t✉r❛ ❛ ❚❡r♠♦ ✭❚❡r♠ ❙tr✉❝✲ t✉r❡✮ r❡❧❛çã♦ ❡♥tr❡ t❛①❛s ❞❡ ❥✉r♦ ❞❡ tít✉❧♦s

  ❝♦♠ ♠❛t✉r✐❞❛❞❡s ❞✐❢❡r❡♥t❡s✱ ❣❡r❛❧♠❡♥t❡✱ r❡✲ ♣r❡s❡♥t❛❞❛ ♥❛ ❢♦r♠❛ ❞❡ ✉♠ ❣rá✜❝♦✱ ❞❡s✐❣✲ ♥❛❞♦ ♣♦r ❝✉r✈❛ ❞❡ ❥✉r♦s ✭❨✐❡❧❞ ❈✉r✈❡✮✳ ❊①♣♦s✐çã♦ ❱✉❧♥❡r❛❜✐❧✐❞❛❞❡ à ♣❡r❞❛ ❞❡✲ ✈✐❞♦ ❛ ❛❝♦♥t❡❝✐♠❡♥t♦s ✐♠♣r❡✈✐sí✈❡✐s✳

  ✶✶✶ ❋

  ❋r❡q✉ê♥❝✐❛ ❞❡ ■♥❝✉♠♣r✐♠❡♥t♦s ❊s♣❡✲ r❛❞♦s ✭❊①♣❡❝t❡❞ ❉❡❢❛✉❧t ❋r❡q✉❡♥❝②✮ ❛ ♣r♦✲ ❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ❞❡✈❡❞♦r s❡ r❡✈❡❧❛r ✐♥❝✉♠✲ ♣r✐❞♦r ❛♥t❡s ❞♦ ✈❡♥❝✐♠❡♥t♦ ❞❛ ♦❜r✐❣❛çã♦✳ ❊st❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ♣♦❞❡♠ s❡r ❡st✐♠❛❞❛s ❛ ♣❛rt✐r ❞❡ ❞❛❞♦s ❤✐stór✐❝♦s ♦✉ ❛♥❛❧✐t✐❝❛♠❡♥t❡✱ ✉s❛♥❞♦ ❛ t❡♦r✐❛ ❞❡ ♦♣çõ❡s✳

  ❍ ❍❡❞❣✐♥❣ ♦✉ ❝♦❜❡rt✉r❛ ❞❡ r✐s❝♦✱ é ✉♠ ♣r♦❝❡ss♦ ♦♥❞❡ ✉♠ r✐s❝♦ ♦✉ ❡①♣♦s✐çã♦ ❡①✐s✲ t❡♥t❡ é r❡❞✉③✐❞♦ ♣❡❧❛ ❛❞✐çã♦ ❞❡ ✉♠❛ ♦✉tr❛ ♣♦s✐çã♦✳ ❖ r✐s❝♦ ♣♦❞❡ ♣r♦✈✐r ❞❡ r✐s❝♦ ❝❛♠✲ ❜✐❛❧✱ ❞❡ r✐s❝♦ ❞❡ t❛①❛ ❞❡ ❥✉r♦✱ ❞❛ ❞❡t❡♥çã♦ ❞❡ ❛❝çõ❡s✱ ❞❡ r✐s❝♦ ❞❡ ❝♦♥tr❛♣❛rt❡✱ ❡t❝✳ ❆ r❡❞✉✲ çã♦ ❞♦ r✐s❝♦ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❛tr❛✈és ❞❡ ♣♦✲ s✐çõ❡s ♥ã♦ ❝♦rr❡❧❛❝✐♦♥❛❞❛s ♦✉ ✐♥✈❡rs❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞❛s✱ ♣❡❧♦ ✉s♦ ❞❡ ❞❡r✐✈❛❞♦s s♦❜r❡ ♦s ♣ró♣r✐♦s ❛❝t✐✈♦s ❝♦♠ r✐s❝♦ ♦✉ s♦❜r❡ ❛❝t✐✲ ✈♦s ❝♦rr❡❧❛❝✐♦♥❛❞♦s ✭s❡ ♦ ❞❡r✐✈❛❞♦ ❣❛♥❤❛r ✈❛❧♦r ❝♦♠ ❛ ♣❡r❞❛ ❞❡ ✈❛❧♦r ❞♦ ❛❝t✐✈♦✮ ♦✉ ✐♥✈❡rs❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞♦s ✭s❡ ♦ ❞❡r✐✈❛❞♦ ❣❛♥❤❛r ✈❛❧♦r ❝♦♠ ❛ s✉❜✐❞❛ ❞❡ss❡s ❛❝t✐✈♦s✮✱ ❡t❝✳

  ▼ ▼❛tr✐③ ❞❡ ❚r❛♥s✐çã♦ ♠❛tr✐③ ❞❡ ♣r♦✲ ❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ♦✉ ❞❡ ♠✐❣r❛çã♦ ❞❡

  ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❝ré❞✐t♦ ♣❛r❛ ♦✉tr❛✳ ▼♦❞❡❧♦ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ s✐♠♣❧✐✜✲ ❝❛❞❛ ❞❡ ❛❧❣♦ r❡❛❧✳ ❖s ♠♦❞❡❧♦s ❢❛❝✐❧✐t❛♠ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ✉♠ ❢❡♥ó♠❡♥♦ ❡✱ ❡✈❡♥✲ t✉❛❧♠❡♥t❡✱ ❛ s✉❛ ❡①♣❧♦r❛çã♦✳ ❆❧❣♦r✐t♠♦s✱ ❢ór♠✉❧❛s✱ s✐st❡♠❛s ♦✉ r❡❣r❛s q✉❡ ✈✐s❛♠ r❡✲ ♣r❡s❡♥t❛r ♣r♦❝❡ss♦s ♦✉ ❛tr✐❜✉t♦s r❡❛✐s r❡✲ ❧❛❝✐♦♥❛❞♦s ❝♦♠ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✳ ▼♦❞❡❧♦ ❞❡ ❘✐s❝♦ r✐s❝♦ ❞❡❝♦rr❡♥t❡ ❞❛ ✉t✐❧✐③❛çã♦ ✐♥❞❡✈✐❞❛ ❞❡ ♠♦❞❡❧♦s ✜♥❛♥❝❡✐r♦s✱ ✐♥❝❧✉✐♥❞♦ ❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛❧✐❜r❛çã♦ ❞♦s ♠♦✲ ❞❡❧♦s ❡ ❡s♣❡❝✐✜❝❛çã♦ ✐♥❝♦rr❡❝t❛✳ ▼♦❞❡❧♦s ❞❡ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ♠♦❞❡✲

  ❧♦s q✉❡ ✈✐s❛♠ q✉❛♥t✐✜❝❛r ♦ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ✲ ♣♦r ❡①❡♠♣❧♦✱ ♠♦❞❡❧♦s ♣❛r❛ ❛ r❡❝✉♣❡r❛çã♦ ❞❡ r❡❝♦♠♣❡♥s❛ ♣r♦♠❡t✐❞❛❀ ♠♦❞❡❧♦s ❞❡ r✐s❝♦ ❞❡ ❞❡❢❛✉❧t✳

  ▼♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ t✐♣♦ ❞❡ ♣r♦✲ ❝❡ss♦ ❞❡ ▼❛r❦♦✈✱ q✉❡ é ✉s❛❞♦ ♣❛r❛ ❡s♣❡❝✐✲ ✜❝❛r ♦ Pr♦❝❡ss♦ ❊st♦❝ást✐❝♦ ❞♦s ♣r❡ç♦s ❞♦s ❛❝t✐✈♦s ✲ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ♣r♦✲ ❝❡ss♦ ❞❡ ❲✐❡♥❡r✳ ▼♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ●❡♦♠é✲ tr✐❝♦ ♠♦❞❡❧♦ ❢r❡q✉❡♥t❡♠❡♥t❡ ✉t✐❧✐③❛❞♦ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❞✐❢✉sã♦ s❡❣✉✐❞♦ ♣❡❧♦s ♣r❡ç♦s ❞♦s ❛❝t✐✈♦s✳ ▼♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ st❛♥❞❛r❞ é ✉♠ ♣r♦❝❡ss♦ ❞❡ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❝♦♠ ✐♥❝r❡✲ ♠❡♥t♦s ●❛✉ss✐❛♥♦s✱ ✐st♦ é✱ ❛s ✈❛r✐❛çõ❡s ♥♦s ♣r❡ç♦s ❞♦s ❛❝t✐✈♦s sã♦ ♥♦r♠❛❧♠❡♥t❡ ❞✐str✐✲ ❜✉í❞♦s✳ ❖ t❡r♠♦ ❣❡♦♠étr✐❝♦ s✐❣♥✐✜❝❛ q✉❡ ❛ ✈❛r✐❛çã♦ ❞♦s r❡t♦r♥♦s ❧♦❣❛rít♠✐❝♦s ✭❡♠ ♦♣♦✲ s✐çã♦ à ✈❛r✐❛çã♦ ❞♦s ♣r❡ç♦s✮ é ♥♦r♠❛❧♠❡♥t❡ ❞✐str✐❜✉í❞❛✳

  ❖ ❖❜r✐❣❛çã♦ ❡♠ ✜♥❛♥ç❛s✱ é ✉♠ tít✉❧♦ ❞❡ ❞í✈✐❞❛✱ ❡♠ q✉❡ ♦ ❡♠✐t❡♥t❡ ❞❡✈❡ ❛♦s t✐t✉✲

  ❧❛r❡s ❞❛ ❞í✈✐❞❛ ❡✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦s t❡r♠♦s ❞♦ tít✉❧♦✱ é ♦❜r✐❣❛❞♦ ❛ ♣❛❣❛r ❥✉r♦s ✭♦s ❝✉✲ ♣õ❡s✮ ❡✴♦✉ ❛ r❡st✐t✉✐r ♦ ♠♦♥t❛♥t❡ ♥✉♠❛ ❞❛t❛ ♣♦st❡r✐♦r✱ ❞❡♥♦♠✐♥❛❞❛ ♠❛t✉r✐❞❛❞❡✳ ❖ ❞❡t❡♥t♦r ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦ ❞❡s✐❣♥❛✲s❡ ♣♦r ♦❜r✐❣❛❝✐♦♥✐st❛ ❡ t♦r♥❛✲s❡ ✉♠ ❝r❡❞♦r ❞❛ ✐♥s✲ t✐t✉✐çã♦ ❡♠✐t❡♥t❡✳ P♦❞❡♠ s❡r ❡♠✐t✐❞❛s ♣♦r ✉♠ ❡st❛❞♦✱ ❣♦✈❡r♥♦ r❡❣✐♦♥❛❧✱ ♠✉♥✐❝í♣✐♦ ♦✉ ❡♠♣r❡s❛✳ ❈♦♥s✐❞❡r❛♥❞♦✲s❡ ✉♠❛ ♦❜r✐❣❛çã♦ ❝♦♠♦ ✉♠ ❡♠♣rést✐♠♦✿ ♦ ❡♠✐ss♦r é ♦ ♠✉✲ t✉ár✐♦ ✭❞❡✈❡❞♦r✮✱ ♦ t✐t✉❧❛r é ♦ ✜♥❛♥❝✐❛❞♦r ✭❝r❡❞♦r✮ ❡✱ ♦ ❝✉♣ã♦ ✭❝♦✉♣♦♥✮ é ♦ ❥✉r♦✳

  ❖❜r✐❣❛çã♦ ❝✉♣ã♦✲③❡r♦ ♦✉ ❖❜r✐❣❛✲ çã♦ s❡♠ ❝✉♣ã♦ ✈❡r ❩❡r♦ ❈♦✉♣♦♥ ❇♦♥❞✳ ❖♣çã♦ ❝♦♥tr❛t♦ ❞❡ q✉❡ ❞á ❛♦ ❝♦♠♣r❛✲ ❞♦r ♦ ❞✐r❡✐t♦✱ ♠❛s ♥ã♦ ❛ ♦❜r✐❣❛çã♦✱ ❞❡ ❝♦♠✲

  ♣r❛r ♦✉ ✈❡♥❞❡r ✉♠ ❛❝t✐✈♦ s✉❜❥❛❝❡♥t❡ ❛ ✉♠

  ✶✶✷ ❇✳ ■♥♣✉ts ❡ ❖✉t♣✉ts ❞♦ ❘

  ❞❡t❡r♠✐♥❛❞♦ ♣r❡ç♦ ✭♣r❡ç♦ ❞❡ ❡①❡r❝í❝✐♦ ♦✉ str✐❦❡ ♣r✐❝❡✮ ❡♠ ♦✉ ❛♥t❡s ❞❡ ✉♠❛ ❞❛t❛ ❛❝♦r✲ ❞❛❞❛ ✭♦ ♣❡rí♦❞♦ ❞❡ ❡①❡r❝í❝✐♦✮✳ P♦r ❡ss❡ ❞✐✲ r❡✐t♦✱ ♦ ❝♦♠♣r❛❞♦r ♣❛❣❛ ✉♠ ♣ré♠✐♦ ❛♦ ✈❡♥✲ ❞❡❞♦r✳ ❖ ✈❡♥❞❡❞♦r ❞❡ ✉♠❛ ♦♣çã♦ t❡♠ ❛ ♦❜r✐❣❛çã♦ ❞❡ ❝♦♠♣r❛r ♦✉ ✈❡♥❞❡r ❛♦ ♣r❡ç♦ ❞❡ ❡①❡r❝í❝✐♦ s❡ ♦ ❝♦♠♣r❛❞♦r ❡①❡r❝❡r ♦ s❡✉ ❞✐r❡✐t♦✳

P

  P❛②✲❖✛ ❜❡♥❡❢í❝✐♦ r❡❝❡❜✐❞♦ ♦✉ r❡❡♠✲ ❜♦❧s♦ ❝♦♠♣❧❡t♦ ❞❡ ✉♠ ❡♠♣rést✐♠♦ ✭♣r✐♥✲ ❝✐♣❛❧ ❡ ❥✉r♦s✮✱ ♦✉ ❛ q✉✐t❛çã♦ ✐♥t❡❣r❛❧ ❞❛ ♦❜r✐❣❛çã♦✱ ♦✉ ♦ r❡t♦r♥♦ ❞❡ ✉♠ ❛❝♦r❞♦✱ ❞❡❝✐✲ sã♦ ♦✉ ✐♥✈❡st✐♠❡♥t♦✳ P❡r❞❛ ❊s♣❡r❛❞❛ ✭❊①♣❡❝t❡❞ ▲♦ss✮ ♣❡r❞❛ ♠é❞✐❛ q✉❡ ♦ ❜❛♥❝♦ ♣♦❞❡ ❡s♣❡r❛r ✈✐r ❛ ✐♥❝♦rr❡r ♥✉♠ ❛❝t✐✈♦ ❞✉r❛♥t❡ ✉♠ ♣❡rí♦❞♦ ♥✉♠ ❤♦r✐③♦♥t❡ ❡s♣❡❝✐✜❝❛❞♦✳ P❡r❞❛ ❊s♣❡r❛❞❛ ❞❡ ✉♠❛ ❈❛rt❡✐r❛ s♦♠❛ s✐♠♣❧❡s ❞❛s ♣❡r❞❛s ❡s♣❡r❛❞❛s ✐♥❞✐✲ ✈✐❞✉❛✐s ❞❡ t♦❞♦s ♦s ❛❝t✐✈♦s ❞❡ r✐s❝♦ ♥✉♠❛ ❝❛rt❡✐r❛✳ P❡r❞❛ ■♥❡s♣❡r❛❞❛ ✐♥❝❡rt❡③❛ ❞♦ ♠♦♥✲ t❛♥t❡ ❞❡ ♣❡r❞❛✱ ❞♦ ✈❛❧♦r ❞❛ ❝❛rt❡✐r❛✱ ❝❛✉s❛❞❛ ♣♦r ❝♦♥❞✐çõ❡s ❞❡ ♠❡r❝❛❞♦✳ ❆s ♣❡r❞❛s ✐♥❡s✲ ♣❡r❛❞❛s sã♦ ♣r♦✈♦❝❛❞❛s ♣❡❧❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞❡❢❛✉❧t ❡ ♠✐❣r❛çõ❡s ❞❡ ❝ré❞✐t♦ ✐♥❡s♣❡r❛❞❛s✳

  P❡r❞❛ ■♥❡s♣❡r❛❞❛ ❞❡ ✉♠❛ ❈❛rt❡✐r❛ ♣❡r❞❛ q✉❡ é ✐♥❡s♣❡r❛❞❛ ❛♦ ♥í✈❡❧ ❞❛ ❝❛r✲ t❡✐r❛✳ ❆♦ ❝♦♥trár✐♦ ❞❛ ♣❡r❞❛ ❡s♣❡r❛❞❛ ❞❡ ✉♠❛ ❝❛rt❡✐r❛✱ ❡st❛ ♣❡r❞❛ ♥ã♦ é ✐❣✉❛❧ à s♦♠❛ ❧✐♥❡❛r ❞❛s ♣❡r❞❛s ✐♥❞✐✈✐❞✉❛✐s ✐♥❡s♣❡r❛❞❛s ❞♦s ❛❝t✐✈♦s ❞❡ r✐s❝♦ q✉❡ ❝♦♠♣õ❡♠ ❛ ❝❛rt❡✐r❛ ❛❣r❡❣❛❞❛✳ ❉❡✈✐❞♦ ❛♦s ❡❢❡✐t♦s ❞❛ ❞✐✈❡rs✐✜❝❛✲ çã♦✱ ❛s ♣❡r❞❛s ✐♥❡s♣❡r❛❞❛s ❞❛ ❝❛rt❡✐r❛ sã♦ ♠✉✐t♦ ♠❡♥♦r❡s ❞♦ q✉❡ ❛ s♦♠❛ ❞❛s ♣❡r❞❛s ✐♥❞✐✈✐❞✉❛✐s ✐♥❡s♣❡r❛❞❛s✳

  P♦rt❢♦❧✐♦ ❝❛rt❡✐r❛ ❞❡ ❛♣❧✐❝❛çõ❡s ✜♥❛♥✲ ❝❡✐r❛s ♦✉ ❝♦❧❡❝çã♦ ❞❡ ✐♥✈❡st✐♠❡♥t♦s ♠❛♥t✐❞❛ ♣♦r ✉♠❛ ✐♥st✐t✉✐çã♦ ♦✉ ✐♥❞✐✈í❞✉♦✳ ▼❛♥t❡r ✉♠ ♣♦rt❢♦❧✐♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❢❛③ ♣❛rt❡ ❞❡ ✉♠❛ ❡str❛té❣✐❛ ❞❡ ❞✐✈❡rs✐✜❝❛çã♦✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❞✐♠✐♥✉✐r r✐s❝♦s✳ ❊ss❡s ✐♥✈❡st✐♠❡♥t♦s ✐♥✲ ❝❧✉❡♠ ❢r❡q✉❡♥t❡♠❡♥t❡ ❛❝çõ❡s✱ q✉❡ sã♦ ♦s ✐♥✲ ✈❡st✐♠❡♥t♦s ❡♠ ❡♠♣r❡s❛s ✐♥❞✐✈✐❞✉❛✐s❀ ♦❜r✐✲ ❣❛çõ❡s✱ q✉❡ sã♦ ♦s ✐♥✈❡st✐♠❡♥t♦s ❡♠ ❞í✈✐❞❛ ❡ q✉❡ sã♦ ♣r♦❥❡❝t❛❞♦s ♣❛r❛ ❣❛♥❤❛r ❥✉r♦s ❡ ❢✉♥❞♦s ♠út✉♦s✳ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡❢❛✉❧t Pr♦❜❛❜✐❧✐✲ ❞❛❞❡ ❞❡ ❢❛❧❤❛r ♦ ❝✉♠♣r✐♠❡♥t♦ ❞♦ ❝♦♥tr❛❝t♦✳ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❚r❛♥s✐çã♦ ♣r♦❜❛✲ ❜✐❧✐❞❛❞❡ ❞❛ q✉❛❧✐❞❛❞❡ ❞❡ ❝ré❞✐t♦ ♠❡❧❤♦r❛r ♦✉ ♣✐♦r❛r❀ q✉❛♥t✐✜❝❛ ❛ ♠✐❣r❛çã♦ ❞❡ ❝ré❞✐t♦

  Pr♦❝❡ss♦ ❊st♦❝ást✐❝♦ ♣r♦❝❡ss♦ q✉❡ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣❡❧❛ ❡✈♦❧✉çã♦ ❞❡ ❛❧❣✉♠❛s ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s s♦❜r❡ ❛❧❣✉♠ ♣❛râ♠❡tr♦✱ q✉❡ ♣♦❞❡ s❡r ❞✐s❝r❡t♦ ♦✉ ❝♦♥tí♥✉♦✳ ❖ ▼♦✈✐✲ ♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ●❡♦♠étr✐❝♦ é ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ Pr♦❝❡ss♦ ❊st♦❝ást✐❝♦ ♣❛r❛♠❡tr✐③❛❞♦ ♣❡❧♦ t❡♠♣♦✳ ❖s Pr♦❝❡ss♦s ❊st♦❝ást✐❝♦s sã♦ ✉s❛❞♦s ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ♠♦❞❡❧♦s ❞❡ ❛♣r❡ç❛✲ ♠❡♥t♦ ❢✉t✉r♦ ❞❡ ✐♥str✉♠❡♥t♦s ❡♠ t❡r♠♦s ❞❡ ♣r❡ç♦ ❡ ❞❡ ❛❧❣✉♠❛s ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s✱ ♦✉✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ ♦ ✈❛❧♦r ❢✉t✉r♦ ❞❡ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦ ♦✉ ❞❛ t❛①❛ ❞❡ ❝â♠❜✐♦✳

  Pr♦♣r✐❡❞❛❞❡ ▼❛rt✐♥❣❛❧❛ ♣♦st✉❧❛ q✉❡ ❛ ❡①♣❡❝t❛t✐✈❛ ❝♦♥❞✐❝✐♦♥❛❧ ❞♦ ✈❛❧♦r ❢✉t✉r♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ é ♦ ✈❛❧♦r ❛❝t✉❛❧✳

  ◗ ◗✉❛❧✐❞❛❞❡ ❞❡ ❈ré❞✐t♦ ✱ é ♠❡❞✐❞♦✱ ❡♠ ❞❡t❡r♠✐♥❛❞♦ ♠♦♠❡♥t♦✱ ♣❡❧❛ ♣r♦❜❛❜✐❧✐❞❛❞❡

  ❞❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♣❛❣❛♠❡♥t♦ ❞❛s ♦❜r✐❣❛çõ❡s ❝♦♥tr❛t✉❛✐s✳ ➱ ❞❡✜♥✐❞❛ ❝♦♠♦ ❞❡❢❛✉❧t ♦✉ ✐♥✲ ❝✉♠♣r✐♠❡♥t♦✳

  ❘ ❘❛t✐♥❣ ❛✈❛❧✐❛çã♦ ❞❛ q✉❛❧✐❞❛❞❡ ❞❡ ✐♥✈❡s✲ t✐♠❡♥t♦ ❞❡ ✉♠ ❡♠✐t❡♥t❡ ♣♦r ✉♠❛ ❛❣ê♥❝✐❛

  ❞❡ r❛t✐♥❣✳

  ✶✶✸ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ ❝♦♥tr❛♣❛rt❡ ❢❛❧❤❛r ✉♠ ♣❛❣❛♠❡♥t♦✳ ◆♦ ✐♥s✲ t❛♥t❡ ❡♠ q✉❡ ❡ss❛ s✐t✉❛çã♦ ♦❝♦rr❡ ❞✐③✲s❡ q✉❡ ♦ ❞❡✈❡❞♦r ♦r✐❣✐♥♦✉ ✉♠ ✐♥❝✉♠♣r✐♠❡♥t♦ ✭❞❡✲ ❢❛✉❧t✮✳ ❘✐s❝♦ ❞❡ ▼❡r❝❛❞♦✭▼❛r❦❡t r✐s❦✮ ❡①✲ ♣♦s✐çã♦ ❛♦ r✐s❝♦ ❞❡ ✈❛r✐❛çã♦ ❞♦ ✈❛❧♦r ❞❡ ❛❧❣✉♠❛s ✈❛r✐á✈❡✐s ❞♦ ♠❡r❝❛❞♦✱ t❛✐s ❝♦♠♦ t❛①❛s ❞❡ ❥✉r♦ ♦✉ ❛❧t❡r❛çã♦ ❞❡ t❛①❛s ❞❡ ❝â♠✲ ❜✐♦ ♦✉ ♣r❡ç♦s ❞❛s ❝♦♠♠♦❞✐t✐❡s✳

  ❘✐s❝♦ ❞❡ ❚❛①❛ ❞❡ ❏✉r♦ r✐s❝♦ ❞❡ ❝♦♥✲ tr❛✐r ❡♠♣rést✐♠♦s ♦✉ ✐♥✈❡st✐♠❡♥t♦s ❞❡❝♦r✲ r❡♥t❡s ❞❡ ❛❧t❡r❛çõ❡s ♥❛ t❛①❛ ❞❡ ❥✉r♦✳ ❘✐s❝♦ ◆❡✉r♦ ✭◆❡✉tr❛❧ ❘✐s❦✮ ❝♦♥❞✐çã♦ t❡ór✐❝❛ ❡♠ q✉❡ ♦s ✐♥✈❡st✐❞♦r❡s ♥ã♦ ❡①✐❣❡♠ ♥❡♥❤✉♠❛ ❝♦♠♣❡♥s❛çã♦ ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ r✐s❝♦✳

  ❙ ❙♣r❡❛❞ ❞❡ ❝ré❞✐t♦ t❛①❛ ❞❡ ❥✉r♦ ❡♥✲ tr❡ ❞✉❛s ❡♠✐ssõ❡s ❞❡ ❞í✈✐❞❛ ❝♦♠ ♠❛t✉r✐❞❛❞❡

  ❡ ❞✉r❛çã♦ s❡♠❡❧❤❛♥t❡s✳ ❘❡✢❡❝t❡ ❛ q✉❛❧✐✲ ❞❛❞❡ ❞❡ ❝ré❞✐t♦ r❡❧❛t✐✈❛ ❞♦s ❡♠✐t❡♥t❡s✳ ❖ s♣r❡❛❞ ❞❡ ❝ré❞✐t♦ é ❢r❡q✉❡♥t❡♠❡♥t❡ ✉s❛❞♦ ❝♦♠♦ ✉♠❛ ♠❡❞✐❞❛ ❞❡ s♦❧✈❛❜✐❧✐❞❛❞❡✳ ❯♠❛ r❡❞✉çã♦ ♥♦ s♣r❡❛❞ ❞♦ ❝ré❞✐t♦ r❡✢❡❝t❡ ✉♠❛ ♠❡❧❤♦r✐❛ ♥❛ q✉❛❧✐❞❛❞❡ ❞❡ ❝ré❞✐t♦ ❞♦ ♠✉t✉á✲ r✐♦✳

  ❚ ❚❛①❛ ❞❡ ❏✉r♦ ❞♦ ❈✉♣ã♦ ✭❈♦✉♣♦♥ ✐♥✲ t❡r❡st r❛t❡✮ P❛❣❛♠❡♥t♦s ❞❡ ✉♠❛ ♦❜r✐❣❛çã♦ ❚❛①❛ ❞❡ ❘❡❝✉♣❡r❛çã♦ ♣❡r❝❡♥t❛❣❡♠ ❞♦ ✈❛❧♦r ❛té ❛♦ q✉❛❧ ♦ ✈❛❧♦r ♥♦♠✐♥❛❧ ❞❡ ✉♠❛

  ♦❜r✐❣❛çã♦ ♣♦❞❡ s❡r r❡❝✉♣❡r❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ ♦ ❞❡✈❡❞♦r ❡stá ❡♠ ✐♥❝✉♠♣r✐♠❡♥t♦✳ ❚❛①❛ ▲✐✈r❡ ❞❡ ❘✐s❝♦ ✭❘✐s❦✲❢r❡❡ r❛t❡✮ ✉♠❛ t❛①❛ ❞❡ ✐♥t❡r❡ss❡ t❡ór✐❝♦ ❡♠ q✉❡

  ✉♠ ✐♥✈❡st✐♠❡♥t♦ ♣♦❞❡ ❣❛♥❤❛r ❥✉r♦s✱ s❡♠ ✐♥❝♦rr❡r ♥❡♥❤✉♠ r✐s❝♦✳ ❚❡r♠ ❙tr✉❝t✉r❡ ✈❡r ❡str✉t✉r❛ ❛ t❡r♠♦✳

  ❱ ❱❛❧✉❡✲❛t✲❘✐s❦ ✭❱❛❘✮ ❧✐♠✐t❡ ♣r♦❜❛❜✐❧ís✲ t✐❝♦ ❞❛s ♣❡r❞❛s ❞❡ ♠❡r❝❛❞♦✱ ❞✉r❛♥t❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ✭♣❡rí♦❞♦ ❞❡ ❡①♣❧♦r❛çã♦✮✱ ❡①♣r❡ss♦ ❡♠ t❡r♠♦s ❞❡ ✉♠ ❞❡✲ t❡r♠✐♥❛❞♦ ❣r❛✉ ❞❡ ❝❡rt❡③❛ ✭♦ ✐♥t❡r✈❛❧♦ ❞❡ ❝♦♥✜❛♥ç❛✮✳ ❖ ❱❛❘ é ❛ ♣✐♦r ♣❡r❞❛ ❡s♣❡r❛❞❛✱ ❛♦ ❧♦♥❣♦ ❞♦ ♣❡rí♦❞♦ ❞❡ ❡①♣❧♦r❛çã♦✱ ❞❡♥tr♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡✜♥✐❞❛s ♣❡❧♦ ✐♥t❡r✈❛❧♦ ❞❡ ❝♦♥✜❛♥ç❛✳ P❡r❞❛s ♠❛✐♦r❡s sã♦ ♣♦ssí✈❡✐s✱ ♠❛s ❝♦♠ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❜❛✐①❛✳

  ❱♦❧❛t✐❧✐❞❛❞❡ ♠❡❞✐❞❛ ❞❡ ✈❛r✐❛❜✐❧✐❞❛❞❡ ✭♠❛s ♥ã♦ ❞❛ ❞✐r❡❝çã♦✮ ❞♦s ♣r❡ç♦s ♦✉ ❞❛s t❛✲ ①❛s ❞❡ ❥✉r♦✳

  ❨ ❨✐❡❧❞ ❈✉r✈❡ ✈❡r ❈✉r✈❛ ❞❡ ❏✉r♦s✳

  ❩ ❩❡r♦ ❈♦✉♣♦♥ ❇♦♥❞ t✐♣♦ ❞❡ ♦❜r✐❣❛çã♦ q✉❡ ♥ã♦ ♣❛❣❛ ❥✉❣♦s ♣❡r✐ó❞✐❝♦s❀ ❥✉r♦ ❡stá ✐♠✲

  ♣❧í❝✐t♦ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♣r❡ç♦ ❞❡ ❡♠✐ssã♦ ✭♦✉ ❞❡ ❛q✉✐s✐çã♦✮ ❡ ♦ ✈❛❧♦r ❞❡ r❡❡♠❜♦❧s♦✳ ❚❛❧ s✐❣♥✐✜❝❛ q✉❡ ❡st❛s ♦❜r✐❣❛çõ❡s sã♦ ❡♠✐t✐✲ ❞❛s ❛ ❞❡s❝♦♥t♦✳

  ❇■❇▲■❖●❘❆❋■❆ ✶✶✺

  ❇✐❜❧✐♦❣r❛✜❛

❬❆♥❞r❛❞❡✱ ✷✵✵✹❪ ❆♥❞r❛❞❡✱ ❋✳ ✭✷✵✵✹✮✳ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ▼♦❞❡❧♦ ❞❡ ❘✐s❝♦ ❞❡ P♦rt❢ó❧✐♦ ♣❛r❛

❈❛rt❡✐r❛s ❞❡ ❈ré❞✐t♦ ❛ P❡ss♦❛s ❋ís✐❝❛s✳ ❉✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦✱ ❊s❝♦❧❛ ❞❡ ❆❞♠✐♥✐str❛çã♦

  ❞❡ ❊♠♣r❡s❛s ❞❡ ❙ã♦ P❛✉❧♦✱ ❋✉♥❞❛çã♦ ●❡tú❧✐♦ ❱❛r❣❛s✱ ❙ã♦ P❛✉❧♦✳

❬❆r❛ú❥♦✱ ✷✵✵✻❪ ❆r❛ú❥♦✱ ❊✳ ✭✷✵✵✻✮✳ ▼♦❞❡❧❛❣❡♠ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✿ ❆♣❧✐❝❛çã♦ ❞❡ ♠♦❞❡❧♦s ❈r❡❞✐t

❙❝♦r✐♥❣ ♥♦ ❋✉♥❞♦ ❘♦t❛t✐✈♦ ❞❡ ❆çã♦ ❞❛ ❈✐❞❛❞❛♥✐❛✲❈r❡❞ ❈✐❞❛❞❛♥✐❛✳ ❉✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦✳

  ❬❆r✈❛♥t✐s ❡ ●r❡❣♦r②✱ ✷✵✵✶❪ ❆r✈❛♥t✐s✱ ❆✳ ❡ ●r❡❣♦r②✱ ❏✳ ✭✷✵✵✶✮✳ ❈r❡❞✐t✳ ❘✐s❦ ❇♦♦❦s✱ ▲♦♥❞♦♥✳

❬❇❛rr❡t♦✱ ✶✾✾✻❪ ❇❛rr❡t♦✱ ■✳ ✭✶✾✾✻✮✳ ▼❛♥✉❛❧ ❞❡ ❋✐♥❛♥ç❛s ✲ ❆ ♠♦❞❡r♥❛ t❡♦r✐❛ ❞❡ ❆ ❛ ❩✳ Pr✐♥t❡r

P♦rt✉❣✉❡s❛✱ ▼❡♠ ▼❛rt✐♥s✳

❬❇❡❞r✐❝❦ ❡ ❍✐❧❧✱ ✶✾✾✵❪ ❇❡❞r✐❝❦✱ ❊✳ ❡ ❍✐❧❧✱ ❏✳ ✭❉❡❝❡♠❜❡r ✶✾✾✵✮✳ ❖✉t❧✐❡r t❡sts ❢♦r ❧♦❣✐st✐❝ r❡❣r❡s✲

s✐♦♥✿ ❆ ❝♦♥❞✐t✐♦♥❛❧ ❛♣♣r♦❛❝❤✳ ❇✐♦♠❡tr✐❦❛ ❚r✉st✱ ✼✼✭✹✮✳ ❬❇❡ss✐s✱ ✶✾✾✽❪ ❇❡ss✐s✱ ❏✳ ✭✶✾✾✽✮✳ ❘✐s❦ ▼❛♥❛❣❡♠❡♥t ✐♥ ❇❛♥❦✐♥❣✳ ❲✐❧❡②✱ ❊♥❣❧❛♥❞✳

  

❬❇❧❛tt✱ ✶✾✾✾❪ ❇❧❛tt✱ ❆✳ ✭✶✾✾✾✮✳ ❆✈❛❧✐❛çã♦ ❉❡ ❘✐s❝♦ ❊ ❉❡❝✐sã♦ ❉❡ ❈ré❞✐t♦✿ ❯♠ ❊♥❢♦q✉❡ Prá✲

t✐❝♦✳ ▲✐✈r❛r✐❛ ◆♦❜❡❧ ❙❛✱ ❙ã♦ P❛✉❧♦✳

❬❈❛ñ✐③❛r❡s ❡t ❛❧✳✱ ✷✵✵✹❪ ❈❛ñ✐③❛r❡s✱ ▲✳✱ ▼♦♥t♦②❛✱ ❆✳✱ ❡ ❩✉r✐❛❣❛✱ ■✳ ✭✷✵✵✹✮✳ ●❧♦s❛r✐♦ ❞❡ ❚ér♠✐✲

♥♦s ❇✉rsát✐❧❡s ❨ ❋✐♥❛♥❝❡✐r♦s✳ ❈✐✈✐t❛s ❊❞✐❝✐♦♥❡s✱ ❙▲✱ ▼❛❞r✐❞✳

❬❈❛♦✉❡tt❡ ❡t ❛❧✳✱ ✶✾✾✽❪ ❈❛♦✉❡tt❡✱ ❏✳✱ ❆❧t♠❛♥✱ ❊✳✱ ❡ ◆❛r❛②❛♥❛♥✱ P✳ ✭✶✾✾✽✮✳ ▼❛♥❛❣✐♥❣ ❈r❡❞✐t

❘✐s❦✿ ❚❤❡ ◆❡①t ●r❡❛t ❋✐♥❛♥❝✐❛❧ ❈❤❛❧❧❡♥❣❡✳ ❲✐❧❡②✱ ❍❛r❞❝♦✈❡r✳

❬❈❤❛✐❛✱ ✷✵✵✸❪ ❈❤❛✐❛✱ ❆✳ ✭✷✵✵✸✮✳ ▼♦❞❡❧♦s ❞❡ ●❡stã♦ ❞♦ r✐s❝♦ ❞❡ ❈ré❞✐t♦ ❡ s✉❛ ❆♣❧✐❝❛❜✐❧✐❞❛❞❡

♥♦ ▼❡r❝❛❞♦ ❇r❛s✐❧❡✐r♦✳ ❉✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦✱ ❋❛❝✉❧❞❛❞❡ ❞❡ ❊❝♦♥♦♠✐❛✱ ❆❞♠✐♥✐str❛çã♦

  ❡ ❈♦♥t❛❜✐❧✐❞❛❞❡✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳ ❬❈r❛✇❧❡②✱ ✷✵✵✼❪ ❈r❛✇❧❡②✱ ▼✳ ✭✷✵✵✼✮✳ ❚❤❡ ❘ ❇♦♦❦✳ ❲✐❧❡②✱ ❊♥❣❧❛♥❞✳

❬❉✉✛②✱ ✶✾✾✵❪ ❉✉✛②✱ ❉✳ ✭❏✉♥❡ ✶✾✾✵✮✳ ♦♥ ❝♦♥t✐♥✉✐t②✲❝♦rr❡❝t❡❞ r❡s✐❞✉❛❧s ✐♥ ❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥✳

❇✐♦♠❡tr✐❦❛ ❚r✉st✱ ✼✼✭✷✮✳

❬❋✐❧❤♦ ❡t ❛❧✳✱ ❪ ❋✐❧❤♦✱ ❊✳ ❋✳✱ ❈❛r✈❛❧❤♦✱ ❆✳ ❈✳ ✳✱ ❡ ▼❛t✐❛s✱ ❆✳ ❇✳ ❯t✐❧✐③❛çã♦ ❞❡ r❡❞❡s ♥❡✉r❛✐s

❛rt✐✜❝✐❛✐s ♥❛ ❛♥á❧✐s❡ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦ ❛ ♣❡ss♦❛s ❢ís✐❝❛s✳

❬●✉❡rr❡✐r♦ ❡ ▼❡①✐❛✱ ✷✵✵✽❪ ●✉❡rr❡✐r♦✱ ●✳ ❡ ▼❡①✐❛✱ ❏✳ ✭✷✵✵✽✮✳ ❙t♦❝❤❛st✐❝ ✈♦rt✐❝❡s ✐♥ ♣❡r✐♦❞✐❝❛❧❧②

r❡❝❧❛ss✐✜❡❞ ♣♦♣✉❧❛t✐♦♥s✳ ❉✐s❝✉ss✐♦♥❡s ▼❛t❤❡♠❛t✐❝❛❡✱ Pr♦❜❛❜✐❧✐t② ❛♥❞ ❙t❛t✐st✐❝s✳

  ✶✶✻ ❇■❇▲■❖●❘❆❋■❆

  

❬●✉❡rr❡✐r♦✱ ✷✵✶✵❪ ●✉❡rr❡✐r♦✱ P✳ ✭✶✾ ❏❛♥❡✐r♦ ❞❡ ✷✵✶✵✮✳ ❉❡s❡♠♣r❡❣❛❞♦s✱ ❞✐✈♦r❝✐❛❞♦s ❡ ❤✐♣♦t❡✲

❝❛❞♦s✳

❬●✉♣t♦♥ ❡t ❛❧✳✱ ✶✾✾✼❪ ●✉♣t♦♥✱ ●✳ ▼✳✱ ❋✐♥❣❡r✱ ❈✳✱ ❡ ❇❤❛t✐❛✱ ▼✳ ✭❆❜r✐❧✴✶✾✾✼✮✳ ❈r❡❞✐t♠❡tr✐❝s

t❡❝❤♥✐❝❛❧ ❞♦❝✉♠❡♥t✳

❬❍❛✐r ❡t ❛❧✳✱ ✷✵✵✾❪ ❍❛✐r✱ ❏✳✱ ❇❧❛❝❦✱ ❲✳✱ ❡ ❆♥❞❡rs♦♥✱ ❘✳ ✭✷✵✵✾✮✳ ▼✉❧t✐✈❛r✐❛t❡ ❉❛t❛ ❆♥❛❧②s✐s✳

❲✐❧❡②✱ ❍❛r❞❝♦✈❡r✳ ❬■◆❊✱ ✷✵✵✷❪ ■◆❊ ✭✷✵✵✷✮✳ ❈❡♥s♦s ✷✵✵✶✳

❬❏❡♥♥✐♥❣s✱ ✶✾✽✻❪ ❏❡♥♥✐♥❣s✱ ❉✳ ✭❉❡❝✳ ✶✾✽✻✮✳ ❖✉❧t✐❡rs ❛♥❞ r❡s✐❞✉❛❧ ❞✐str✐❜✉t✐♦♥s ✐♥ ❧♦❣✐st✐❝

r❡❣r❡ss✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛❧ ❙t❛t✐st✐❝❛❧ ❆ss♦❝✐❛t✐♦♥✱ ✽✶✭✸✾✻✮✳

❬❏♦✉✐♥✐ ❡t ❛❧✳✱ ✷✵✵✶❪ ❏♦✉✐♥✐✱ ❊✳✱ ❈✈✐t❛♥✐❝✱ ❏✳✱ ❡ ▼✉s✐❡❧❛✱ ▼✳ ✭✷✵✵✶✮✳ ❖♣t✐♦♥ Pr✐♥❝✐♥❣✱ ■♥t❡r❡st

❘❛t❡s ❛♥❞ ❘✐s❦ ▼❛♥❛❣❡♠❡♥t✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t②✱ ❈❛♠❜r✐❞❣❡✳

❬▲❡✇✐s✱ ✶✾✾✷❪ ▲❡✇✐s✱ ❊✳ ✭✶✾✾✷✮✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈r❡❞✐t s❝♦r✐♥❣✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈r❡❞✐t

s❝♦r✐♥❣✱ ❈❛❧✐❢♦r♥✐❛✳

❬▼❝❈✉❧❧❛❣❤ ❡ ◆❡❧❞❡r✱ ✶✾✽✾❪ ▼❝❈✉❧❧❛❣❤✱ P✳ ❡ ◆❡❧❞❡r✱ ❏✳ ✭✶✾✽✾✮✳ ●❡♥❡r❛❧✐③❡❞ ▲✐♥❡❛r ▼♦❞❡❧s✳

❈❤❛♣♠❛♥ ✫ ❍❛❧❧✴❈❘❈✱ ❯❙❆✳

❬▼❝◆❡✐❧ ❡t ❛❧✳✱ ✷✵✵✺❪ ▼❝◆❡✐❧✱ ❆✳✱ ❋r❡②✱ ❘✳✱ ❡ ❊♠❜r❡❝❤ts✱ P✳ ✭✷✵✵✺✮✳ ◗✉❛♥t✐t❛t✐✈❡ ❘✐s❦ ▼❛♥❛✲

❣❡♠❡♥t✳ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❯❙❆✳ ❬▼♦r❣❛♥✱ ✶✾✾✼❪ ▼♦r❣❛♥✱ ❏✳ ✭❆❜r✐❧✴✶✾✾✼✮✳ ❈r❡❞✐t♠❡tr✐❝s t❡❝❤♥✐❝❛❧ ❞♦❝✉♠❡♥t✳

  

❬▼✉rt❡✐r❛ ❡t ❛❧✳✱ ✷✵✵✷❪ ▼✉rt❡✐r❛✱ ❇✳✱ ❘✐❜❡✐r♦✱ ❈✳✱ ❙✐❧✈❛ ❡✱ ❏✳✱ ❡ P✐♠❡♥t❛✱ ❈✳ ✭✷✵✵✷✮✳ ■♥tr♦❞✉çã♦

à ❊st❛tíst✐❝❛✳ ▼❈❝ ●r❛✇ ❍✐❧❧✱ P♦rt✉❣❛❧✳ ❬◆❛tár✐♦✱ ✷✵✵✾❪ ◆❛tár✐♦✱ ■✳ ✭✸✵ ❏✉❧❤♦ ❞❡ ✷✵✵✾✮✳ Pr♦❜❛❜✐❧✐❞❛❞❡s ❡ ❡st❛tíst✐❝❛ ❞✳

❬◆❡❧❞❡r ❡ ❲❡❞❞❡r❜✉r♥✱ ✶✾✼✷❪ ◆❡❧❞❡r✱ ❏✳ ❡ ❲❡❞❞❡r❜✉r♥✱ ❘✳ ✭❉❡❝✳ ✶✾✼✷✮✳ ●❡♥❡r❛❧✐③❡❞ ❧✐♥❡❛r

  ♠♦❞❡❧s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❘♦②❛❧ ❙t❛t✐st✐❝❛❧ ❙♦❝✐❡t②✱ ❆✭✶✸✺✮✳

❬◆❡t♦✱ ✷✵✵✷❪ ◆❡t♦✱ ❆✳ ✭✷✵✵✷✮✳ ▼♦❞❡❧❛❣❡♠ ❞❡ r✐s❝♦ ❞❡ ❝ré❞✐t♦✿ ✉♠ ❡st✉❞♦ ❞♦ s❡❣♠❡♥t♦ ❞❡

♣❡ss♦❛s ❢ís✐❝❛s ❡♠ ✉♠ ❜❛♥❝♦ ❞❡ ✈❛r❡❥♦✳ ❉✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦✳

  ❬❖❦s❡♥❞❛❧✱ ✷✵✵✼❪ ❖❦s❡♥❞❛❧✱ ❇✳ ✭✷✵✵✼✮✳ ❙t♦❝❤❛st✐❝ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ❙♣r✐♥❣❡r✳ ❬P❡r❡✐r❛✱ ✷✵✵✾❪ P❡r❡✐r❛✱ ❏✳ ✭✷✵✵✾✮✳ ❈r❡❞✐t r✐s❦✳

❬❘✐❜❡✐r♦ ❡t ❛❧✳✱ ✷✵✵✻❪ ❘✐❜❡✐r♦✱ ❊✳ ▼✳✱ ◆❡t♦✱ ❏✳ ❉✳✱ ▼❡❧r♦✱ ❊✳ ▼✳✱ ❡ ▼❡❧❧♦✱ ❈✳ ❘✳ ✭✷✵✵✻✮✳ ❆♣❧✐✲

❝❛çã♦ ❞❛s r❡❞❡s ♥❡✉r❛✐s ♥❛ ❝♦♥❝❡ssã♦ ❞❡ ❝ré❞✐t♦ ✲ ✉♠ ❡st✉❞♦ ❞❡ ✉♠ ❝❛s♦ ❡♠ ✉♠❛ ❡♠♣r❡s❛

  ❞❡ ❝♦♥sór❝✐♦✳

❬❘✉t❦♦✇s❦✐ ❡ ▼✉s✐❡❧❛✱ ✶✾✾✽❪ ❘✉t❦♦✇s❦✐✱ ▼✳ ❡ ▼✉s✐❡❧❛✱ ▼✳ ✭✶✾✾✽✮✳ ▼❛rt✐♥❣❛❧❡ ▼❡t❤♦❞s ✐♥

❋✐♥❛♥❝✐❛❧ ▼♦❞❡❧❧✐♥❣✳ ❙♣r✐♥❣❡r✳

❬❙❛✉♥❞❡rs ❡ ❆❧❧❡♥✱ ✷✵✵✷❪ ❙❛✉♥❞❡rs✱ ❆✳ ❡ ❆❧❧❡♥✱ ▲✳ ✭✷✵✵✷✮✳ ❈r❡❞✐t ❘✐s❦ ▼❡❛s✉r❡♠❡♥t✿ ◆❡✇

❆♣♣r♦❛❝❤❡s ❚♦ ❱❛❧✉❡✲❆t✲r✐s❦ ❆♥❞ ❖t❤❡r P❛r❛❞✐❣♠s✳ ❏♦❤♥ ❲✐❧❡②✱ ❍❛r❞❝♦✈❡r✳

  ❇■❇▲■❖●❘❆❋■❆ ✶✶✼

  

❬❙❝❤r✐❝❦❡❧✱ ✷✵✵✵❪ ❙❝❤r✐❝❦❡❧✱ ❲✳ ✭✷✵✵✵✮✳ ❆♥á❧✐s❡ ❞❡ ❝ré❞✐t♦✿ ❝♦♥❝❡ssã♦ ❡ ❣❡rê♥❝✐❛ ❞❡ ❡♠♣rést✐✲

♠♦s✳ ❆t❧❛s✱ ❙ã♦ P❛✉❧♦✳

❬❙❡❝✉r❛t♦✱ ✷✵✵✷❪ ❙❡❝✉r❛t♦✱ ❏✳ ✭✷✵✵✷✮✳ ❈ré❞✐t♦✿ ❆♥á❧✐s❡ ❡ ❆✈❛❧✐❛çã♦ ❞♦ ❘✐s❝♦✿ P❡ss♦❛s ❋ís✐❝❛s

❡ ❏✉rí❞✐❝❛s✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈r❡❞✐t s❝♦r✐♥❣✱ ❙ã♦ P❛✉❧♦✳ ❬❙✐❧✈❛✱ ✷✵✵✵❪ ❙✐❧✈❛✱ ❏✳ ✭✷✵✵✵✮✳ ●❡stã♦ ❡ ❆♥á❧✐s❡ ❞♦ ❘✐s❝♦ ❞❡ ❈ré❞✐t♦✳ ❆t❧❛s✱ ❙ã♦ P❛✉❧♦✳

❬❚✉r❦♠❛♥ ❡ ❙✐❧✈❛✱ ✷✵✵✵❪ ❚✉r❦♠❛♥✱ ▼✳ ❡ ❙✐❧✈❛✱ ❏✳ ✭✷✵✵✵✮✳ ▼♦❞❡❧♦s ❧✐♥❡❛r❡s ❣❡♥❡r❛❧✐③❛❞♦s✲❞❛

t❡♦r✐❛ à ♣rát✐❝❛✳

❬❱❛s❝♦♥❝❡❧❧♦s✱ ✷✵✵✷❪ ❱❛s❝♦♥❝❡❧❧♦s✱ ▼✳ ✭✷✵✵✷✮✳ Pr♦♣♦st❛ ❞❡ ▼ét♦❞♦ ♣❛r❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥❝❡ssã♦ ❞❡

❝ré❞✐t♦ ❛ ♣❡ss♦❛s ❢ís✐❝❛s✳ ❉✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦✱ ❋❛❝✉❧❞❛❞❡ ❞❡ ❊❝♦♥♦♠✐❛✱❆❞♠✐♥✐str❛çã♦ ❡ ❈♦♥t❛❜✐❧✐❞❛❞❡✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳

  

❬❲❛♥❣ ❡ ❙❛❧♦✉s✱ ✷✵✵✾❪ ❲❛♥❣✱ ❩✳ ❡ ❙❛❧♦✉s✱ ❙✳ ✭❋❡✈✳ ✷✵✵✾✮✳ ❙♣❡❝tr✉♠ ♦❝❝✉♣❛♥❝② st❛t✐st✐❝s ❛♥❞

t✐♠❡ s❡r✐❡s ♠♦❞❡❧s ❢♦r ❝♦❣♥✐t✐✈❡ r❛❞✐♦✳ ❏♦✉r♥❛❧ ♦❢ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣ ❙②st❡♠s✳ ❬❲✐❦✐♣é❞✐❛✱ ❪ ❲✐❦✐♣é❞✐❛✳ ❱❛s✐❝❡❦ ♠♦❞❡❧✳ ❬❲✐❦✐♣é❞✐❛✱ ✷✵✵✾❛❪ ❲✐❦✐♣é❞✐❛ ✭✷✵✵✾❛✮✳ ❆❦❛✐❦❡ ✐♥❢♦r♠❛t✐♦♥ ❝r✐t❡r✐♦♥✳ ❬❲✐❦✐♣é❞✐❛✱ ✷✵✵✾❜❪ ❲✐❦✐♣é❞✐❛ ✭✷✵✵✾❜✮✳ ❆♥á❧✐s❡ ❞❡ ❝ré❞✐t♦✳ ❬❲✐❦✐♣é❞✐❛✱ ✷✵✵✾❝❪ ❲✐❦✐♣é❞✐❛ ✭✷✵✵✾❝✮✳ ❘❡❣r❡ssã♦ ❧♦❣íst✐❝❛✳ ❬❲✐❦✐♣é❞✐❛✱ ✷✵✵✾❞❪ ❲✐❦✐♣é❞✐❛ ✭✷✵✵✾❞✮✳ ❩✲s❝♦r❡ ✜♥❛♥❝✐❛❧ ❛♥❛❧②s✐s t♦♦❧✳ ❬❩❛❣st✱ ✷✵✵✷❪ ❩❛❣st✱ ❘✳ ✭✷✵✵✷✮✳ ■♥t❡r❡st ❘❛t❡ ▼❛♥❛❣❡♠❡♥t✳ ❙♣r✐♥❣❡r✱ ●❡r♠❛♥②✳

❬❩❡r❜✐♥✐✱ ✷✵✵✵❪ ❩❡r❜✐♥✐✱ ▼✳ ✭✷✵✵✵✮✳ ❚rês ❡♥s❛✐♦s s♦❜r❡ ❝ré❞✐t♦✳ ❚❡s❡ ❞♦✉t♦r❛♠❡♥t♦✱ ❋❛❝✉❧❞❛❞❡

❞❡ ❊❝♦♥♦♠✐❛✱❆❞♠✐♥✐str❛çã♦ ❡ ❈♦♥t❛❜✐❧✐❞❛❞❡✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳