Modelos multivariados binários com funções de ligação assimétricas

▼♦❞❡❧♦s ♠✉❧t✐✈❛r✐❛❞♦s ❜✐♥ár✐♦s

❝♦♠ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛s

❘❛❢❛❡❧ ❇rá③ ❆③❡✈❡❞♦ ❋❛r✐❛s

❚❡s❡ ❛♣r❡s❡♥t❛❞❛

❛♦

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

♣❛r❛

♦❜t❡♥çã♦ ❞♦ tít✉❧♦

❞❡

❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s

Pr♦❣r❛♠❛✿ ❊st❛tíst✐❝❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r❛✳ ▼ár❝✐❛ ❉❵❊❧✐❛ ❇r❛♥❝♦

❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ t❡s❡ ♦ ❛✉t♦r r❡❝❡❜❡✉ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞♦ ❈◆Pq ❡ ❞❛ ❋❆P❊❙P

▼♦❞❡❧♦s ♠✉❧t✐✈❛r✐❛❞♦s ❜✐♥ár✐♦s

❝♦♠ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛s

❊st❛ t❡s❡ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ r❡❛❧✐③❛❞❛ ♣♦r ❘❛❢❛❡❧ ❇rá③ ❆③❡✈❡❞♦ ❋❛r✐❛s ❡♠ ✷✺✴✵✺✴✷✵✶✷✳ ❖ ♦r✐❣✐♥❛❧ ❡♥❝♦♥tr❛✲s❡ ❞✐s♣♦♥í✈❡❧ ♥♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

• Pr♦❢❛✳ ❉r❛✳ ▼ár❝✐❛ ❉❵❊❧✐❛ ❇r❛♥❝♦ ✭♦r✐❡♥t❛❞♦r❛✮ ✲ ■▼❊✲❯❙P • Pr♦❢✳ ❉r❛✳ ❘♦sâ♥❣❡❧❛ ❍❡❧❡♥❛ ▲♦s❝❤✐ ✲ ❯❋▼●

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦s ♠❡✉s ♣❛✐s✱ ❆❢♦♥s♦ ❡ ❋át✐♠❛✱ ♣❡❧♦ ❛♠♦r✱ ❝❛r✐♥❤♦✱ ❡ ♣♦r ♠❡ ❛♣♦✐❛r❡♠ ❡♠ ♠✐♥❤❛s ❞✐❢í❝❡✐s ❞❡❝✐sõ❡s✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦ ♠❡✉s q✉❡r✐❞♦s✦

➚ ♠✐♥❤❛ ✐r♠ã ❉❛♥✐❡❧❡ ♣❡❧♦ ❛♣♦✐♦✱ ❜r✐♥❝❛❞❡✐r❛s ❞❡ ✐♥❢â♥❝✐❛ ❡ ❝♦♥s❡❧❤♦s ♥❛ ✈✐❞❛ ❛❞✉❧t❛✳ ➚ ♠✐♥❤❛ ♣r✐♠✐♥❤❛ ❈✐♥t❤✐❛ ✭❛❣♦r❛ ♥ã♦ tã♦ ♣❡q✉❡♥❛✮ ♣♦r t♦❞♦ ❝❛r✐♥❤♦✳

➚ ●✐s❡❧❡ ♣❡❧♦ ❛♣♦✐♦✱ ❝❛r✐♥❤♦ ❡ ♣♦r ♠❡ ❝♦♠♣r❡❡♥❞❡r ♥❛s ✐♥ú♠❡r❛s ✈❡③❡s q✉❡ ❛ t❡s❡ ♥ã♦ ♠❡ ♣❡r♠✐t✐❛ s❛✐r ♥♦s ✜♥❛✐s ❞❡ s❡♠❛♥❛✳

➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ▼ár❝✐❛ ❉✬❊❧✐❛ ❇r❛♥❝♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛✱ ❝♦♥s❡❧❤♦s✱ ❛❥✉❞❛ ❡ ♣❡❧❛ ót✐♠❛ ♦r✐✲ ❡♥t❛çã♦ q✉❡ r❡❝❡❜✐ ❛♦ ❧♦♥❣♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦✦

❆♦ ♣r♦❢❡ss♦r ❉✐♣❛❦ ❉❡② ♣♦r s✉❛ ♦r✐❡♥t❛çã♦ ❡ ❛❝♦❧❤✐❞❛ ❞✉r❛♥t❡ ♠❡✉ ❡stá❣✐♦ ♥♦ ❡①t❡r✐♦r✳ ❆♦s ❢✉♥❝✐♦♥ár✐♦s ❡ ❝♦❧❡❣❛s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❡ ❯❈❖◆◆ ❡ ❞♦ ■♥t❡r♥❛t✐♦♥❛❧ ❈❡♥t❡r ❞❡ ❯❈❖◆◆ ♣❡❧❛ ❛❥✉❞❛ ❡ ❤♦s♣✐t❛❧✐❞❛❞❡✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞♦ ■▼❊✲❯❙P ♣❡❧♦s ✈❛❧✐♦s♦s ❡♥s✐♥❛♠❡♥t♦s r❡❝❡✲ ❜✐❞♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ à ▼ô♥✐❝❛ ❙❛♥❞♦✈❛❧✱ ❏ú❧✐❛ P❛✈❛♥✱ ❈❤❛♥❣ ❈❤✐❛♥♥✱ ❆♥❛t♦❧✐ ■❛♠❜❛rts❡✈✱ ●✐❧❜❡rt♦ ❆❧✈❛r❡♥❣❛✱ ▼ár❝✐❛ ❇r❛♥❝♦✱ ❙✐❧✈✐❛ ❋❡rr❛r✐✱ ❆♥t♦♥✐♦ ❈❛r❧♦s✱ ❆❞✐❧s♦♥ ❙✐♠♦♥✐s✱ ❱✐✈✐❛♥❡ ●✐❛♠♣❛♦❧✐✱ ❙❡r✲ ❣✐♦ ❲❡❝❤s❧❡r✱ ❈❛r❧✐♥❤♦s P❡r❡✐r❛ ❡ ❏ú❧✐♦ ❙✐♥❣❡r ♣❡❧♦s ❝✉rs♦s ♠✐♥✐str❛❞♦s✳ ❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞❛ ❯❙P ♣❡❧❛ ♣r♦♥t✐❞ã♦ ❡♠ ❞✐✈❡rs♦s ♠♦♠❡♥t♦s ❡ ❡s❝❧❛r❡❝✐♠❡♥t♦s ♣r❡st❛❞♦s✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❡ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❞❛ ❯❋❈ ♣♦r ♠❡ ❢♦r♥❡✲ ❝❡r❡♠ ✉♠❛ ❜❛s❡ só❧✐❞❛ ♥❛ ♠✐♥❤❛ ❝❛♠✐♥❤❛❞❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♦s ♣r♦❢❡ss♦r❡s ❏♦ã♦ ▼❛✉rí❝✐♦ ❡ ❏♦ã♦ ❲❡❧❧✐❛♥❞r❡✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❛ ➚r❡❛ ❞❡ ▼♦❞❡❧❛❣❡♠ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈♦♥tr♦❧❡ ❞❡ ❘✐s❝♦s ❞♦ ❇❛♥❝♦ ❇r❛❞❡s❝♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ❝♦❧❡❣❛s ▲✉❝✐❛♥♦ ❱❛❧✐❡♥s✐ ❡ ❉❛♥✐❧♦ ❉❛♠❛s❝❡♥♦ ♣❡❧❛s ✈ár✐❛s t❛r❞❡s q✉❡ ♣❡r♠✐t✐r❛♠ q✉❡ ❡✉ ♠❡ ❛✉s❡♥t❛ss❡ ♣❛r❛ t❡r♠✐♥❛r ❡st❛ t❡s❡✳

❆♦s ❛♠✐❣♦s ❋❛❜✐♦ ❉❡♠❛rq✉✐✱ ▼❛r❝♦s Pr❛t❡s✱ ❏✐♥❤②♦✉❦ ❏✉♥❣✱ ❱✐❝t♦r ❍✉❣♦ ▲❛❝❤♦s ❡ ❢❛♠í❧✐❛ ♣♦r t♦r♥❛r❡♠ ♠✐♥❤❛ ❡st❛❞❛ ♥♦s ❊st❛❞♦s ❯♥✐❞♦s ❜❡♠ ♠❛✐s ❛❣r❛❞á✈❡❧✳

❆♦s ❛♠✐❣♦s ❞❡ ❣r❛❞✉❛çã♦ ✃♥✐♦ ▲♦♣❡s✱ ❈❤❛❣❛s ❆❧♠❡✐❞❛✱ ▼✐❝❤❡❧ ❍❡❧❝✐❛s✱ ❋❛❜✐❡♥♥❡ ❘♦❞r✐❣✉❡s ❡ ❊✈❡❧✐♥② ❇❛rr♦s♦✱ ♣❡❧❛s ❝♦♥✈❡rs❛s ❡ ♣❡❧♦s ✐♥ú♠❡r♦s ♠♦♠❡♥t♦s ❛❣r❛❞á✈❡✐s✳ ❆♦s ❛♠✐❣♦s ❏✉✈ê♥❝✐♦

✐✐

◆♦❜r❡✱ ❏❛❝q✉❡❧✐♥❡ ❇❛t✐st❛✱ ❈❛✐♦ ❆③❡✈❡❞♦✱ ■❡s✉s ❉✐♥✐③ ❡ ▼ár❝✐♦ ▼❡❞❡✐r♦s✱ ♣❡❧❛ ❛❝♦❧❤✐❞❛ ❞✉r❛♥t❡ ♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞❡ ♠❡str❛❞♦✱ ❡ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ ❡ ❛♣r❡♥❞✐③❛❞♦✳

❆♦s ❝♦❧❡❣❛s ❞♦ ■▼❊✲❯❙P ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ♣❡❧❛ ❛❥✉❞❛ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s ❡ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✱ ❞❡♥tr❡ ♦s q✉❛✐s ❞❡st❛❝♦✿ ●❡r♠á♥ ▼♦r❡♥♦✱ ❆rt✉r ▲❡♠♦♥t❡✱ ▼✐❝❤❡❧ ❍❡❧❝✐❛s✱ ❚✐❛❣♦ ▼❛❣❛❧❤ã❡s✱ ◆✉❜✐❛ ❊st❡❜❛♥✱ ▲✉③ ▼❛r✐♥❛✱ ❇❡ts❛❜é ●r✐♠❛❧❞❛✱ ❆♠❛♥❞❛ ●♦♠❡s✱ ❏❛❧♠❛r ❋❛r✈❛♥✱ ■✈❛♥ ❊♥r✐q✉❡③✱ ❋r❛♥❝✐s❝♦ ▼❛r❝❡❧♦✱ ❋r❡❞❡r✐❝♦ P♦❧❡t♦✱ ❊❧✐❛r❞♦ ❡ ●✉st❛✈♦✳

❆♦s ♠❡✉s t✐♦s ❈❧❡✐❞❡✱ ▼✐❧t♦♥✱ ❩✉❧❡✐❞❡ ❡ ❈❤✐❝♦ ▼❡♥❞❡s✱ ♣❡❧♦ ❝❛r✐♥❤♦✱ ❝♦♠♣r❡❡♥sã♦ ❡ ❛♠✐③❛❞❡ ❞✉r❛♥t❡ ♠✐♥❤❛ ✐♥❢â♥❝✐❛ ❡ ❛❞♦❧❡s❝ê♥❝✐❛✳

❆ t♦❞♦s ♦s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ ❋♦rt❛❧❡③❛✱ ❙ã♦ P❛✉❧♦ ❡ ❈♦♥♥❡❝t✐❝✉t✲❊❯❆ q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❡ss❡ ♠♦♠❡♥t♦✳

❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ♣❡❧❛s s✉❣❡stõ❡s ❡ ❝♦♠❡♥tár✐♦s ✈❛❧✐♦s♦s ♣❛r❛ ♦ ❛♣r✐♠♦r❛✲ ♠❡♥t♦ ❞♦ t❡①t♦✳

❘❡s✉♠♦

▼♦❞❡❧♦s ♠✉❧t✐✈❛r✐❛❞♦s ❜✐♥ár✐♦s ❝♦♠ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛s

❈♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ❝♦♠ r❡s♣♦st❛s ♠✉❧t✐✈❛r✐❛❞❛s ❛♣❛r❡❝❡♠ ❢r❡q✉❡♥t❡♠❡♥t❡ ❡♠ ♣❡sq✉✐s❛s ❡♠ q✉❡ ♦s ❞❛❞♦s sã♦ ♣r♦✈❡♥✐❡♥t❡s ❞❡ q✉❡st✐♦♥ár✐♦s✳ ❆s ♣❡sq✉✐s❛s ❞❡ ♦♣✐♥✐ã♦ sã♦ ❡①❡♠♣❧♦s ❝♦♠✉♥s ❞❡ ❡st✉❞♦s ❡♠ q✉❡ ❛s r❡s♣♦st❛s sã♦ ♠✉❧t✐✈❛r✐❛❞❛s✳ ❯♠ t✐♣♦ ♣❡sq✉✐s❛ ❞❡ ♦♣✐♥✐ã♦ q✉❡ ❣❛♥❤❛ ❣r❛♥❞❡ ❞❡st❛q✉❡ ♥♦ ❇r❛s✐❧ ❡♠ ❛♥♦s ❞❡ ❡❧❡✐çõ❡s✱ é ❛ ♣❡sq✉✐s❛ ❞❡ ✐♥t❡çã♦ ❞❡ ✈♦t♦s✳ ❊♥tr❡t❛♥t♦✱ ❛♣❡s❛r ❞❛ ♠❛✐♦r ✈✐s✐❜✐❧✐❞❛❞❡ ❞♦s ❡st✉❞♦s ❞❡ ♣r♦❣♥óst✐❝♦s ❡❧❡✐t♦r❛✐s✱ ❛ ♣❡sq✉✐s❛ ❞❡ ♦♣✐♥✐ã♦ ♣ú❜❧✐❝❛ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❞❡t❡❝t❛r t❡♥❞ê♥❝✐❛s ❡ ♣♦s✐çõ❡s ❞♦s ❞✐❢❡r❡♥t❡s s❡❣♠❡♥t♦s s♦❝✐❛✐s ❡♠ ❞✐✈❡rs♦s t❡♠❛s✱ s❡❥❛♠ ❡❧❡s✱ ♣♦❧ít✐❝♦s✱ s♦❝✐❛✐s ❡ ❣♦✈❡r♥❛♠❡♥t❛✐s✳ ◆ós ✐♥tr♦❞✉③✐♠♦s ♥❡st❛ t❡s❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ♠✉❧t✐✈❛r✐❛❞♦s ❝♦♠ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛s ♣❛r❛ ♦ ❛❥✉st❡ ❞❡ ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ❝♦♠ r❡s♣♦st❛s ♠✉❧t✐✈❛r✐❛❞❛s ❜✐♥ár✐❛s✳ ❆s ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❝♦♥s✐❞❡r❛❞❛s sã♦ ❜❛st❛♥t❡ ✢❡①í✈❡✐s ❡ r♦❜✉st❛s✱ ❝♦♥t❡♠♣❧❛♥❞♦ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ s✐♠étr✐❝❛s ❝♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✳ ❉❡✈✐❞♦ ❛ ❝♦♠♣❧❡①✐✲ ❞❛❞❡ ❞♦ ♠♦❞❡❧♦✱ ♥ós ❞✐s❝✉t✐♠♦s ❛ s✉❛ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡✳ ❆ ❛❜♦r❞❛❣❡♠ ❇❛②❡s✐❛♥❛ ❢♦✐ ❝♦♥s✐❞❡r❛❞❛ ❡ ❛❧❣✉♥s ❛❧❣♦r✐t♠♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✈✐❛ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ✭▼❈▼❈✮ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s✳ ❊st✉❞♦s ❞❡ s✐♠✉❧❛çã♦ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ❝♦♠ ❞♦✐s ♦❜❥❡t✐✈♦s❀ ✐✮ ✈❡r✐✜❝❛r ❛ q✉❛❧✐❞❛❞❡ ❞♦s ❛❧❣♦r✐t♠♦s ❞❡s❡♥✈♦❧✲ ✈✐❞♦s ❡ ✐✐✮ ✈❡r✐✜❝❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦✳ ◆♦ ✜♥❛❧ ❞❛ t❡s❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s r❡❛❧ é ❝♦♥s✐❞❡r❛❞❛ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ✐❧✉str❛r ❛s ♠❡t♦❞♦❧♦❣✐❛s ❡ té❝♥✐❝❛s ❛♣r❡s❡♥t❛❞❛s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ♠♦❞❡❧♦s ❜✐♥ár✐♦s✱ r❡❣r❡ssã♦ ♠✉❧t✐✈❛r✐❛❞❛✱ ❞✐str✐❜✉✐çõ❡s ❡❧í♣t✐❝❛s ❛ss✐♠étr✐❝❛s✳

❆❜str❛❝t

▼✉❧t✐✈❛r✐❛t❡ ❜✐♥❛r② ♠♦❞❡❧s ✇✐t❤ s❦❡✇❡❞ ❧✐♥❦ ❢✉♥❝t✐♦♥s

❉❛t❛ s❡ts ✇✐t❤ ♠✉❧t✐✈❛r✐❛t❡ r❡s♣♦♥s❡s ♦❢t❡♥ ❛♣♣❡❛r ✐♥ s✉r✈❡②s ✇❤❡r❡ t❤❡ ❞❛t❛ ❝❛♠❡ ❢r♦♠ q✉❡s✲ t✐♦♥♥❛✐r❡s✳ ❖♣✐♥✐♦♥ ♣♦❧❧✱ s♦♠❡t✐♠❡s s✐♠♣❧② r❡❢❡rr❡❞ t♦ ❛s ❛ ♣♦❧❧✱ ❛r❡ ❝♦♠♠♦♥ ❡①❛♠♣❧❡s ♦❢ st✉❞✐❡s ✐♥ ✇❤✐❝❤ t❤❡ r❡s♣♦♥s❡s ❛r❡ ♠✉❧t✐✈❛r✐❛t❡✳ ❖♥❡ t②♣❡ ♣♦❧❧ t❤❛t ❣❛✐♥ ❣r❡❛t ♣r♦♠✐♥❡♥❝❡ ✐♥ ❇r❛③✐❧ ✐♥ ❡❧❡❝t✐♦♥ ②❡❛rs✱ ✐s t❤❡ s✉r✈❡② ♦❢ ✈♦t❡ ✐♥t❡♥t✳ ❍♦✇❡✈❡r✱ ❞❡s♣✐t❡ t❤❡ ❤✐❣❤❡r ✈✐s✐❜✐❧✐t② ♦❢ ♣r♦❣♥♦st✐❝ st✉❞✐❡s ♦❢ ❡❧❡❝t✐♦♥✱ ♦♣♥✐♦♥ ♣♦❧❧s ✐s ❛ t♦♦❧ ✇✐❞❡❧② ✉s❡❞ t♦ ❞❡t❡❝t tr❡♥❞s ❛♥❞ ♣♦s✐t✐♦♥s ♦❢ ❞✐✛❡r❡♥t s♦❝✐❛❧ s❡❣♠❡♥ts ♦♥ ✈❛r✐♦✉s t♦♣✐❝s✱ ❜❡ t❤❡② ♣♦❧✐t✐❝❛❧✱ s♦❝✐❛❧ ♦r ❣♦✈❡r♥♠❡♥t❛❧✳ ❲❡ ✐♥tr♦❞✉❝❡ ✐♥ t❤✐s ✇♦r❦ ❛ ❝❧❛ss ♦❢ ♠✉❧t✐✈❛r✐❛t❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s ✇✐t❤ ❛s②♠♠❡tr✐❝ ❧✐♥❦ ❢✉♥❝t✐♦♥s t♦ ✜t ❞❛t❛ s❡ts ✇✐t❤ ♠✉❧t✐✈❛r✐❛t❡ ❜✐♥❛r② r❡s♣♦♥s❡s✳ ❚❤❡ ❧✐♥❦ ❢✉♥❝t✐♦♥s ❤❡r❡ ❝♦♥s✐❞❡r❡❞ ❛r❡ q✉✐t❡ ✢❡①✐❜❧❡ ❛♥❞ r♦❜✉st✱ ❝♦♥t❡♠♣❧❛t✐♥❣ s②♠♠❡tr✐❝❛❧ ❧✐♥❦ ❢✉♥❝t✐♦♥s ❛s s♣❡❝✐❛❧ ❝❛s❡s✳ ❉✉❡ t♦ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♠♦❞❡❧✱ ✇❡ ❞✐s❝✉ss ✐ts ✐❞❡♥✲ t✐✜❛❜✐❧✐t②✳ ❚❤❡ ❇❛②❡s✐❛♥ ❛♣♣r♦❛❝❤ ✇❛s ❝♦♥s✐❞❡r❡❞ ❛♥❞ s♦♠❡ ▼♦♥t❡ ❈❛r❧♦ ▼❛r❦♦✈ ❈❤❛✐♥ ✭▼❈▼❈✮ ❛❧❣♦r✐t❤♠s ❤❛✈❡ ❜❡❡♥ ❞❡✈❡❧♦♣❡❞✳ ❙✐♠✉❧❛t✐♦♥ st✉❞✐❡s ❤❛✈❡ ❜❡❡♥ ❞❡✈❡❧♦♣❡❞ ✇✐t❤ t✇♦ ♦❜❥❡❝t✐✈❡s✿ ✐✮ ✈❡r✐❢② t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s ❞❡✈❡❧♦♣❡❞ ❛♥❞ ✐✐✮ t♦ ✈❡r✐❢② t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ❝❤♦♦s✐♥❣ t❤❡ ❧✐♥❦ ❢✉♥❝t✐♦♥✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤✐s ✇♦r❦ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ❛ r❡❛❧ ❞❛t❛ s❡t ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ ♦r❞❡r t♦ ✐❧❧✉str❛t❡ t❤❡ ♠❡t❤♦❞♦❧♦❣✐❡s ❛♥❞ t❡❝❤♥✐q✉❡s ♣r❡s❡♥t❡❞✳

❑❡②✇♦r❞s✿ ❜✐♥❛r② ♠♦❞❡❧s✱ ♠✉❧t✐✈❛r✐❛t❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✱ s❦❡✇✲❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s✳

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s ✐①

▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐

▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✐✐✐

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✷ ▼♦❞❡❧♦s ▼✉❧t✐✈❛r✐❛❞♦s ❇✐♥ár✐♦s ✼

✷✳✶ ▼♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ❝♦♠ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❡❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ ▼♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ❝♦♠ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❡❧í♣t✐❝❛ ❛ss✐♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷✳✶ ❊①❡♠♣❧♦s ❞❡ ❞✐str✐❜✉✐çõ❡s ❛ss✐♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸ ■♥❢❡rê♥❝✐❛ ❡♠ ▼♦❞❡❧♦s ❆ss✐♠étr✐❝♦s ✷✶

✸✳✶ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✶✳✶ ❈❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞❡ ❢❛❧t❛ ❞❡ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷ ❆❧❣♦r✐t♠♦s ▼❈▼❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✸ ❉✐str✐❜✉✐çã♦ ❝♦♥❞✐❝✐♦♥❛❧ ❝♦♠♣❧❡t❛ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ r❡❣r❡ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✹ ❉✐str✐❜✉✐çã♦ ❝♦♥❞✐❝✐♦♥❛❧ ❝♦♠♣❧❡t❛ ❞❛ ♠❛tr✐③ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✹✳✶ ❊①♣❛♥sã♦ ♣❛r❛♠étr✐❝❛ ♣❛r❛ ❞❛❞♦s ❛✉♠❡♥t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✺ ❉✐str✐❜✉✐çã♦ ❝♦♥❞✐❝✐♦♥❛❧ ❝♦♠♣❧❡t❛ ❞❛ ✈❛r✐á✈❡❧ ❞❡ ♠✐st✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✹ ❊st✉❞♦s ❞❡ ❙✐♠✉❧❛çã♦ ✸✼

✹✳✶ ❊st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✷ ❊st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✺ ❆♣❧✐❝❛çã♦ ✹✾

✺✳✶ ❈♦♠♣❛r❛çã♦ ❞❡ ♠♦❞❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✷ ❉✐❛❣♥óst✐❝♦ ❞❡ ✐♥✢✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✻ ❈♦♠❡♥tár✐♦s ❋✐♥❛✐s ✻✺

❆ ❙❡❧❡çã♦ ❞❡ ▼♦❞❡❧♦s ✻✼

❆✳✶ ▼❡❞✐❞❛s ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ ♠♦❞❡❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ❆✳✷ ❆♥á❧✐s❡ ❞❡ ♦❜s❡r✈❛çõ❡s ✐♥✢✉❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵

✈✐✐✐ ❙❯▼➪❘■❖

▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s

▼❈▼❈ ▼♦♥t❡ ❈❛r❧♦ ✈✐❛ ❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ✭▼♦♥t❡ ❈❛r❧♦ ▼❛r❦♦✈ ❈❤❛✐♥✮ ▼❊◆ ▼✐st✉r❛ ❞❡ ❊s❝❛❧❛ ❞❡ ◆♦r♠❛✐s

▼❊◆❆ ▼✐st✉r❛ ❞❡ ❊s❝❛❧❛ ❞❡ ◆♦r♠❛✐s ❆ss✐♠étr✐❝❛s ▼▲● ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s

▼▼P ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ Pr♦❜✐t♦ ▼▼❚ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ t✲❙t✉❞❡♥t ▼▼▲ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ▲♦❣íst✐❝♦ ▼▼❈ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❈❛✉❝❤②

▼▼❊❉ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❊①♣♦♥ê♥❝✐❛❧ ❉✉♣❧♦ ▼▼❊P ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❊①♣♦♥ê♥❝✐❛❧ P♦tê♥❝✐❛ ▼▼❙▲ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❙❧❛s❤

▼▼❙ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❙t❛❜❧❡

▼▼P❆ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ Pr♦❜✐t♦ ❆ss✐♠étr✐❝♦ ▼▼❚❆ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ t✲❙t✉❞❡♥t ❆ss✐♠étr✐❝♦ ▼▼▲❆ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ▲♦❣íst✐❝♦ ❆ss✐♠étr✐❝♦ ▼▼❈❆ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❈❛✉❝❤② ❆ss✐♠étr✐❝♦

▼▼❊❉❆ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❊①♣♦♥ê♥❝✐❛❧ ❉✉♣❧♦ ❆ss✐♠étr✐❝♦ ▼▼❊P❆ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❊①♣♦♥ê♥❝✐❛❧ P♦tê♥❝✐❛ ❆ss✐♠étr✐❝♦ ▼▼❙▲❆ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❙❧❛s❤ ❆ss✐♠étr✐❝♦

▼▼❙❆ ▼♦❞❡❧♦ ▼✉❧t✐✈❛r✐❛❞♦ ❙t❛❜❧❡ ❆ss✐♠étr✐❝♦

P❳✲❉❆ ❊①♣❛♥sã♦ P❛r❛♠étr✐❝❛ ♣❛r❛ ❉❛❞♦s ❆✉♠❡♥t❛❞♦s

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ▼♦❞❡❧♦s ♣r♦❜✐t♦ ❡ ❝❧♦❣❧♦❣ ❛❥✉st❛❞♦s à ♣r♦♣♦rçã♦ ❞❡ ❜❡s♦✉r♦s ♠♦rt♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✹✳✶ ❇♦①♣❧♦ts ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦st❡r✐♦r✐ ❞❡ β11 ❝♦♠δ ❡n ✈❛r✐❛♥❞♦✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✹✳✷ ❇♦①♣❧♦ts ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦st❡r✐♦r✐ ❞❡ β12 ❝♦♠δ ❡n ✈❛r✐❛♥❞♦✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹✳✸ ❇♦①♣❧♦ts ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦st❡r✐♦r✐ ❞❡ β21 ❝♦♠δ ❡n ✈❛r✐❛♥❞♦✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹✳✹ ❇♦①♣❧♦ts ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦st❡r✐♦r✐ ❞❡ β22 ❝♦♠δ ❡n ✈❛r✐❛♥❞♦✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✹✳✺ ❇♦①♣❧♦ts ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦st❡r✐♦r✐ ❞❡ σ12 ❝♦♠δ ❡n ✈❛r✐❛♥❞♦✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✺✳✶ ❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦st❡r✐♦r✐ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ✐♥t❡r❡ss❡✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✷ ❊st✐♠❛t✐✈❛s ♣♦♥t✉❛✐s ❞❛s ❝♦rr❡❧❛çõ❡s ✭❛✮ ❡ ❣rá✜❝♦s ❜♦①♣❧♦t ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦s✲

t❡r✐♦r✐ ❞❛s ❝♦rr❡❧❛çõ❡s ✭❜✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✸ ❊st✐♠❛t✐✈❛s ♣♦♥t✉❛✐s ❞❛s ❝♦rr❡❧❛çõ❡s ✭❛✮ ❡ ❣rá✜❝♦s ❜♦①♣❧♦t ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦s✲

t❡r✐♦r✐ ❞❛s ❝♦rr❡❧❛çõ❡s ✭❜✮ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ♠♦❞❡❧♦ ♣r♦❜✐t♦ ❛ss✐♠étr✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✺✳✹ ❖ ✐♥✈❡rs♦ ❞♦ ❈P❖ ❛♣❧✐❝❛❞♦ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✈♦t♦s

♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦② ♣❛r❛ ♦s ♠♦❞❡❧♦sM1✱M2✱M3✱M4✱M5 ❡ M6✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✺✳✺ ❖ ✐♥✈❡rs♦ ❞♦ ❈P❖ ❛♣❧✐❝❛❞♦ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✈♦t♦s ♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦② ♣❛r❛ ♦s ♠♦❞❡❧♦sM7✱M8✱M9✱M10✱M21 ❡M40✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

✺✳✻ ❆ ❞✐✈❡r❣ê♥❝✐❛ ❑✲▲ ❛♣❧✐❝❛❞❛ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✈♦t♦s ♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦② ♣❛r❛ ♦s ♠♦❞❡❧♦sM1✱M2✱M3✱M4✱M5 ❡ M6✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✺✳✼ ❆ ❞✐✈❡r❣ê♥❝✐❛ ❑✲▲ ❛♣❧✐❝❛❞❛ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✈♦t♦s ♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦② ♣❛r❛ ♦s ♠♦❞❡❧♦sM7✱M8✱M9✱M10✱M21 ❡M40✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✺✳✽ ❆ ❝❛❧✐❜r❛çã♦ ❞❛ ❞✐✈❡r❣ê♥❝✐❛ ❑✲▲ ❛♣❧✐❝❛❞❛ ❛♦ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s s♦❜r❡ ♦ ❝♦♠♣♦r✲ t❛♠❡♥t♦ ❞❡ ✈♦t♦s ♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦② ♣❛r❛ ♦s ♠♦❞❡❧♦sM1 ❡ M11✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✷✳✶ ❋✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ ❛❧❣✉♠❛s ❞✐str✐❜✉✐çõ❡s ❡❧í♣t✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷ ❆❧❣✉♠❛s ❢✉♥çõ❡s ❞❡ ♠✐st✉r❛ ❡ s✉❛s r❡s♣❡❝t✐✈❛s ✈❛r✐á✈❡✐s ♠✐st✉r❛❞♦r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✹✳✶ ❱❛❧♦r❡s ♠é❞✐♦s ❞♦s ♣❛râ♠❡tr♦s ❡st✐♠❛❞♦s ♣❛r❛ ❝❛❞❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✷ ❱✐és❡s r❡❧❛t✐✈♦s ❞♦s ♣❛râ♠❡tr♦s ❡st✐♠❛❞♦s ♣❛r❛ ❝❛❞❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ♣❛râ♠❡tr♦ ❞❡

❢♦r♠❛ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✸ ❊rr♦s q✉❛❞rát✐❝♦s ♠é❞✐♦s r❡❧❛t✐✈♦s ✭❊◗▼R✮ ❞♦s ♣❛râ♠❡tr♦s ❡st✐♠❛❞♦s ♣❛r❛ ❝❛❞❛

❝♦♥✜❣✉r❛çã♦ ❞❡ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✹ ❱❛❧♦r❡s ♠é❞✐♦s ❞♦s ♣❛râ♠❡tr♦s ❡st✐♠❛❞♦s ♣❛r❛ ❝❛❞❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ♣❛râ♠❡tr♦ ❞❡

❢♦r♠❛ ❡ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛✱ ❝♦♥s✐❞❡r❛❞♦δ= (0,0)♥♦ ♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛çã♦✳✳ ✳ ✳ ✳ ✹✻ ✹✳✺ ❊rr♦ q✉❛❞rát✐❝♦ ♠é❞✐♦✱ ❆■❈ ❡ ❇■❈ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ♣r♦❜✐t♦ s✐♠étr✐❝♦✳ ✳ ✳ ✹✻ ✹✳✻ ❊rr♦ q✉❛❞rát✐❝♦ ♠é❞✐♦✱ ❆■❈ ❡ ❇■❈ ♣❛r❛ ❝❛❞❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛ ❡

t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛ ♣❛r❛ ♦s ❡st✉❞♦s ❞❡ s✐♠✉❧❛çã♦ ■ ❡ ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✺✳✶ ❊st❛tíst✐❝❛s ❛ ♣♦st❡r✐♦r✐ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❜✐✈❛r✐❛❞♦ ♣r♦❜✐t♦ ❛❥✉st❛❞♦ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✈♦t♦s ♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✷ ❈♦♥✜❣✉r❛çõ❡s ❞♦ ✈❡t♦r❡s δk = (δk1, δk2)′ ♣❛r❛ ❝❛❞❛ ✉♠ ♠♦❞❡❧♦s ❝♦♥❝♦rr❡♥t❡s Mk✱

k= 1, . . . ,40✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✸ ❈r✐tér✐♦s ❞❡ s❡❧❡çã♦ ❆■❈✱ ❇■❈✱ _D¯_{✱ pD✱ ❉■❈✱ ▲P▼▲ ❡ Ps❇❋ ♣❛r❛ ♦s ✹✵ ♠♦❞❡❧♦s}

❝♦♥❝♦rr❡♥t❡s ❛♣❧✐❝❛❞♦s ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✈♦t♦s ♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✹ ❱❛❧♦r❡s ❞❛s ♠❡❞✐❞❛s ❞❡ ✐♥✢✉ê♥❝✐❛ ❞❛s ♦❜s❡r✈❛çõ❡s ✐♥✢✉❡♥t❡s ♣❛r❛ ♦ ♠♦❞❡❧♦ M1

❝♦♠♣❛r❛❞❛ ❝♦♠ ❛s ✈❛❧♦r❡s ♦❜t✐❞♦s ♣❡❧♦ ♠♦❞❡❧♦ M10 ✭c ✐♥❞✐❝❛ q✉❡ ❛ ♦❜s❡r✈❛çã♦ é

✐♥✢✉❡♥t❡ s❡❣✉♥❞♦ ❛ ❈P❖ ❡ k ✐♥❞✐❝❛ q✉❡ ❛ ♦❜s❡r✈❛çã♦ é ✐♥✢✉❡♥t❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐✈❡r❣ê♥❝✐❛ ❑✲▲✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✺✳✺ ❖❜s❡r✈❛çõ❡s ✐♥✢✉❡♥t❡s ♣❛r❛ ♦ ♠♦❞❡❧♦M1❡ s✉❛s r❡s♣❡❝t✐✈❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❛❥✉st❛❞❛s

♣❛r❛ ♦s ♠♦❞❡❧♦sM1 ❡M10✭c✐♥❞✐❝❛ q✉❡ ❛ ♦❜s❡r✈❛çã♦ é ✐♥✢✉❡♥t❡ s❡❣✉♥❞♦ ❛ ❈P❖−1

❡ k✐♥❞✐❝❛ q✉❡ ❛ ♦❜s❡r✈❛çã♦ é ✐♥✢✉❡♥t❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐✈❡r❣ê♥❝✐❛ ❑✲▲✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✺✳✻ ❊st❛tíst✐❝❛s ❛ ♣♦st❡r✐♦r✐ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❜✐✈❛r✐❛❞♦ ♣r♦❜✐t♦ ❛ss✐♠étr✐❝♦ ❝♦♠ ✈❡t♦r ❞❡

♣❛râ♠❡tr♦s ❞❡ ❢♦r♠❛δ= (0,0; 0,2)′ _{❛❥✉st❛❞♦ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s s♦❜r❡ ♦ ❝♦♠♣♦r✲}

t❛♠❡♥t♦ ❞❡ ✈♦t♦s ♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❈♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ❝♦♠ r❡s♣♦st❛s ♠✉❧t✐✈❛r✐❛❞❛s ❛♣❛r❡❝❡♠ ❢r❡q✉❡♥t❡♠❡♥t❡ ❡♠ ♣❡sq✉✐s❛s ❡♠ q✉❡ ♦s ❞❛❞♦s sã♦ ♣r♦✈❡♥✐❡♥t❡s ❞❡ q✉❡st✐♦♥ár✐♦s✳ ❆s ♣❡sq✉✐s❛s ❞❡ ♦♣✐♥✐ã♦ sã♦ ❡①❡♠♣❧♦s ❝♦♠✉♥s ❞❡ ❡st✉❞♦s ❡♠ q✉❡ ❛s r❡s♣♦st❛s sã♦ ♠✉❧t✐✈❛r✐❛❞❛s✳ ❯♠ t✐♣♦ ♣❡sq✉✐s❛ ❞❡ ♦♣✐♥✐ã♦ q✉❡ ❣❛♥❤❛ ❣r❛♥❞❡ ❞❡st❛q✉❡ ♥♦ ❇r❛s✐❧ ❡♠ ❛♥♦s ❞❡ ❡❧❡✐çõ❡s✱ é ❛ ♣❡sq✉✐s❛ ❞❡ ✐♥t❡♥çã♦ ❞❡ ✈♦t♦s✳ ❊♥tr❡t❛♥t♦✱ ❛♣❡s❛r ❞❛ ♠❛✐♦r ✈✐s✐❜✐❧✐❞❛❞❡ ❞♦s ❡st✉❞♦s ❞❡ ♣r♦❣♥óst✐❝♦s ❡❧❡✐t♦r❛✐s✱ ❛ ♣❡sq✉✐s❛ ❞❡ ♦♣✐♥✐ã♦ ♣ú❜❧✐❝❛ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❞❡t❡❝t❛r t❡♥❞ê♥❝✐❛s ❡ ♣♦s✐çõ❡s ❞♦s ❞✐❢❡r❡♥t❡s s❡❣♠❡♥t♦s s♦❝✐❛✐s ❡♠ ❞✐✈❡rs♦s t❡♠❛s✱ s❡❥❛♠ ❡❧❡s✱ ♣♦❧ít✐❝♦s✱ s♦❝✐❛✐s ❡ ❣♦✈❡r♥❛♠❡♥t❛✐s✳ ●r❡❡♥❡ ✭✶✾✾✼✮ ❛♣r❡s❡♥t❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s r❡❢❡r❡♥t❡ ❛ ✉♠❛ ♣❡sq✉✐s❛ s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ✈♦t♦s ❞❡ r❡s✐❞❡♥t❡s ❞❛ ❝✐❞❛❞❡ ❞❡ ❚r♦②✱ ♥♦ ❡st❛❞♦ ❞❡ ▼✐❝❤✐❣❛♥✱ ❊st❛❞♦s ❯♥✐❞♦s ❞❛ ❆♠ér✐❝❛✳ ◆❡st❡ ❡st✉❞♦✱ ✾✺ ♠♦r❛❞♦r❡s ❢♦r❛♠ q✉❡st✐♦♥❛❞♦s ❛ r❡s♣♦♥❞❡r ❞✉❛s ♣❡r❣✉♥t❛s✿ ✶a_{✮ s❡ t❡♠ ♦✉ ♥ã♦✱ ♣❡❧♦ ♠❡♥♦s ✉♠ ✜❧❤♦ ♥❛ ❡s❝♦❧❛ ♣ú❜❧✐❝❛ ❡❀ ✷}a_{✮ s❡ ✈♦t❛} ❛ ❢❛✈♦r ♦✉ ❝♦♥tr❛ ❛ ❝r✐❛çã♦ ❞❡ ✉♠ ❢✉♥❞♦ ♦rç❛♠❡♥tár✐♦ ♣❛r❛ ❛ ❡s❝♦❧❛✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❛ ♣❡sq✉✐s❛ ❢♦✐ ❡st✉❞❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♥tr❡ ❛s ❞✉❛s ✈❛r✐á✈❡✐s r❡s♣♦st❛ ❜✐♥ár✐❛s ❡ ❛❧❣✉♠❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ✭♣r❡❞✐t♦r❛s✮✳ ❆s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❝♦♥s✐❞❡r❛❞❛s ❢♦r❛♠ ❛ r❡♥❞❛ ❢❛♠✐❧✐❛r ❛♥✉❛❧✱ ♦ ✈❛❧♦r ♣❛❣♦ ♣♦r ❛♥♦ ❡♠ ✐♠♣♦st♦s s♦❜r❡ ✐♠ó✈❡✐s ♣♦r ❝❛❞❛ ❢❛♠í❧✐❛ ❡ ♦ ♥ú♠❡r♦ ❞❡ ❛♥♦s q✉❡ ❛ ❢❛♠í❧✐❛ r❡s✐❞❡ ♥❛ ❝✐❞❛❞❡ ❞❡ ❚r♦②✳ ➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡✱ ✉♠❛ ✈❡③ q✉❡ ♦ ♠❡s♠♦ ✐♥❞✐✈í❞✉♦ r❡s♣♦♥❞❡ t♦❞♦s ♦s ✐t❡♥s ❞♦ q✉❡st✐♦♥ár✐♦✱ é ♣r♦✈á✈❡❧ q✉❡ s✉❛s r❡s♣♦st❛s s❡❥❛♠ ❝♦rr❡❧❛❝✐♦♥❛❞❛s✳ ■st♦ é✱ ♦ ❢❛t♦ ❞❡ ✉♠ ✐♥❞✐✈í❞✉♦ t❡r ✜❧❤♦s ♥❛ ❡s❝♦❧❛ ❞❡✈❡ s❡r ❝♦rr❡❧❛❝✐♦♥❛❞♦ ❝♦♠ s✉❛ ❞✐s♣♦s✐çã♦ ❛ ✈♦t❛r ❛ ❢❛✈♦r ♦✉ ❝♦♥tr❛ ❛ ❝r✐❛çã♦ ❞❡ ✉♠ ❢✉♥❞♦ ♦rç❛♠❡♥tár✐♦ ♣❛r❛ ❛ ❡s❝♦❧❛✳ ❉❡ ❛♥t❡♠ã♦✱ ♣♦❞❡✲s❡ ❡s♣❡r❛r q✉❡ ♣❡ss♦❛s ❝♦♠ ✜❧❤♦s ♥❛ ❡s❝♦❧❛ s❡❥❛♠ ♠❛✐s ♣r♦♣❡♥s❛s ❛ ✈♦t❛r ♥❛ ❝r✐❛çã♦ ❞❡ ✉♠ ❢✉♥❞♦ ♦rç❛♠❡♥tár✐♦ ♣❛r❛ ❛ ❡s❝♦❧❛ ❞♦ q✉❡ ♣❡ss♦❛s q✉❡ ♥ã♦ t❡♠ ✜❧❤♦s ♥❛ ❡s❝♦❧❛✳

❖s ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s ✭▼▲●✮ sã♦ ❢❡rr❛♠❡♥t❛s ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞❛s q✉❛♥❞♦ ❡①✐st❡ ♦ ✐♥t❡r❡ss❡ ❡♠ ❛♥❛❧✐s❛r ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s r❡s♣♦st❛s ❡ ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ✭❝♦✈❛✲ r✐á✈❡✐s✮✱ ❛s q✉❛✐s ♣♦❞❡♠ s❡r ❝♦♥tí♥✉❛s ♦✉ ❞✐s❝r❡t❛s ✭♥♦♠✐♥❛✐s ♦✉ ♦r❞✐♥❛✐s✮✳ ❖ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧♦❣íst✐❝❛ é ♦ ▼▲● ✉t✐❧✐③❛❞♦ ❝♦♠ ♠❛✐♦r ❢r❡q✉ê♥❝✐❛ ♥♦ ❛❥✉st❡ ❞❡ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❝♦♠ r❡s♣♦st❛s ❜✐♥ár✐❛s✳ ❯♠ ❞♦s ♠♦t✐✈♦s ❞❡st❛ ♣♦♣✉❧❛r✐❞❛❞❡ é ❛ ❢á❝✐❧ ✐♥t❡r♣r❡t❛çã♦ ❞❡ s❡✉s ♣❛râ♠❡tr♦s r❡❣r❡ss♦✲ r❡s ❛tr❛✈és ❞❡ r❛③õ❡s ❞❡ ❝❤❛♥❝❡s✳ ❖✉tr♦s ♠♦❞❡❧♦s✱ t❛♠❜é♠ ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞♦s✱ sã♦ ♦s ♠♦❞❡❧♦s ♣r♦❜✐t♦ ❡ ❝♦♠♣❧❡♠❡♥t♦ ❧♦❣✲❧♦❣✳ ◆♦ ❡♥t❛♥t♦✱ ❡♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❡st✉❞♦ s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ✈♦t♦s ❞❡ r❡s✐❞❡♥t❡s ❞❛ ❝✐❞❛❞❡ ❞❡ ❚r♦② ❛♣r❡s❡♥t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ✐♥❞❡♣❡♥❞ê♥✲ ❝✐❛ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s r❡s♣♦st❛s ♥ã♦ é ✉♠❛ s✉♣♦s✐çã♦ r❛③♦á✈❡❧✳ ❊st❛ ❢❛❧t❛ ❞❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s r❡s♣♦st❛ é ❢❛❝✐❧♠❡♥t❡ ♥♦t❛❞♦ ❡♠ ❡st✉❞♦s ❧♦♥❣✐t✉❞✐♥❛✐s✱ ❡♠ q✉❡ ❛s ♠ú❧t✐♣❧❛s ♠❡❞✐❞❛s sã♦ ❝♦❧❡t❛❞❛s ❞♦ ♠❡s♠♦ ✐♥❞✐✈í❞✉♦✳ ❯♠❛ té❝♥✐❝❛ ❜❛st❛♥t❡ ❣❡r❛❧ ♣❛r❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❝♦rr❡❧❛❝✐♦♥❛❞♦s é

✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✵

❛ ❛❜♦r❞❛❣❡♠ ❛tr❛✈és ❞❛s ❡q✉❛çõ❡s ❞❡ ❡st✐♠❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ✭❊❊●✮ ✐♥tr♦❞✉③✐❞❛ ♣♦r▲✐❛♥❣ ❡ ❩❡❣❡r

✭✶✾✽✻✮ ❡ ❩❡❣❡r ❡ ▲✐❛♥❣✭✶✾✽✻✮✳ ❯♠❛ ❜♦❛ r❡✈✐sã♦ ❞❡ ✈ár✐❛s ❡str❛té❣✐❛s ❞❡ ♠♦❞❡❧❛❣❡♠ ✉t✐❧✐③❛♥❞♦ r❡✲

❣r❡ssã♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞❛ ❝♦♠ ❛ ❛❜♦r❞❛❣❡♠ ♣♦r ❡q✉❛çõ❡s ❞❡ ❡st✐♠❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ❛♥❛❧✐s❛r ❞❛❞♦s ❜✐♥ár✐♦s ❝♦rr❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❝♦✈❛r✐á✈❡✐s ❛ss♦❝✐❛❞❛s ❛ ❝❛❞❛ r❡s♣♦st❛ é ❛♣r❡s❡♥t❛❞❛ ❡♠Pr❡♥t✐❝❡

✭✶✾✽✽✮✳ ❆ ♣r✐♥❝✐♣❛❧ ❞❡s✈❛♥t❛❣❡♠ ❞❛ ❛❜♦r❞❛❣❡♠ ✉s❛♥❞♦ ❡q✉❛çõ❡s ❞❡ ❡st✐♠❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛s é q✉❡ ❡❧❛ é ❜❛s❡❛❞❛ ❡♠ t❡♦r✐❛ ❛ss✐♥tót✐❝❛✱ ❡ ♣♦rt❛♥t♦✱ ✉♠ t❛♠❛♥❤♦ ❣r❛♥❞❡ ❞❡ ❛♠♦str❛ é ♥❡❝❡ssár✐♦✳

◆♦ ❝♦♥t❡①t♦ ❇❛②❡s✐❛♥♦✱ ❞✐❢❡r❡♥t❡s ❛❜♦r❞❛❣❡♥s ♣❛r❛ ♠♦❞❡❧♦s ❝♦♠ r❡s♣♦st❛s ❜✐♥ár✐❛s sã♦ ❛♣r❡✲ s❡♥t❛❞❛s ❡♠ ❩❡❧❧♥❡r ❡ ❘♦ss✐ ✭✶✾✽✹✮ ❡ ❆❧❜❡rt ❡ ❈❤✐❜ ✭✶✾✾✸✮✳ ❆❧❜❡rt ❡ ❈❤✐❜ ✭✶✾✾✸✮ ♣r♦♣õ❡♠ ♦ ✉s♦ ❞❡ ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s ❡ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ●✐❜❜s ✭●❡❧❢❛♥❞ ❡ ❙♠✐t❤✱ ✶✾✾✵✮ ♥❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡✲ tr♦s ❞❡ ♠♦❞❡❧♦s ❝♦♠ r❡♣♦st❛ ❜✐♥ár✐❛✱ ♦ q✉❡ t♦r♥❛ ♦ ♠♦❞❡❧♦ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ♠❛✐s ❛tr❛t✐✈♦✳ ❊st❛ ♣r♦♣♦st❛ ♣❡r♠✐t❡ ❞❡✜♥✐r r❡sí❞✉♦s ❧❛t❡♥t❡s ❝♦♥tí♥✉♦s✱ ♦ q✉❡ ♥ã♦ ♦❝♦rr❡ ♥❛ ❛❜♦r❞❛❣❡♠ ❞❡ ❡s✲ t✐♠❛çã♦ ✈✐❛ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ ❋❛r✐❛s ❡ ❇r❛♥❝♦ ✭✷✵✶✷✮ s✉❣❡r❡♠ ♦ ✉s♦ ❞❡ ❞✐✈❡rs♦s t✐♣♦s ❞❡ r❡sí❞✉♦s ❧❛t❡♥t❡s ❜❛②❡s✐❛♥♦s ♣❛r❛ ♦s ♠♦❞❡❧♦s ♣r♦❜✐t♦✱ ❧♦❣íst✐❝♦ ❡ ♣r♦❜✐t♦✲❛ss✐♠étr✐❝♦✳ ❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♦❜t❡r ❛❧❣♦r✐t♠♦s ♠❛✐s ❡✜❝✐❡♥t❡s✱ ❈♦✇❧❡s ✭✶✾✾✻✮ ❡◆❛♥❞r❛♠ ❡ ❈❤❡♥ ✭✶✾✾✻✮ ♣r♦♣õ❡♠ ❛❧❣♦r✐t♠♦s ❛❧t❡r♥❛t✐✈♦s ♣❛r❛ ❛❝❡❧❡r❛r ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ●✐❜❜s ♣r♦♣♦st♦ ❆❧❜❡rt ❡ ❈❤✐❜ ✭✶✾✾✸✮✳ ❘❡❝❡♥t❡♠❡♥t❡✱ ❍♦❧♠❡s ❡ ❍❡❧❞ ✭✷✵✵✻✮ ♣r♦♣õ❡♠ ❛❧t❡r♥❛t✐✈❛s ♠❛✐s ❡✜❝✐❡♥t❡s ❞❡ q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ●✐❜❜s ❝♦♥✈❡♥❝✐♦♥❛❧ ♥♦ ❝♦♥t❡①t♦ ❞❡ r❡❣r❡ssã♦ ❜✐♥ár✐❛ ❝♦♠ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ♣r♦❜✐t♦ ❡ ❧♦❣✐t♦✳ ❖s ❛✉t♦r❡s ✉t✐❧✐③❛♠ ♦ ❛♠♦str❛❞♦r ❞❡ ●✐❜❜s ❡♠ ❜❧♦❝♦s ✭▲✐✉✱✶✾✾✹✮ ♣❛r❛ t♦r♥❛r ♦ ❛❧❣♦r✐t♠♦ ♠❛✐s ❡✜❝✐✲ ❡♥t❡✱ r❡❞✉③✐♥❞♦ ❛ ❛✉t♦❝♦rr❡❧❛çã♦ s❡r✐❛❧ ❞❛ ❛♠♦str❛ s✐♠✉❧❛❞❛✳ ❆ ❝♦♥str✉çã♦ ❞❡st❡s ♥♦✈♦s ❛❧❣♦r✐t♠♦s ❞❡♣❡♥❞❡ ❞❛ ♦❜t❡♥çã♦ ❞❡ ❢♦r♠❛s ❝♦♥❤❡❝✐❞❛s ♣❛r❛ ❛s ❞✐str✐❜✉✐çõ❡s ♠❛r❣✐♥❛✐s ❞❡ ❛❧❣✉♥s ♣❛râ♠❡tr♦s ❛♦ ✐♥✈és ❞❛s ❞✐str✐❜✉✐çõ❡s ❝♦♥❞✐❝✐♦♥❛✐s ❝♦♠♣❧❡t❛s✳ ❋❛r✐❛s ❡ ❇r❛♥❝♦ ✭✷✵✶✶✮ ♣r♦♣õ❡♠ ✉♠ ❛❧❣♦r✐t♠♦ s✐♠✐❧❛r ♣❛r❛ tr❛❜❛❧❤❛r ❝♦♠ ♦ ♠♦❞❡❧♦ ♣r♦❜✐t♦✲❛ss✐♠étr✐❝♦ ♣r♦♣♦st♦ ♣♦r❈❤❡♥✱ ❉❡② ❡ ❙❤❛♦✭✶✾✾✾✮✱ ❡ ♠♦str❛♠ q✉❡ ♦ ❛❧❣♦r✐t♠♦ ♣r♦♣♦st♦ é ♠❛✐s ❡✜❝✐❡♥t❡ q✉❡ ♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ✭s❡♠ ❜❧♦❝♦s✮✳ ◆♦ ❡♥t❛♥t♦✱

❋❛r✐❛s ❡ ❇r❛♥❝♦ ✭✷✵✶✶✮ ♥ã♦ ✐♥❝❧✉ír❛♠ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s r❡s♣♦st❛s✳

❈❤✐❜ ❡ ●r❡❡♥❜❡r❣ ✭✶✾✾✽✮ ♣r♦♣✉s❡r❛♠ ✉♠ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ♣r♦❜✐t♦ ✭▼▼P✮ ♣❛r❛ ❛♥❛❧✐s❛r

❞❛❞♦s ❝♦♠ r❡s♣♦st❛s ❜✐♥ár✐❛s ❝♦rr❡❧❛❝✐♦♥❛❞❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❛❞♦s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ❡st✉❞♦s ❧♦♥✲ ❣✐t✉❞✐♥❛✐s✳ ❖s ❛✉t♦r❡s ❞✐s❝✉t❡♠ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡st❡ ♠♦❞❡❧♦ ✈✐❛ ✐♥❢❡rê♥❝✐❛ ❜❛②❡s✐❛♥❛ ❡ ♦ ♠é✲ t♦❞♦ ❞❛ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ ❈❤❡♥ ❡ ❉❡② ✭✶✾✾✽✮ ❡st❡♥❞❡♠ ♦ ♠♦❞❡❧♦ ▼▼P ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ✉♠❛ r✐❝❛ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s ♦❜t✐❞❛s ❛tr❛✈és ❞❡ ♠✐st✉r❛ ♥♦ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ❞❛ ❞✐s✲ tr✐❜✉✐çã♦ ♥♦r♠❛❧ ✭❆♥❞r❡✇s ❡ ▼❛❧❧♦✇s✱ ✶✾✼✹✮✱ ❛ q✉❛❧ ❝♦♥t❡♠♣❧❛ ♦s ♠♦❞❡❧♦s ♠✉❧t✐✈❛r✐❛❞♦s ♣r♦❜✐t♦ ✭▼▼P✮✱t−❙t✉❞❡♥t ✭▼▼❚✮ ❡ ❧♦❣íst✐❝♦ ✭▼▼▲✮✳❈❤❡♥ ❡ ❉❡②✭✷✵✵✸✮ ❞✐s❝✉t❡♠ ♦ ♣r♦❜❧❡♠❛ ❞❡ s❡❧❡çã♦

❞❡ ✈❛r✐á✈❡✐s ♥❡st❛ ❛❜♦r❞❛❣❡♠✳ ❖✉tr❛s ❛❜♦r❞❛❣❡♥s ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ♠✉❧t✐✈❛r✐❛✲ ❞♦s ❝♦♠ r❡s♣♦st❛ ❜✐♥ár✐❛ ❡ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❧♦❣íst✐❝❛ sã♦ ❛♣r❡s❡♥t❛❞❛s ♣♦r ❈❛r❡② ❡t ❛❧✳ ✭✶✾✾✸✮✱

❧❡ ❈❡ss✐❡ ❡ ✈❛♥ ❍♦✉✇❡❧✐♥❣❡♥ ✭✶✾✾✹✮ ❡ ●❧♦♥❡❦ ❡ ▼❝❈✉❧❧❛❣❤✭✶✾✼✹✮✳ ◆♦ ❡♥t❛♥t♦✱ ❛♣❡♥❛s ♠♦❞❡❧♦s s✐✲

♠étr✐❝♦s ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞♦s ❡♠ t♦❞❛s ❡st❛s ❛❜♦r❞❛❣❡♥s✳ ❖✉ s❡❥❛✱ ♠♦❞❡❧♦s ❡♠ q✉❡ ❛ r❡❧❛çã♦ ✭❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦✮ ❡♥tr❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ✐♥t❡r❡ss❡ ❡ ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s é s✐♠étr✐❝❛✳ ❊st❛s ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ s✐♠étr✐❝❛s ❝♦♠✉♠❡♥t❡ ✉t✐❧✐③❛❞❛s✱ t❛✐s ❝♦♠♦ ♦ ♣r♦❜✐t♦ ❡ ❧♦❣✐t♦✱ ♥❡♠ s❡♠♣r❡ ❢♦r♥❡❝❡♠ ♦ ♠❡❧❤♦r ❛❥✉st❡✳

✶✳✵ ✸

♥❡❝❡ssár✐♦ ❢❛③❡r ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ♣❛r❛ ♦ ✉s♦ ❞♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡♠❡♥t♦ ❧♦❣❧♦❣✳ ❈♦♥t✉❞♦✱ ❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❝♦♠♣❧❡♠❡♥t♦ ❧♦❣❧♦❣ t❡♠ ✉♠❛ ❛ss✐♠❡tr✐❛ ✜①❛✱ ♦ q✉❡ ♣r❡❥✉❞✐❝❛ ♦ ❛❥✉st❡ q✉❛♥❞♦ ❛ ❛ss✐♠❡tr✐❛ ❞❛ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ é ♠✉✐t♦ ❡❧❡✈❛❞❛✳ ◆❡st❡ ❝♦♥t❡①t♦✱❈③❛❞♦ ❡ ❙❛♥t♥❡r

✭✶✾✾✷✮✱ ❈❤❡♥ ✭✷✵✵✹✮✱ ❇á③❛♥✱ ❇r❛♥❝♦ ❡ ❇♦❧❢❛r✐♥❡ ✭✷✵✵✺✮ ❡❲❛♥❣ ❡ ❉❡② ✭✷✵✶✵✮ ❛♣r❡s❡♥t❛♠ ❞✐✈❡rs❛s

r❛③õ❡s ❡♠ ❢❛✈♦r ❞♦ ✉s♦ ❞❡ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ q✉❛♥❞♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s✉❝❡ss♦ ❛♣r♦①✐♠❛✲s❡ ❞❡ ③❡r♦ ♥✉♠❛ t❛①❛ ❞✐❢❡r❡♥t❡ ❞❛ q✉❛❧ ❛♣r♦①✐♠❛✲s❡ ❞❡ ✉♠✳ ❈❤❡♥ ✭✷✵✵✹✮ ❡♠♣r❡❡♥❞❡✉ ✉♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ♣❛r❛ ✐♥✈❡st✐❣❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ♥❛ ♣r❡❞✐çã♦ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❡♠ ♠♦❞❡❧♦s ❜✐♥ár✐♦s ✉♥✐✈❛r✐❛❞♦s✳ ❖ ❛✉t♦r ❝♦♥s✐❞❡r♦✉ ❞♦✐s ❡sq✉❡♠❛s ❞❡ s✐♠✉❧❛çã♦✿ ✭✐✮ ♦s ❞❛❞♦s sã♦ ❣❡r❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♠♦❞❡❧♦ ♣r♦❜✐t♦❀ ❡ ✭✐✐✮ ♦s ❞❛❞♦s sã♦ ❣❡r❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡♠❡♥t♦ ❧♦❣❧♦❣ ✭❝❧♦❣❧♦❣✮✳ ❊♠ ❛♠❜❛s ❛s s✐t✉❛çõ❡s✱ ❢♦r❛♠ ❛❥✉st❛❞♦s ♦s ♠♦❞❡❧♦s ♣r♦❜✐t♦✱ ❧♦❣✐t♦ ❡ ❝♦♠♣❧❡♠❡♥t♦ ❧♦❣❧♦❣✳ ❖ ❛✉t♦r ♦❜s❡r✈♦✉ q✉❡✱ q✉❛♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ✈❡r❞❛❞❡✐r❛ é ❛ ♣r♦❜✐t♦✱ q✉❛s❡ ♥ã♦ ❤á ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♠♦❞❡❧♦ ♣r♦❜✐t♦ ❡ ❧♦❣✐t♦✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ♠♦❞❡❧♦ ❝♦♠♣❧❡♠❡♥t♦ ❧♦❣❧♦❣ ♠♦str♦✉✲s❡ ✐♥❛❞❡q✉❛❞♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ✈❡r❞❛❞❡✐r❛ é ❛ ❝♦♠♣❧❡♠❡♥t♦ ❧♦❣❧♦❣✱ ♦s ♠♦❞❡❧♦s s✐♠étr✐❝♦s ❢♦r❛♠ ❝❧❛r❛♠❡♥t❡ ✐♥❛❞❡q✉❛❞♦s✳ ❖ ❛✉t♦r ❝♦♥❝❧✉✐✉ ♥❡st❡ ❡st✉❞♦ ❡♠♣ír✐❝♦ q✉❡ ❛ ❡s❝♦❧❤❛ ❞❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❡✱ ❝❛s♦ s❡❥❛ ♠❛❧ ❡s♣❡❝✐✜❝❛❞❛✱ ♣♦❞❡ ❢♦r♥❡❝❡r ♣r❡❞✐çõ❡s ♣♦❜r❡s✳

P❛r❛ ✐❧✉str❛r ♦ ❢❛t♦ ❞❡ q✉❡ ♠♦❞❡❧♦s ❛ss✐♠étr✐❝♦s ♣♦❞❡♠ ❛♣r❡s❡♥t❛r ♠❡❧❤♦r ❛❥✉st❡ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❞❡ q✉❡ ♠♦❞❡❧♦s s✐♠étr✐❝♦s✱ ♥ós ✉t✐❧✐③❛r❡♠♦s ✉♠ ❡st✉❞♦ ❜❛st❛♥t❡ s✐♠♣❧❡s ❡ ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛ q✉❡ ♠♦str❛ ❝❧❛r❛♠❡♥t❡ q✉❡ ✉♠ ♠♦❞❡❧♦ ❛ss✐♠étr✐❝♦ é ♣r❡❢❡rí✈❡❧✳ ❊st❡ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s é ❛♣r❡s❡♥t❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❡♠❇❧✐ss ✭✶✾✸✺✮ ❡ r❡❢❡r❡✲s❡ ❛♦ ♥ú♠❡r♦ ❞❡ ✐♥s❡t♦s ❛❞✉❧t♦s ♠♦rt♦s ❛♣ós ✺ ❤♦r❛s ❞❡ ❡①♣♦s✐çã♦ ❛♦ ❣ás ❝❛r❜ô♥✐❝♦ ♣❛r❛ ✈ár✐❛s ❝♦♥❝❡♥tr❛çõ❡s ❞❡st❡ ❣ás✳ ❊st❡ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❢♦✐ ❡st✉❞❛❞♦ ♣♦r ✈ár✐♦s ❛✉t♦r❡s q✉❡ ♣r♦♣✉s❡r❛♠ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛s ❡♠ r❡❣r❡ssã♦ ❜✐♥ár✐❛✱ ❞❡♥tr❡ ❡❧❡s✱ ❝✐t❛♠♦sPr❡♥t✐❝❡✭✶✾✼✻✮✱❙t✉❦❡❧ ✭✶✾✽✽✮ ❡ ❇á③❛♥✱ ❇r❛♥❝♦ ❡ ❇♦❧❢❛r✐♥❡✭✷✵✵✺✮✳ ❚♦❞♦s ❡st❡s ❛✉t♦r❡s ❝♦♥❝❧✉ír❛♠ q✉❡ ♠♦❞❡❧♦s ❛ss✐♠étr✐❝♦s sã♦ ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡s q✉❡ ♠♦❞❡❧♦s s✐♠étr✐❝♦s ♣❛r❛ ♦ ❛❥✉st❡ ♥❡st❡ ❝❛s♦✳ ❆s ♣r♦♣♦rçõ❡s ❞❡ ✐♥s❡t♦s ♠♦rt♦s ❡ ❛s ❝✉r✈❛s ❞❛s ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ♣r♦❜✐t♦ ❡ ❝♦♠♣❧❡♠❡♥t♦ ❧♦❣❧♦❣ ✭❝❧♦❣❧♦❣✮ ❛❥✉st❛❞❛s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛ ❋✐❣✉r❛ ✶✳✶✳ ◆♦t❡✱ ♥❛ ❋✐❣✉r❛ ✶✳✶✱ q✉❡ ♦ ♠♦❞❡❧♦ ❛ss✐♠étr✐❝♦ ✉t✐❧✐③❛❞♦✱ ♦ ♠♦❞❡❧♦ ❝✲❧♦❣❧♦❣✱ ❛♣❛r❡♥t❛ ❢♦r♥❡❝❡r ✉♠ ♠❡❧❤♦r ❛❥✉st❡ q✉❡ ♦ s✐♠étr✐❝♦✱ ♦ ♠♦❞❡❧♦ ♣r♦❜✐t♦✳

❋✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛s sã♦ ♠❛✐s ✢❡①í✈❡✐s q✉❡ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ s✐♠étr✐❝❛s✱ ♦ q✉❡ ❛s t♦r♥❛♠ ❜❛st❛♥t❡ út❡✐s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ q✉❛♥❞♦ ❡st❛s ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ t❡♠ ❝♦♠♦ ❝❛s♦s ❡s♣❡❝✐✲ ❛✐s ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ s✐♠étr✐❝❛s ❝♦♥❤❡❝✐❞❛s✳ ❈❤❡♥✱ ❉❡② ❡ ❙❤❛♦ ✭✶✾✾✾✮ ♣r♦♣✉s❡r❛♠ ✉♠❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛ ♣❛r❛ ♠♦❞❡❧♦s ❝♦♠ r❡s♣♦st❛s ❜✐♥ár✐❛s q✉❡ t❡♠ ❝♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❛s ❢✉♥çã♦ ♣r♦❜✐t♦✱ t✲❙t✉❞❡♥t✳ ❇á③❛♥✱ ❇r❛♥❝♦ ❡ ❇♦❧❢❛r✐♥❡ ✭✷✵✵✺✮ ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♣r♦❜✐t♦ ❛ss✐♠étr✐❝♦ ♣r♦♣♦st♦ ♣♦r ❈❤❡♥✱ ❉❡② ❡ ❙❤❛♦ ✭✶✾✾✾✮✳ ❆ r❡❧❛çã♦ ❡♥tr❡ ❡st❡s ❞♦✐s ♠♦❞❡❧♦s ♣r♦❜✐t♦ ❛ss✐♠étr✐❝♦ é ❞✐s❝✉t✐❞♦ ❡♠ ❞❡t❛❧❤❡s ♣♦r ❇á③❛♥✱ ❇♦❧❢❛♥✐♥❡ ❡ ❇r❛♥❝♦ ✭✷✵✶✵✮✳ ❊st❡s ❛✉t♦r❡s ♠❡♥❝✐♦♥❛♠ q✉❡ ♠♦❞❡❧♦s ❝♦♠ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❛ss✐♠étr✐❝❛s sã♦ ♠❛✐s ❛tr❛t✐✈❛s ❡♠ ❞✐✈❡rs♦s ❛s♣❡❝t♦s✳

✹ ■◆❚❘❖❉❯➬➹❖ ✶✳✶

1.70 1.75 1.80 1.85 1.90

0.0

0.2

0.4

0.6

0.8

1.0

Dose de carbono gasoso (log10CS2)

Proporção de besouros mor

tos

Modelo probito Modelo cloglog

❋✐❣✉r❛ ✶✳✶✿ ▼♦❞❡❧♦s ♣r♦❜✐t♦ ❡ ❝❧♦❣❧♦❣ ❛❥✉st❛❞♦s à ♣r♦♣♦rçã♦ ❞❡ ❜❡s♦✉r♦s ♠♦rt♦s✳

♣❛r❛ ♠♦❞❡❧♦s ♠✉❧t✐✈❛r✐❛❞♦s s✐♠étr✐❝♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♣r♦❜✐t♦ ❡ ❧♦❣✐t♦✱ ♣❛r❛ ♠♦❞❡❧♦s ❝♦♠ ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ♠✉❧t✐✈❛r✐❛❞❛s ❛ss✐♠étr✐❝❛s✱ t❛✐s ❝♦♠♦ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ♣r♦❜✐t♦ ❛ss✐♠étr✐❝♦ ✭▼▼P❆✮ ❡ ♦ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ❧♦❣íst✐❝♦ ❛ss✐♠étr✐❝♦ ✭▼▼▲❆✮✳ ❆ ❛❜♦r❞❛❣❡♠ ✉t✐❧✐③❛❞❛ ♣❛r❛ r❡❛❧✐③❛çõ❡s ❞❡ ✐♥❢❡rê♥❝✐❛s é ❛ ❛❜♦r❞❛❣❡♠ ❜❛②❡s✐❛♥❛✳

✶✳✶ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦

❊st❛ t❡s❡ ✐♥tr♦❞✉③ ♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ♠✉❧t✐✈❛r✐❛❞♦ ❜✐♥ár✐♦ ❝♦♠ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ♠✉❧t✐✈❛r✐✲ ❛❞❛ ❛ss✐♠étr✐❝❛✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐♠♦s ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ♠✉❧t✐✈❛r✐❛❞♦ ❣❡r❛❧ ♣❛r❛ ♦ ❛❥✉st❡ ❞❡ ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ❝♦♠ r❡♣♦st❛s ❜✐♥ár✐❛s ❡ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ r❡✈✐sã♦ ❞♦ ♠♦❞❡❧♦ ❜✐♥ár✐♦ ❝♦♠ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❡❧í♣t✐❝❛ ❡ ♣r♦♣♦♠♦s ✉♠❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ♠✉❧t✐✈❛r✐❛❞❛ ❛ss✐♠étr✐❝❛ q✉❡ t❡♠ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❛s ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ♠✉❧t✐✈❛r✐❛❞❛s s✐♠étr✐❝❛s ♣r♦♣♦st❛s ♣♦r❈❤✐❜ ❡ ●r❡❡♥❜❡r❣

✭✶✾✾✽✮ ❡❈❤❡♥ ❡ ❉❡② ✭✷✵✵✸✮✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❛❜♦r❞❛❣❡♥s ✐♥❢❡rê♥❝✐❛✐s ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ❜✐♥ár✐♦ ❝♦♠ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❡❧í♣t✐❝❛ ❛ss✐♠étr✐❝❛✳ ❊st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ ❞❡ ✐❞❡♥t✐✜❝❛✲ ❜✐❧✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ❜✐♥ár✐♦ ❡ ♦❜t❡♠ ❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦st❡r✐♦r✐ ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ❛❧❣♦r✐t♠♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✈✐❛ ❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ✭▼❈▼❈✮✱ ❡♠ ♣❛rt✐❝✉❧❛r ♦ ❛❧❣♦r✐t♠♦ ❞❡ ●✐❜❜s✳

✶✳✶ ❖❘●❆◆■❩❆➬➹❖ ❉❖ ❚❘❆❇❆▲❍❖ ✺

❈❛♣ít✉❧♦ ✷

▼♦❞❡❧♦s ▼✉❧t✐✈❛r✐❛❞♦s ❇✐♥ár✐♦s

❈♦♥s✐❞❡r❡ Yij ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❜✐♥ár✐❛ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ r❡s♣♦st❛ ❞♦ i−és✐♠♦ ✐♥❞✐✈í❞✉♦ ♣❛r❛ ❛ j−és✐♠❛ ✈❛r✐á✈❡❧✱ ❡ s❡❥❛ Yi = (Yi1, Yi2, . . . , YiJ)′(i= 1, . . . , n) ♦ ✈❡t♦r ❛❧❡❛tór✐♦ ❝♦♠ t♦❞❛s ❛sJ r❡s♣♦st❛s ❛ss♦❝✐❛❞❛s ❛♦ i✲és✐♠♦ ✐♥❞✐✈í❞✉♦✳ ❈♦♥s✐❞❡r❡ t❛♠❜é♠ q✉❡ ❛ ❝❛❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛YiJ ❡stá ❛ss♦❝✐❛❞♦ ✉♠ ✈❡t♦rx_ij = (xij1, . . . , xijkj)

′ _{❞❡ ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ✜①❛s ❞❡ ❞✐♠❡♥sã♦ k}

j ❡ ✉♠ ✈❡t♦rβj = (βj1,βj2, . . . ,βjkj)

′_{❞❡ ❝♦❡✜❝✐❡♥t❡s ❞❡s❝♦♥❤❡❝✐❞♦s ❞❛ r❡❣r❡ssã♦✳ ❆ss✉♠✐♥❞♦ q✉❡ ♣❛r❛ ❝❛❞❛}

✐♥❞✐✈í❞✉♦ i✱ ❛s ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s Yi1, Yi2, . . . , YiJ sã♦ ❝♦rr❡❧❛❝✐♦♥❛❞♦s ❡ ♦s ✈❡t♦r❡s Y1,Y2, . . . ,Yn sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ♠✉❧t✐✈❛r✐❛❞♦ ♣❛r❛ r❡s♣♦st❛s ❜✐♥ár✐❛s ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

P(Yi1 =yi1, Yi2=yi2,· · · , YiJ =yiJ |Xi,β) =

Z

AiJ

. . .

Z

Ai1

fJ(ti1,· · ·, tiJ)dti1· · ·dtiJ, ✭✷✳✶✮

♣❛r❛ i = 1, . . . n ❡ j = 1, . . . J✱ ❡♠ q✉❡ yij é ♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ❞❛ ✈❛r✐á✈❡❧ Yij ∈ {0,1}✱ fJ é ✉♠❛ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✭❢❞♣✮ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ J✲✈❛r✐❛❞❛ ❡ ♦s ✐♥t❡r✈❛❧♦s ❞❡

✐♥t❡❣r❛çã♦Aij sã♦

Aij =

(

(_−∞,x′_ijβj] s❡yij = 1✱ (x′_ijβj,+∞) s❡yij = 0✱

✭✷✳✷✮

❝♦♠β= (β1,β2, . . . ,β_J)′ ❞❡♥♦t❛♥❞♦ ♦ ✈❡t♦r ❝♦♠ t♦❞♦s ♦sk=PJ_j=1kj ❝♦❡✜❝✐❡♥t❡s r❡❣r❡ss♦r❡s ❡

Xi =

        

x′_i1 0 0 · · · 0 0 x′_i2 0 _{· · ·} 0 ✳✳✳ · · · ✳✳✳ · · · ✳✳✳ 0 · · · 0 x′

i,J−1 0

0 _{· · ·} 0 0 x′_iJ

        

, ✭✷✳✸✮

❞❡♥♦t❛♥❞♦ ❛ ♠❛tr✐③ J ×k ❞❡ ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ✭❝♦✈❛r✐á✈❡✐s✮ ❛ss♦❝✐❛❞❛ ❛♦ i−és✐♠♦ ✐♥❞✐✈í❞✉♦✳

◆♦t❡ q✉❡ ❝❛❞❛ ✐♥t❡r✈❛❧♦ ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✷✮ é ❞❡t❡r♠✐♥❛❞♦ ♣♦r s❡✉ ♣ró♣r✐♦ ❝♦♥❥✉♥t♦ ❞❡ ❝♦✈❛r✐á✈❡✐sxij ❡ ❞❡ r❡❣r❡ss♦r❡sβ_j✳ ❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡fJ ❞❡✜♥❡ ❝♦♠♣❧❡t❛♠❡♥t❡ ❛ ❢♦r♠❛ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❡①♣❧✐❝❛t✐✈❛s ❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ✐♥t❡r❡ss❡✳ ❆ ❢✉♥çã♦ ❞❡

✽ ▼❖❉❊▲❖❙ ▼❯▲❚■❱❆❘■❆❉❖❙ ❇■◆➪❘■❖❙ ✷✳✵

✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞❡st❡ ♠♦❞❡❧♦ é ❞❛❞❛ ♣♦r

L(β;y,X) = n

Y

i=1

P(Yi1 =yi1,|Xi,β) = n

Y

i=1

Z

AiJ

. . .

Z

Ai1

fJ(ti1,· · · , tiJ)dti1· · ·dtiJ. ✭✷✳✹✮

➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ ❡st❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞❡ ♦✉tr♦s ♣❛râ♠❡tr♦s ❞❡s❝♦♥❤❡❝✐❞♦s ❛❧é♠ ❞♦ ✈❡t♦rβ✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛râ♠❡tr♦s ❞❡ ❢♦r♠❛ ❡ ❞❡ ❞❡♣❡♥❞ê♥❝✐❛✳ ❊st❡s ♣❛râ♠❡tr♦s ❛❞✐❝✐♦♥❛✐s✱

❝❛s♦ ❡①✐st❛♠✱ sã♦ ❡s♣❡❝✐✜❝❛❞♦s ♣❡❧❛ ❢❞♣fJ✳ ❖ ❢❛t♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭✷✳✹✮ ❞❡♣❡♥❞❡r ❞❡ ✐♥t❡❣r❛✐s ♠ú❧t✐♣❧❛s ♣♦❞❡ ♣r❡❥✉❞✐❝❛r ❛ ❡✜❝✐ê♥❝✐❛ ❞♦s ❛❧❣♦r✐t♠♦s ❞❡ ❡st✐♠❛çã♦✳ ❙❡❥❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♦✉ ♥❛ ♦❜t❡♥çã♦ ❞❛s ❞✐str✐❜✉✐çõ❡s ❛ ♣♦st❡r✐♦r✐ ♣❛r❛ ♦ ✉s♦ ❞♦ ▼♦♥t❡ ❈❛r❧♦ ✈✐❛ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ✭▼❈▼❈✮✳

❆❧❜❡rt ❡ ❈❤✐❜✭✶✾✾✸✮ ♦❜s❡r✈❛r❛♠ q✉❡ ♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❜✐♥ár✐❛ ♣♦❞❡ s❡r ❡s♣❡❝✐✜❝❛❞♦ ❡♠ t❡r✲

♠♦s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❧❛t❡♥t❡ ♥♦r♠❛❧♠❡♥t❡ ❞✐str✐❜✉í❞❛✳ ❊st❛ ❢♦r♠✉❧❛çã♦ ❛❧t❡r♥❛t✐✈❛ ❢❛❝✐❧✐t❛ ♥❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ❛❧❣♦r✐t♠♦s ▼❈▼❈✱ ♦ q✉❡ t♦r♥❛ ♦ ♠♦❞❡❧♦ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ♠❛✐s ❛tr❛t✐✈♦ ♣❛r❛ ♦ ✉s♦ ❞❡ ✐♥❢❡rê♥❝✐❛ ❜❛②❡s✐❛♥❛ ❡ ♥❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❊▼ ✭❉❡♠♣st❡r✱ ▲❛✐r❞ ❡ ❘✉❜✐♥✱

✶✾✼✼✮ ♥❛ ❛❜♦r❞❛❣❡♠ ❢r❡q✉❡♥t✐st❛✳❆❧❜❡rt ❡ ❈❤✐❜✭✶✾✾✸✮ ❞✐s❝✉t✐r❛♠ ❡st❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦s ♠♦❞❡✲ ❧♦s ♣r♦❜✐t♦ ❡ t✲❙t✉❞❡♥t✳ ▼❛✐s r❡❝❡♥t❡♠❡♥t❡✱❈❤✐❜ ❡ ●r❡❡♥❜❡r❣✭✶✾✾✽✮ ❝♦♥s✐❞❡r❛r❛♠ ❡st❛ ❛❜♦r❞❛❣❡♠ ♥♦s ♠♦❞❡❧♦s ♠✉❧t✐✈❛r✐❛❞♦s ♣r♦❜✐t♦ ✭▼▼P✮ ❡❈❤❡♥ ❡ ❉❡② ✭✷✵✵✸✮ ♥♦s ♠♦❞❡❧♦s ♠✉❧t✐✈❛r✐❛❞♦s ♣r♦❜✐t♦ ✭▼▼P✮✱t−❙t✉❞❡♥t ✭▼▼❚✮ ❡ ❧♦❣íst✐❝♦ ✭▼▼▲✮✳

P❛r❛ ❛ ❡s♣❡❝✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✶✮ ❛tr❛✈és ❞❡ ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s✱ ❝♦♥s✐❞❡r❡ Zi = (Zi1, . . . , ZiJ)′ ✉♠ ✈❡t♦r ❛❧❡❛tór✐♦J✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s t❛❧ q✉❡

Z_i =X_iβ₋ǫ_i, ❝♦♠ǫ_{i ∼}FJ, i= 1, . . . , n, ✭✷✳✺✮

❡♠ q✉❡ FJ é ❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ ✭❢❞❛✮ q✉❡ ❞❡✜♥❡ ❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ✉t✐❧✐③❛❞❛ ♥♦ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ❜✐♥ár✐♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ✭✷✳✶✮✳ ❚❡♠♦s✱ ❝♦♥❞✐❝✐♦♥❛♥❞♦ ♦ ✈❛❧♦r ❞❡ Yij ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ s✐♥❛❧ ❞❡Zij✱ q✉❡ ♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ♠✉❧t✐✈❛r✐❛❞❛ ❝♦♠ r❡s♣♦st❛ ❜✐♥ár✐❛ ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✶✮

♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❛tr❛✈és ❞❡ ✈❛r✐á✈❡✐s ❧❛t❡♥t❡s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

Yij =

(

✶ s❡Zij >0✱ ✵ s❡Zij ≤0✱

✭✷✳✻✮

❝♦♠

Z_i = X_iβ₋ǫ_i ✭✷✳✼✮

ǫ_{i ∼} FJ. ✭✷✳✽✮

❡♠ q✉❡ǫi, i= 1, . . . , nsã♦ ✈❡t♦r❡s ❛❧❡❛tór✐♦s ✐♥❞❡♣❡♥❞❡♥t❡s✳

❖ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ♣r♦❜✐t♦ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❈❤✐❜ ❡ ●r❡❡♥❜❡r❣ ✭✶✾✾✽✮ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ♠♦❞❡❧♦ ♠✉❧t✐✈❛r✐❛❞♦ ❜✐♥ár✐♦ ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✻✮✲✭✷✳✽✮ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ ♦s ✈❡t♦r❡s ❛❧❡❛tór✐♦sJ✲

✷✳✵ ✾

❡ ✭✷✳✻✮✲✭✷✳✽✮ é ❞❡♠♦♥str❛❞❛ ❛❞✐❛♥t❡✱ ✐♥✐❝✐❛❧♠❡♥t❡ ❛♣r❡s❡♥t❛❞❛ ♣❛r❛ ♦ ❝❛s♦ ♠❛r❣✐♥❛❧ ✭✉♥✐✈❛r✐❛❞♦✮✱ ❡ ❞❡♣♦✐s ❡st❡♥❞✐❞❛ ♣❛r❛ ♦ ❝❛s♦ ♠✉❧t✐✈❛r✐❛❞♦✳

◆♦t❡ q✉❡ ♦ ✈❛❧♦r ❞❛ ✈❛r✐á✈❡❧Yij ♥♦ ♠♦❞❡❧♦ ✭✷✳✻✮✲✭✷✳✽✮ é ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ ✈❛❧♦r ❞❛ ✈❛r✐á✈❡❧ ❧❛t❡♥t❡Zij ❡ q✉❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s ❡✈❡♥t♦sYij = 1❡Yij = 0sã♦ ❞❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r

P(Yij = 1|xi,βj) =P(Zij >0|xijβj) ❡ P(Yij = 0|xij,βj) =P(Zij ≤0|xij,βj).

❊st❛s ❞✉❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ♣♦❞❡♠ s❡r ❡s❝r✐t❛s ❡♠ ✉♠❛ só ❡①♣r❡ssã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

P(Yij =yij |xij,βj) =P(Zij ∈Bij |xij,βj), ✭✷✳✾✮

❡♠ q✉❡

Bij =

(

(0,∞) s❡yij = 1✱ (_−∞,0] s❡yij = 0✱

♣❛r❛i= 1, . . . , n ❡j = 1, . . . , J✳ ❆ss✉♠✐♥❞♦ ❛❣♦r❛ q✉❡Zij =x′_ijβj−ǫij ❡ ǫij ∼F1j✱ ❡♠ q✉❡ F1j é ❛ j−és✐♠❛ ❞✐str✐❜✉✐çã♦ ♠❛r❣✐♥❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ ❝♦♥❥✉♥t❛FJ ❞❡✜♥✐❞❛ ❡♠ ✭✷✳✽✮✱ t❡♠♦s q✉❡

P(Yij = 1|xij,βj) =P(Zij >0|xij,βj) =P(x′ijβj−ǫij >0) =P(ǫij <x′ijβj) =

Z x′

ijβj

−∞

f1j(t)dt

❡

P(Yij = 0|xij,βj) =P(Zij ≤0|xij,βj) =P(x′ijβj−ǫij ≤0) =P(ǫij ≥Xiβ) =

Z ∞

x′

ijβj

f1j(t)dt,

q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ✉♠❛ só ❡①♣r❡ssã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

P(Yij =yij |xij,βj) =

Z

Aij

f1j(t)dt,

❝♦♠

Aij =

(

(_−∞,x′_ijβj] s❡yij = 1✱ (x′_ijβj,+∞) s❡yij = 0✳

❡ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ♦s ♠♦❞❡❧♦s ✭✷✳✶✮ ❡ ✭✷✳✻✮✲✭✷✳✽✮ ❡stá ❞❡♠♦♥str❛❞❛ ♣❛r❛ ♦ ❝❛s♦ ♠❛r❣✐♥❛❧✳

❊st❡♥❞❡♥❞♦ ❛❣♦r❛ ✭✷✳✾✮ ♣❛r❛ ❝❛s♦ ♠✉❧t✐✈❛r✐❛❞♦✱ t❡♠♦s q✉❡

P(Yi1 =yi1,· · ·, YiJ =yiJ |Xi,β) =P(Zi1 ∈Bi1,· · · , ZiJ ∈BiJ |Xi,β), ✭✷✳✶✵✮

❡♠ q✉❡ ♦s ✐♥t❡r✈❛❧♦s Bij(i= 1, . . . n;j = 1, . . . J)✱ sã♦ ❞❡✜♥✐❞♦s ❝♦♠♦

Bij =

(