Universidade Federal de São Carlos

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛

❊st✐♠❛❞♦r❡s ♥ã♦ ♣❛r❛♠étr✐❝♦s

♣❛r❛ ❞❛❞♦s ❝♦♠ ❝❡♥s✉r❛

P❛✉❧♦ ❘✐❝❛r❞♦ ❙✐♠✐♦♥✐

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛

❊st✐♠❛❞♦r❡s ♥ã♦ ♣❛r❛♠étr✐❝♦s

♣❛r❛ ❞❛❞♦s ❝♦♠ ❝❡♥s✉r❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❊st❛tíst✐❝❛✳

P❛✉❧♦ ❘✐❝❛r❞♦ ❙✐♠✐♦♥✐

❖r✐❡♥t❛❞♦r✿

❆❞r✐❛♥♦ P♦❧♣♦ ❞❡ ❈❛♠♣♦s

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Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar

S589ep

Simioni, Paulo Ricardo.

Estimadores não paramétricos para dados com censura / Paulo Ricardo Simioni. -- São Carlos : UFSCar, 2013. 66 f.

Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2013.

1. Estatística. 2. Estimadores de Bayes. 3. Sistema em paralelo. 4. Sistema em série. I. Título.

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❘❡s✉♠♦

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❆❜str❛❝t

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❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ♣✳ ✼

✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ♣✳ ✾

✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾

✷✳✷ ❆ ❚❛①❛ ❞❡ ❋❛❧❤❛ ❡ ❛ ❚❛①❛ ❞❡ ❋❛❧❤❛ ❘❡✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✵

✷✳✸ ❋✉♥çõ❡s ❞❡ ❉✐str✐❜✉✐çã♦ ❡ ❙✉❜✲❞✐str✐❜✉✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✶

✸ ❊st✐♠❛❞♦r ❇❛②❡s✐❛♥♦ ♥ã♦ ♣❛r❛♠étr✐❝♦ ♣✳ ✶✼

✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✼

✸✳✷ ❙✐st❡♠❛ ❡♠ ❙ér✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✽

✸✳✸ ❙✐st❡♠❛ ❡♠ P❛r❛❧❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✵

✹ ●❡♥❡r❛❧✐③❛çã♦ ♣✳ ✷✷

✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✷

✹✳✷ ❙✐st❡♠❛ ❡♠ ❙ér✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✸

✹✳✸ ❙✐st❡♠❛ ❡♠ P❛r❛❧❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✻

✺ ❊st✉❞♦ ❞❡ ❙✐♠✉❧❛çã♦ ♣✳ ✺✵

✺✳✶ ❙✐st❡♠❛ ❝♦♠ ❈♦♠♣♦♥❡t❡s ❡♠ ❙ér✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✶

✺✳✷ ❙✐st❡♠❛ ❝♦♠ ❈♦♠♣♦♥❡t❡s ❡♠ P❛r❛❧❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✶

✻ ❈♦♠❡♥tár✐♦s ❋✐♥❛✐s ♣✳ ✺✽

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✼

✶ ■♥tr♦❞✉çã♦

❊♠ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❛❞♦s ❝❡♥s✉r❛❞♦s à ❞✐r❡✐t❛ ❢♦✐ ❡st✉❞❛❞♦ ♣♦r ✈ár✐♦s ❛✉t♦r❡s✳ P❛r❛ ✉♠❛ r❡✈✐sã♦✱ ✈❡❥❛ ■❜r❛❤✐♠ ❡t ❛❧✳ ❬✹❪✱ s♦❜ ✉♠❛ ♣❡rs♣❡❝t✐✈❛ ❇❛②❡s✐❛♥❛ ❡ ▲❛✇❧❡ss ❬✼❪✱ ♣❛r❛ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❢r❡q✉❡♥t✐st❛✳ P❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❛❞♦s ❝❡♥s✉r❛❞♦s à ❡sq✉❡r❞❛✱ ❛té ♦♥❞❡ s❛❜❡♠♦s✱ ❡①✐st❡♠ ♣♦✉❝♦s tr❛❜❛❧❤♦s✳ P♦❞❡♠♦s ❝✐t❛r P♦❧♣♦ ❡ P❡r❡✐r❛ ❬✶✵❪ ❡ P♦❧♣♦ ❡t ❛❧✳ ❬✾❪✳ ❆❧é♠ ❞❡st❡s✱ t❛♠❜é♠ ❝✐t❛♠♦s ❇❛r❧♦✇ ❡ Pr♦❝❤❛♥ ❬✶❪✱ ❝♦♠♦ ✉♠❛ ❜♦❛ r❡❢❡rê♥❝✐❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡✳

◆❡st❡ tr❛❜❛❧❤♦ s❡❣✉✐♠♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡✿ P❡t❡rs♦♥ ❬✽❪✱ q✉❡ ❞❡s❝r❡✈❡ ✉♠❛ r❡❧❛çã♦ ❢✉♥✲ ❝✐♦♥❛❧ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❛❞♦s ❝❡♥s✉r❛❞♦s ❛ ❞✐r❡✐t❛❀ ❙❛❧✐♥❛s ❡t ❛❧✳ ❬✶✷❪ q✉❡ ❛♣r❡s❡♥t❛ ♦ ❡st✐♠❛❞♦r ❇❛②❡s✐❛♥♦ ♥ã♦✲♣❛r❛♠étr✐❝♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ s✐st❡♠❛s ❡♠ sér✐❡ ❡ P♦❧♣♦ ❡ ❙✐♥❤❛ ❬✶✶❪ q✉❡ ♣♦st❡r✐♦r♠❡♥t❡ ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ ❝♦rr❡çã♦ ❞♦ tr❛❜❛❧❤♦ ❞❡ ❙❛❧✐♥❛s ❡t ❛❧✳ ❬✶✷❪❀ ❡ P♦❧♣♦ ❡ P❡r❡✐r❛ ❬✶✵❪ q✉❡ ❛♣r❡s❡♥t❛r❛♠ ✉♠ ❡st✐♠❛❞♦r ❇❛②❡s✐❛♥♦ ♥ã♦✲♣❛r❛♠étr✐❝♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ❡♠ ♣❛r❛❧❡❧♦✳

❊♠ ✉♠ s✐st❡♠❛ ❡♠ sér✐❡✱ t♦❞♦s ♦s ❝♦♠♣♦♥❡♥t❡s ❞❡✈❡♠ ❡st❛r ❢✉♥❝✐♦♥❛♥❞♦ ♣❛r❛ q✉❡ ♦ s✐st❡♠❛ ❢✉♥❝✐♦♥❡✳ ❉❡♥♦t❛♥❞♦ ♣♦r Xj ♦ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ❞❡ ❞♦ j✲és✐♠♦ ❝♦♠♣♦♥❡♥t❡✱

j = 1, . . . , k✱ ♦ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ s❡rá ❞❛❞♦ ♣♦r T = min(X1, . . . , Xk)✳ ❊q✉✐✈❛✲

❧❡♥t❡♠❡♥t❡✱ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❡♠ sér✐❡ ❢❛❧❤❛rá ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ♦ ♣r✐♠❡✐r♦ ❝♦♠♣♦♥❡♥t❡ ❢❛❧❤❛r✳

◆♦ ❝❛s♦ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❡♠ ♣❛r❛❧❡❧♦✱ ❜❛st❛ q✉❡ ✉♠ ❞♦s ❝♦♠♣♦♥❡♥t❡s ❡st❡❥❛ ❢✉♥❝✐♦♥❛♥❞♦ ♣❛r❛ ♦ s✐st❡♠❛ ❢✉♥❝✐♦♥❛r✳ ❊♥tã♦✱ ♦ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ s❡rá ❞❛❞♦ ♣♦r

T = max(X1, . . . , Xk)✳ ❯♠ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦ ❢❛❧❤❛ q✉❛♥❞♦ ♦ s❡✉ ú❧t✐♠♦ ❝♦♠♣♦♥❡♥t❡

❢❛❧❤❛r✳

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✽

♣♦❞❡ s❡r ❝❡♥s✉r❛❞❛ à ❞✐r❡t❛ ✭♥♦ ❝❛s♦ ❞❡ ✉♠ s✐st❡♠❛ ❡♠ sér✐❡✮✱ à ❡sq✉❡r❞❛ ✭♥♦ ❝❛s♦ ❞❡ ✉♠ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦✮ ♦✉ ♥ã♦ ❝❡♥s✉r❛❞❛✳

❊st❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♥♦ ❈❛♣ít✉❧♦ ✷✱ ❞✐s❝✉t✐♠♦s ❛ t❡♦r✐❛ ❜❛s❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❡st✐♠❛❞♦r❡s ♥ã♦ ♣❛r❛♠étr✐❝♦s❀ ♥♦ ❈❛♣ít✉❧♦ ✸ ❞❡s❡♥✈♦❧✲ ✈❡♠♦s ♦s ❡st✐♠❛❞♦r❡s ❇❛②❡s✐❛♥♦s ♥ã♦ ♣❛r❛♠étr✐❝♦s ♣❛r❛ ♦s ❞♦✐s t✐♣♦s ❞❡ s✐st❡♠❛s❀ ♥♦ ❈❛♣ít✉❧♦ ✹ ❛♣r❡s❡♥t❛♠♦s ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❈❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ❛❧é♠ ❞❡ ❡①❡♠♣❧♦ ♣❛r❛ ♦s s✐st❡♠❛s ❡♠ sér✐❡ ❡ ❡♠ ♣❛r❛❧❡❧♦❀ ♥♦ ❈❛♣ít✉❧♦ ✺ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ❝♦♠♣❛r❛♥❞♦ ♦ ❡st✐♠❛❞♦r ❇❛②❡s✐❛♥♦ ♥ã♦ ♣❛r❛♠étr✐❝♦ ❝♦♠ ♦ ❡st✐♠❛❞♦r ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r ❬✺❪❀ ♣♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✻✱ ❞✐s❝✉t✐♠♦s ♦s ♣r✐♥❝✐♣❛✐s ❛s♣❡❝t♦s ❞♦ tr❛❜❛❧❤♦✳ ◆♦ ❛♣ê♥❞✐❝❡ ❆ t❡♠♦s ♦ ❝ó❞✐❣♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ♥❛ ❧✐♥❣✉❛❣❡♠ ❘✱ ❝♦♠ ❛s ❢✉♥çõ❡s ✉t✐❧✐③❛❞❛s ♥♦ tr❛❜❛❧❤♦✳

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✾

✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s

✷✳✶ ■♥tr♦❞✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❜❛s❡ t❡ór✐❝❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❡st✐♠❛❞♦r❡s ♥ã♦✲♣❛r❛♠étr✐❝♦s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✳ ❙❡❣✉✐♠♦s ❛q✉✐ ♦s ♣❛ss♦s ❛♣r❡s❡♥t❛✲ ❞♦s ♣♦r P❡t❡rs♦♥ ❬✽❪ ❡ P♦❧♣♦ ❡ P❡r❡✐r❛ ❬✶✵❪✳ P❛r❛ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦ ❛❞♦t❛♠♦s ♦ ✉s♦ ❞❛s ✭s✉❜✲✮❞✐str✐❜✉✐çõ❡s✱ r❡❡s❝r❡✈❡♥❞♦ ❛♣r♦♣r✐❛❞❛♠❡♥t❡✱ q✉❛♥❞♦ ♥❡❝❡ssár✐♦✱ ♦s r❡s✉❧t❛❞♦s ❛♣r❡✲ s❡♥t❛❞♦s ♣♦r P❡t❡rs♦♥ ❬✽❪✱ q✉❡ ❞❡s❝r❡✈❡ s❡✉ tr❛❜❛❧❤♦ ❡♠ t❡r♠♦s ❞❡ ✭s✉❜✲✮s♦❜r❡✈✐✈ê♥❝✐❛s✳

❆♥t❡s✱ ♣♦ré♠✱ ❛❜❛✐①♦ ❞❛♠♦s ❞✉❛s ❞❡✜♥✐çõ❡s q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♠❡❧❤♦r ❝♦♠✲ ♣r❡❡♥sã♦ ❞♦ t❡①t♦✳

✷✳✶✳✶ ❉❡✜♥✐çã♦✳ P♦♥t♦ ❞❡ s❛❧t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ g é ♦ ♣♦♥t♦ s ❡♠ q✉❡ g(s) 6= g(s+₎ _♦✉

g(s)6=g(s−)✳

✷✳✶✳✷ ❉❡✜♥✐çã♦✳ ❙❡❥❛ I ✉♠ ✐♥t❡r✈❛❧♦ ❞❛ r❡t❛ r❡❛❧ R✳ ❯♠❛ ❢✉♥çã♦ f : I → R é ❛❜s♦❧✉✲ t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ I s❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ ε✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ δ t❛❧

q✉❡ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ✜♥✐t❛ ❞✐s❥✉♥t❛ ❞❡ s✉❜✲✐♥t❡r✈❛❧♦s (ak, bk) ❞❡ I s❛t✐s❢❛ç❛♠ X

k

|bk−ak|< δ

❡♥tã♦

X

k

|f(bk)−f(ak)|< ε.

❆ ❢✉♥çã♦ f(x) = 0 s❡x= 0 ❡ f(x) = xsin(1/x) s❡ x6= 0✱ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ✜♥✐t♦ ❝♦♥✲

(12)

✶✵

✷✳✷ ❆ ❚❛①❛ ❞❡ ❋❛❧❤❛ ❡ ❛ ❚❛①❛ ❞❡ ❋❛❧❤❛ ❘❡✈❡rs❛

❆ t❛①❛ ❞❡ ❢❛❧❤❛ é ✈✐st❛ ❝♦♠♦ ❛ ❝❤❛♥❝❡ ❞❡ q✉❡ ❛ ❢❛❧❤❛ ♦❝♦rr❛ ♥✉♠ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ ❧♦❣♦ ❛♣ós t ❞❛❞♦ q✉❡ ♥ã♦ ♦❝♦rr❡✉ ❛♥t❡s ❞❡ t✳ ❆ t❛①❛ ❞❡ ❢❛❧❤❛ r❡✈❡rs❛ é ❛ ❝❤❛♥❝❡ ❞❡ ❡❧❡

❢❛❧❤❛r ♥✉♠ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t ❞❛❞♦ q✉❡ ❢✉♥❝✐♦♥❛rá ❛té ♥♦ ♠á①✐♠♦t✳

❆ t❛①❛ ❞❡ ❢❛❧❤❛ r❡✈❡rs❛ r❡❧❛❝✐♦♥❛✲s❡ ❝♦♠ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❛ss✐♠ ❝♦♠♦ ❛ t❛①❛ ❞❡ ❢❛❧❤❛ r❡❧❛❝✐♦♥❛✲s❡ ❝♦♠ ❛ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✳

❉❡♥♦t❛♥❞♦ ♣♦r f ❛ ❞❡♥s✐❞❛❞❡ ❡ ♣♦r F ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦✱ t❡♠♦s ❛ t❛①❛ ❞❡ ❢❛❧❤❛

✭❝♦♥tí♥✉❛ ❡ ❞✐s❝r❡t❛✮

✭✷✳✷✳✶✮ λc(t) =

f(t)

1−F(t−₎ ❡ λd(t) =

Pr(T =t) 1−F(t−₎ =

Pr(T =t) Pr(T ≥t) ❡ ❛ t❛①❛ ❞❡ ❢❛❧❤❛ r❡✈❡rs❛ ✭❝♦♥tí♥✉❛ ❡ ❞✐s❝r❡t❛✮

✭✷✳✷✳✷✮ µc(t) =

f(t)

F(t) ❡ µd(t) =

Pr(T =t)

F(t) =

Pr(T =t) Pr(T ≤t).

✐✮ P❛r❛ ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s

✭✷✳✷✳✸✮ F(t) = 1−exp

Z t

0

λc(y)dy = exp − Z ∞ t

µc(y)dy

.

✐✐✮ P❛r❛ ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❞✐s❝r❡t❛s

✭✷✳✷✳✹✮ F(t) = 1−Y

y≤t

[1−λd(y)] = Y

y>t

[1−µd(y)].

✐✐✐✮ P❛r❛ ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❝♦♠ ♣❛rt❡ ❝♦♥tí♥✉❛ ❡ ♣❛rt❡ ❞✐s❝r❡t❛

✭✷✳✷✳✺✮ F(t) = 1−exp[−Λ(t)] = exp[−M(t)].

❆q✉✐✱ Λ(.) é ❛ t❛①❛ ❞❡ ❢❛❧❤❛ ❛❝✉♠✉❧❛❞❛ ❡ M(.) é ❛ t❛①❛ ❞❡ ❢❛❧❤❛ r❡✈❡rs❛ ❛❝✉♠✉❧❛❞❛✳ ❚❛✐s ❢✉♥çõ❡s sã♦ ❞❡✜♥✐❞❛s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

✭✷✳✷✳✻✮ Λ(t) =

Z t

0

λc(y)dy+ X

y≤t

[−ln(1−λd(y))]

❡

✭✷✳✷✳✼✮ M(t) =

Z ∞

t

µc(y)dy+ X

y>t

(13)

✶✶

❡♠ q✉❡ ❛ ✐♥t❡❣r❛❧ é s♦❜r❡ ✐♥t❡r✈❛❧♦s ❛❜❡rt♦s ❡ ❞✐s❥✉♥t♦s q✉❡ ♥ã♦ ✐♥❝❧✉❡♠ ♦s ♣♦♥t♦s ❞❡ s❛❧t♦s ❞❡F✳

✷✳✸ ❋✉♥çõ❡s ❞❡ ❉✐str✐❜✉✐çã♦ ❡ ❙✉❜✲❞✐str✐❜✉✐çã♦

❈♦♥s✐❞❡r❛♥❞♦ q✉❡ t❡♠♦s ❛♣❡♥❛s ❞♦✐s ❝♦♠♣♦♥❡♥t❡s ♥♦ s✐st❡♠❛ ❡ q✉❡ ❛ ✐♥❢♦r♠❛çã♦ ❞✐s♣♦♥í✈❡❧ é ♦ t❡♠♣♦ ❞❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ❡ ❛ ❝♦♠♣♦♥❡♥t❡ q✉❡ ✏♣r♦❞✉③✐✉✑ ❛ ❢❛❧❤❛✳ ■st♦ é✱ ♦❜s❡r✈❛♠♦sT ❡ δ✱ ❡♠ q✉❡ T = min(X1, X2) ♦✉ T = max(X1, X2) ✭s❡ ♦ s✐st❡♠❛ t❡♠ s❡✉s

❝♦♠♣♦♥❡♥t❡s ❡♠ sér✐❡ ♦✉ ♣❛r❛❧❡❧♦✮ ❡ δ = 1 s❡ ♦ ❝♦♠♣♦♥❡♥t❡ ✶ ♣r♦❞✉③✐✉ ❛ ❢❛❧❤❛ ♦✉ δ = 2 s❡ ❢♦✐ ♦ ❝♦♠♣♦♥❡♥t❡ ✷ ♦ r❡s♣♦♥sá✈❡❧✳ ❆ss✉♠✐♠♦s q✉❡X1 ❡X2 sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳

❆❣♦r❛✱ ❞❡✜♥✐♠♦s ♣❛r❛ ❛ j✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡✱ j = 1,2✱ ❛ s✉❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦

♣♦rFj(t) = Pr(Xj ≤t)❡ ❛ s✉❛ ❢✉♥çã♦ ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦ ♣♦r Fj∗(t) = Pr(T ≤t, δ =j)✱

q✉❡ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ✉♠❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❝♦♥❥✉♥t❛ ❞♦ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ✭s❡r ♠❡♥♦r q✉❡ ✉♠ t❡♠♣♦ t > 0✮ ❡ ❞❛ ❢❛❧❤❛ ♦❝♦rr❡r ♣❡❧❛ j✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡✳ ❖ ♥♦ss♦

♣r✐♥❝✐♣❛❧ ✐♥t❡r❡ss❡ ❝♦♥s✐st❡ ❡♠ ❡st✐♠❛r ❛Fj(·)✳ P♦ré♠✱ t❛♥t♦ ♥♦ s✐st❡♠❛ ❡♠ sér✐❡✱ q✉❛♥t♦

♥♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦✱ ❡st❛♠♦s s✉❥❡✐t♦s ❛ ❝❡♥s✉r❛ ♥❛ ♦❜s❡r✈❛çã♦ ❞♦s ❞❛❞♦s✳ ❉❡st❛ ❢♦r♠❛ ♥❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ♦❜s❡r✈❛r ❞✐r❡t❛♠❡♥t❡ ❛s r❡❛❧✐③❛çõ❡s ❞❛s ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ ✐♥t❡r❡ss❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ♦❜s❡r✈❛çã♦ ❞❡ r❡❛❧✐③❛çõ❡s ❞❛ ❢✉♥çã♦ ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦ é ❢❛❝✐❧♠❡♥t❡ ♦❜t✐❞❛✱ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ ♦❜s❡r✈❛çã♦ ❞❛s r❡❛❧✐③❛çõ❡s ❞♦ ♣❛r (T, δ)✳

❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❢✉♥çõ❡s ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦ F∗

1(·) ❡ F2∗(·) sã♦ ❞❛❞❛s ♣♦r✿

✷✳✸✳✶ Pr♦♣r✐❡❞❛❞❡✳ ❆ ❢✉♥çã♦ ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦F∗

1(·)♣♦❞❡ s❡r ❡①♣r❡ss❛ ❡♠ t❡r♠♦s ❞❛s

❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦F1 ❡ F2 ❝♦♠♦

✭✷✳✸✳✷✮ F∗

1(t) =

Z t

0

[1−F2(s)][dF1(s)],

♣❛r❛ ♦ s✐st❡♠❛ ❡♠ sér✐❡ ❡

✭✷✳✸✳✸✮ F∗

1(t) =

Z t

0

F2(s)dF1(s),

♣❛r❛ ♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ♦ ❙✐st❡♠❛ ❡♠ ❙ér✐❡✱ t❡♠♦s✿

F∗

1(t) = Pr(T ≤t, δ= 1)

= Pr(T ≤t, X1 < X2)

= Z ∞

0

(14)

✶✷

❖❜s❡r✈❡♠♦s q✉❡

Pr(T ≤t, X2 > s|X1 =s) =

(

0, s❡ s > t,

Pr(X2 > s|X1 =s), s❡ s≤t.

❆ss✐♠✱

F∗

1(t) =

Z t

0

Pr(X2 > s|X1 =s)dF1(s)

= Z t

0

Pr(X2 > s)dF1(s)

= Z t

0

(1−F2(s))dF1(s).

P❛r❛ ♦ ❙✐st❡♠❛ ❡♠ P❛r❛❧❡❧♦✱ t❡♠♦s✿

F∗

1(t) = Pr(T ≤t, δ= 1)

= Pr(T ≤t, X1 > X2)

= Z ∞

0

Pr(T ≤t, X2 < s|X1 =s)dF1(s)

❖❜s❡r✈❡♠♦s q✉❡

Pr(T ≤t, X2 < s|X1 =s) =

(

0, s❡ s > t,

Pr(X2 < s|X1 =s), s❡ s≤t.

❆ss✐♠✱

F∗

1(t) =

Z t

0

Pr(X2 < s|X1 =s)dF1(s)

= Z t

0

Pr(X2 < s)dF1(s)

= Z t

0

F2(s)dF1(s).

✷✳✸✳✹ ❉❡✜♥✐çã♦✳ ❆ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡st❛ ♣r♦✈♦❝❛r ❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛✱ ✐st♦ é✱ ❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❡ Xj é ❞❛❞❛ ♣♦r Pr(δ=j)

❉❡st❛ ❢♦r♠❛✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡

✷✳✸✳✺ Pr♦♣r✐❡❞❛❞❡✳ ❆s ❢✉♥çõ❡s ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦ ❞❛ ❝♦♠♣♦♥❡♥t❡ j✱ q✉❛♥❞♦ t → ∞✱

❝♦♥✈❡r❣❡♠ ♣❛r❛ ❛ s✉❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡ ❛ s♦♠❛ ❞❛s s✉❜✲❞✐str✐❜✉✐çõ❡s é ✐❣✉❛❧ ❛ ✶✳

✶✳ F∗

(15)

✶✸

✷✳ F∗

2(+∞) = Pr(δ= 2)❀

✸✳ F∗

1(+∞) +F2∗(+∞) = 1.

❉❡♠♦♥str❛çã♦✳

✶✳ F∗

1(+∞) = Pr(T <∞, δ= 1)

= Pr(T <∞|δ = 1) Pr(δ = 1) = Pr(δ = 1).

✷✳ F∗

2(+∞) = Pr(T <∞, δ= 2)

= Pr(T <∞|δ = 2) Pr(δ = 2) = Pr(δ = 2).

✸✳ F∗

1(+∞) +F

∗

2(+∞) = Pr(δ = 1) + Pr(δ = 2) = 1.

✷✳✸✳✻ Pr♦♣r✐❡❞❛❞❡✳ ❆ s♦♠❛ ❞❛s ❢✉♥çõ❡s ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦✱ ♣❛r❛t >0✱ é ✐❣✉❛❧ ❛ ❢✉♥çã♦

❞❡ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛ ✭F(t)✮✳

F∗

1(t) +F

∗

2(t) = Pr [(T ≤t, δ= 1)∪(T ≤t, δ = 2)]

= Pr(T ≤t) = F(t).

◆♦t❡ q✉❡✱ ❛ ♥♦♠❡♥❝❧❛t✉r❛ ✏s✉❜✲❞✐str✐❜✉✐çã♦✑ é ♠♦t✐✈❛❞❛ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳✻✱ ✉♠❛ ✈❡③ q✉❡ ❛s s✉❜✲❞✐str✐❜✉✐çõ❡s ♣♦❞❡♠ s❡r ✈✐st❛s ❝♦♠♦ ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛✳

✷✳✸✳✼ Pr♦♣r✐❡❞❛❞❡✳

✶✳ ❖s ♣♦♥t♦s ❞❡ s❛❧t♦ ❞❡ Fj(·) sã♦ ♣♦♥t♦s ❞❡ s❛❧t♦ ❞❡Fj∗(·) ❡ ✈✐❝❡✲✈❡rs❛✱ j = 1,2❀

✷✳ ❙❡ F1(·) ❡ F2(·) ♥ã♦ tê♠ ♣♦♥t♦s ❞❡ s❛❧t♦ ❡♠ ❝♦♠✉♠✱ ❡♥tã♦ F1∗(·) ❡ F2∗(·) t❛♠❜é♠

♥ã♦ t❡♠❀

✷✳✸✳✽ Pr♦♣r✐❡❞❛❞❡✳ ❆♦ ♠❡♥♦s ✉♠❛ ❞❛s ❞✉❛s ❢✉♥çõ❡s ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦ F∗

1(·) ❡ F2∗(·)

é ♣♦s✐t✐✈❛ ♣❛r❛ t < t∗_{✱ ❡ ❛♠❜❛s sã♦ ✉♠ ♣❛r❛} _t_≥_t∗_{✱ ❡♠ q✉❡} _t∗ _{❞❡✜♥❡ ♦ s✉♣♦rt❡ ❞❛ ❢✉♥çã♦}

❞❡ ❞✐str✐❜✉✐çã♦ ❞❡Fj✱ j = 1,2✳

❙❡❣✉♥❞♦ ❇r❡s❧♦✇ ❡ ❈r♦✇❧❡② ❬✷❪✱ ♣❛r❛ s✐st❡♠❛ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❡♠ sér✐❡✱ s❡ ❛s ❢✉♥çõ❡s

F∗

(16)

✶✹

❞❛s ❢✉♥çõ❡s ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦ F∗

1(·) ❡ F2∗(·)♣♦r✱

✭✷✳✸✳✾✮ F1(t) =

Z ∞

t

dF∗

1(s)

F∗

1(s) +F2∗(s)

.

P♦ré♠✱ t❛❧ ❡q✉❛çã♦ ♥ã♦ s❡rá ✈á❧✐❞❛ q✉❛♥❞♦ ❛s ❢✉♥çõ❡s ❞❡ s✉❜✲❞✐str✐❜✉✐çã♦ F∗

1(·) ❡

F∗

2(·) ♥ã♦ ❢♦r❡♠ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s✳

❖ ❢✉♥❝✐♦♥❛❧ ♣r♦♣♦st♦ ♣♦r P❡t❡rs♦♥ ❬✽❪ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❊q✉❛çã♦ ✭✷✳✸✳✾✮✱ ♣❛r❛ ♦ s✐st❡♠❛ ❡♠ sér✐❡✱ ❡ r❡t✐r❛ ❛ r❡str✐çã♦ ❞❡ F∗

1(·) ❡ F2∗(·) s❡r❡♠ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s✱

✜❝❛♥❞♦ ❛♣❡♥❛s ❝♦♠ ❛ r❡str✐çã♦ ❞❡ q✉❡F∗

1(·)❡F2∗(·)♥ã♦ ♣♦❞❡♠ t❡r ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐✲

❞❛❞❡ ❡♠ ❝♦♠✉♠✳ P♦❧♣♦ ❡ P❡r❡✐r❛ ❬✶✵❪✱ ❛♣r❡s❡♥t❛r❛♠ ❛ ✈❡rsã♦ ❞✉❛❧ ❞❡ P❡t❡rs♦♥ ❬✽❪✱ ♣❛r❛ ♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦✳ ❊st❡s ❢✉♥❝✐♦♥❛✐s sã♦ ❞❛❞♦s ♥♦ t❡♦r❡♠❛ ❛❜❛✐①♦✳

✷✳✸✳✶✵ ❚❡♦r❡♠❛✳ ❆s ❢✉♥çõ❡s s✉❜✲❞✐str✐❜✉✐çã♦F∗

1 ❡F2∗✱ ❞❡t❡r♠✐♥❛♠ ✭✉♥✐❝❛♠❡♥t❡✮ ❛ ❢✉♥✲

çã♦ ❞✐str✐❜✉✐çã♦ F1 ♣❛r❛ t < t∗✳ ■st♦ é✱ ♣❛r❛ ♦ s✐st❡♠❛ ❡♠ sér✐❡ t❡♠♦s

F1(t) = Φs(F1∗, F

∗

2, t)

✭✷✳✸✳✶✶✮

= 1−exp "Z t

0

−dF∗

1(s)

1−(F∗

1(s) +F2∗(s))

+X

s≤t

ln

1−(F∗

1(s+) +F2∗(s+))

1−(F∗

1(s−) +F2∗(s−))

#

❡ ♣❛r❛ ♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦✱ t❡♠♦s

F1(t) = Φp(F1∗, F2∗, t)

✭✷✳✸✳✶✷✮

= exp "Z ∞

t

−dF∗

1(s)

F∗

1(s) +F2∗(s)

+X

s>t

ln

F∗

1(s−) +F2∗(s−)

F∗

1(s+) +F2∗(s+)

#

.

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ q✉❡✱ s❡ F1(t∗) = 0✱ ❡♥tã♦ F1(t) = 0♣❛r❛ t♦❞♦ t > t∗✱ ✐st♦ é✱ ♦ ♠❛✐♦r

t❡♠♣♦ ❞❡ ❢❛❧❤❛ ♣♦ssí✈❡❧ é ❞♦ ❝♦♠♣♦♥❡♥t❡ ✶ ét∗_✳

❚♦♠❛♥❞♦ ❛ P s♦❜ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ s❛❧t♦ ❞❡ F∗

1✱ ❞❛s ❡q✉❛çõ❡s ✭✷✳✷✳✺✮✱ ✭✷✳✷✳✻✮ ❡

✭✷✳✷✳✼✮ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡✱ ♣❛r❛ ♦ s✐st❡♠❛ ❡♠ sér✐❡✱

✭✷✳✸✳✶✸✮ Z t

0

dF∗

1(s)

1−(F∗

1(s) +F2∗(s))

= Z t

0

λ1

c(s)ds,

❡

✭✷✳✸✳✶✹✮ X

s≤t

ln

1−(F∗

1(s+) +F2∗(s+))

1−(F∗

1(s−) +F2∗(s−))

=X

s≤t

ln[1−λ1_d(s)].

❡♠ q✉❡λ1

(17)

✶✺

X1✳ P❛r❛ ♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦✱ ❜❛st❛ ♣r♦✈❛r q✉❡

✭✷✳✸✳✶✺✮ Z ∞

t

dF∗

1(s)

F∗

1(s) +F2∗(s)

= Z ∞

t

µ1_c(s)ds,

❡ ✭✷✳✸✳✶✻✮ X s>t ln F∗

1(s−) +F2∗(s−)

F∗

1(s+) +F2∗(s+)

=X

s>t

ln[1−µ1_d(s)].

❡♠ q✉❡µ1

c ❡µ1d sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛s t❛①❛s ❞❡ ❢❛❧❤❛ ❝♦♥tí♥✉❛ ❡ ❞✐s❝r❡t❛ ❞♦ ❝♦♠♣♦♥❡♥t❡

X1✳

❆ ❊q✉❛çã♦ ✭✷✳✸✳✶✸✮ é ❝♦♥s❡q✉ê♥❝✐❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ✷✳✸✳✶ ❡ ✷✳✸✳✻ ❡ ❞❡ Z t

0

dF∗

1(s)

1−(F∗

1(s) +F2∗(s))

= Z t

0

(1−F2(s))dF1(s)

1−F(s)

= Z t

0

(1−F2(s))dF1(s)

(1−F1(s))(1−F2(s))

= Z t

0

dF1(s)

1−F1(s)

= Z t

0

λ1_c(s)ds.

❆ ❊q✉❛çã♦ ✭✷✳✸✳✶✻✮ é ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳✻✱ ❞♦ ❢❛t♦ ❞❡ q✉❡F2(s−) = F2(s+)

q✉❛♥❞♦s é ♣♦♥t♦ ❞❡ s❛❧t♦ ❞❡ F1(·) ❡ ❞❡

X

s>t

ln

1−(F∗

1(s−) +F2∗(s−))

1−(F∗

1(s+) +F2∗(s+))

= X

s>t

ln

1−F(s−₎

1−F(s+₎

= X

s>t

ln

(1−F1(s−))(1−F2(s−))

(1−F1(s+))(1−F2(s+))

= X

s>t

ln

(1−F1(s−))

(1−F1(s+))

= X

s>t

ln1−λ1_d(s).

❆ ❊q✉❛çã♦ ✭✷✳✸✳✶✺✮ é ❝♦♥s❡q✉ê♥❝✐❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ✷✳✸✳✶ ❡ ✷✳✸✳✻ ❡ ❞❡ Z ∞

t

dF∗

1(s)

F∗

1(s) +F2∗(s)

= Z ∞

t

F2(s)dF1(s)

F1(s)F2(s)

= Z ∞

t

dF1(s)

F1(s)

= Z ∞

t

µ1_c(s)ds.

(18)

✶✻

q✉❛♥❞♦s é ♣♦♥t♦ ❞❡ s❛❧t♦ ❞❡ F1(·) ❡ ❞❡

X

s>t

ln

F∗

1(s−) +F2∗(s−)

F∗

1(s+) +F2∗(s+)

= X

s>t

ln

F1(s−)F2(s−)

F1(s+)F2(s+)

= X

s>t

ln

F1(s−)

F1(s+)

= X

s>t

ln1−µ1_d(s).

❉❡st❛ ❢♦r♠❛✱ ♦s ❢✉♥❝✐♦♥❛✐s Φs(·) ❡ Φp(·) sã♦ ❛s ✐♥✈❡rs❛s ❞❛s ❡q✉❛çõ❡s ✭✷✳✸✳✷✮ ❡

✭✷✳✸✳✸✮✳ ❈♦♠ ✐ss♦ t❡♠♦s ❛s ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ F1 ❡ F2 ❡①♣r❡ss❛s ❡♠ t❡r♠♦s ❞❛s

s✉❜✲❞✐str✐❜✉✐çõ❡sF∗

1 ❡ F2∗✳ ❖❜✈✐❛♠❡♥t❡✱ ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦ s❡♠❡❧❤❛♥t❡ ♣❛r❛ ♦ ❝♦♠♣♦✲

(19)

✶✼

✸ ❊st✐♠❛❞♦r ❇❛②❡s✐❛♥♦ ♥ã♦

♣❛r❛♠étr✐❝♦

✸✳✶ ■♥tr♦❞✉çã♦

❈♦♥s✐❞❡r❛♠♦s ❛❣♦r❛ ✉♠❛ ❛♠♦str❛ ❞❡ t❛♠❛♥❤♦ n ❞♦ ♣❛r (T, δ)✳ ❖❜s❡r✈❛r ♦s ❞❛❞♦s (T1, δ1), . . . ,(Tn, δn) ❡q✉✐✈❛❧❡ ❛ ♦❜s❡r✈❛r✱ ♣❛r❛ ❝❛❞❛ t > 0✱ ♦ ✈❡t♦r ❞❡ ❝♦♥t❛❣❡♠ ❛❧❡❛tór✐❛

nF∗

n(t) = (nF

∗

1n(t), nF2∗n(t), n(1−Fn(t))) q✉❡ ❛ss✉♠✐♠♦s t❡r ❞✐str✐❜✉✐çã♦ tr✐♥♦♠✐❛❧ ❝♦♠

❛♠♦str❛ ❞❡ t❛♠❛♥❤♦ n ❡ ♣❛râ♠❡tr♦s(F∗

1(t), F2∗(t); 1−F(t))✱ ❡♠ q✉❡

F∗

n(t) =

1

n

n X

i=1

I(Ti ≤t),

✭✸✳✶✳✶✮

F∗

jn(t) =

1

n

n X

i=1

I(Ti ≤t, δi =j), j = 1,2,

✭✸✳✶✳✷✮

❡♠ q✉❡ F∗

n(t) é ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❡♠♣ír✐❝❛ ❞♦ s✐st❡♠❛✱ Fjn∗ (t) é ❛ ❢✉♥çã♦ s✉❜✲

❞✐str✐❜✉✐çã♦ ❡♠♣ír✐❝❛ ❞♦ j✲és✐♠♦ ❝♦♠♣♦♥❡♥t❡✱ I(A) é ❛ ❢✉♥çã♦ ✐♥❞✐❝❛❞♦r❛ ❞♦ ❝♦♥❥✉♥t♦

A❡ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳✻✱F∗

1(t) +F2∗(t) + (1−F(t)) = 1✳ ◆♦t❡ q✉❡✱ nFjn∗ (t) é ❛ q✉❛♥t✐✲

❞❛❞❡ ❞❡ ❢❛❧❤❛s ❞♦ s✐st❡♠❛ ❛té ♦ t❡♠♣♦t❞❡✈✐❞♦ ❛♦ ❝♦♠♣♦♥❡♥t❡j❡n(1−Fn(t))é ♦ t♦t❛❧ ❞❡

❛♠♦str❛s ❞♦ s✐st❡♠❛ q✉❡ ♥ã♦ ❢❛❧❤❛r❛♠ ❛té ♦ t❡♠♣♦t✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ✐♥❢♦r♠❛çã♦ r❡❢❡r❡♥t❡

❛♦s t❡♠♣♦s ❞❡ ❢❛❧❤❛ ✭❞♦ s✐st❡♠❛ ❡ ❝♦♠♣♦♥❡♥t❡s✮ ❝♦♥t✐❞❛ ❡♠ nF∗

n(t) é ❡q✉✐✈❛❧❡♥t❡ à ✐♥✲

❢♦r♠❛çã♦ ❝♦♥t✐❞❛ ❡♠ (T1, δ1), . . . ,(Tn, δn)✳ ❆❧é♠ ❞✐ss♦✱ nFjn∗ (t) ♦❝♦rr❡ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡

❞❛❞❛ ♣❡❧❛ ✈❡r❞❛❞❡✐r❛ s✉❜✲❞✐str✐❜✉✐çã♦F∗

j(t)✱ ❞❡s❝♦♥❤❡❝✐❞❛✳

✸✳✶✳✸ ❉❡✜♥✐çã♦✳ ❙❡❥❛ (X;A) ✉♠ ❡s♣❛ç♦ ♠❡♥s✉rá✈❡❧ ❡ α1, α2 ♠❡❞✐❞❛s ✜♥✐t❛s✱ ♥ã♦✲

♥❡❣❛t✐✈❛s ❡ ♥ã♦✲♥✉❧❛s ❡♠ (X;A)✳ ❙❡❥❛ ρ = (ρ1, ρ2)❀ P1, P2 ❡❧❡♠❡♥t♦s ❛❧❡❛tór✐♦s ♠✉t✉✲

❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡✜♥✐❞♦s ❡♠ (Ω;F;Q)✳ ❙✉♣♦♥❤❛ q✉❡ ρ t❡♥❤❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡

❉✐r✐❝❤❧❡t D(α1(X), α2(X)) ❡ Pj é ✉♠ ♣r♦❝❡ss♦ ❞❡ ❉✐r✐❝❤❧❡t ❝♦♠ ♣❛râ♠❡tr♦ αj✱ ♦✉ s❡❥❛✱

P j ∼D(αj)✱ ❥ ❂ ✶✱✷✳ ❉❡✜♥✐♠♦s P∗ = (P1∗, P2∗) = (ρ1P1, ρ2P2)✳ ❊♥tã♦ P∗ é ✉♠ ♣r♦❝❡ss♦

❞❡ ❉✐r✐❝❤❧❡t ❜✐✈❛r✐❛❞♦ ❝♦♠ ♣❛râ♠❡tr♦ α = (α1, α2)✱ ♦✉ s❡❥❛✱ P∗ ∼ DM2(α) ✭❝❢✳✱ P♦❧♣♦ ❡

(20)

✶✽

◆♦ ❝♦♥t❡①t♦ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✱ s❡❥❛♠ α1, α2 ♠❡❞✐❞❛s ✜♥✐t❛s✱ ♥ã♦✲♥❡❣❛t✐✈❛s ❡ ♥ã♦✲

♥✉❧❛s ❡♠ ((0,∞),B(0,∞))✳ ❙❡❥❛♠ t❛♠❜é♠ ρ = (ρ1, ρ2) = (Pr(δ = 1),Pr(δ = 2)) ∼

Ds(α1(0,∞), α2(0,∞))✱Pj∗ = Pr(T ≤t|δ =j)✱P∗ ∼D(αj)✱ ❥❂✶✱✷✳ ❙✉♣♦♥❤❛ q✉❡ρ,P∗₁,P∗₂

sã♦ ♠✉t✉❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❊♥tã♦✱ ❛ ♣r✐♦r✐ ♣❛r❛ ❋∗ _{é ❞❛❞❛ ♣♦r ❋}∗

= (ρ1T1∗, ρ2T2∗)∼

DM2(α1, α2)✳

❆ ♣r✐♦r✐ ✐♥❞✉③✐❞❛ ♣❛r❛ F∗

1 é ❞❛❞❛ ♣♦r✿

✭✸✳✶✳✹✮ F∗

1(t)∼Beta(c2F1∗,0(t), c2(1−F1∗,0(t))), t >0,

❡♠ q✉❡ c2 = P2_j₌₁αj(0,∞) ❡ F1∗,0(t) = α1(0, t)/c2 é ❛ ♠é❞✐❛ ❛ ♣r✐♦r✐ ❞❡ F1∗✳ ❚❛♠❜é♠✱

F0(t) = F1,0(t) +F2,0(t) é ❛ ♠é❞✐❛ ♣r✐♦r✐ ❞❡ F(·)✳

❈♦♥s✐❞❡r❛♥❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ❉✐r✐❝❤❧❡t ❝♦♠♦ ♣r✐♦r✐ ♣❛r❛ F∗₍_t)✱ ❛ ❞✐str✐❜✉✐çã♦ ❛ ♣♦st❡✲

r✐♦r✐ ❞❡F∗₍_t₎ _{é ✉♠ ♣r♦❝❡ss♦ ❞❡ ❉✐r✐❝❤❧❡t ❜✐✈❛r✐❛❞♦ ❞❛❞♦ ♣♦r✿}

F∗(t)|n_F∗

n(t)∼P D(α1(t,∞) +nF1∗n(t), α2(t,∞) +nF2∗n(t);

α1(0, t] +α2(0, t] +n(1−Fn(t))),

✭✸✳✶✳✺✮

❡♠ q✉❡ F∗

jn é ❝♦♠♦ ❞❡✜♥✐❞♦ ♥❛ ❊q✉❛çã♦ ✭✸✳✶✳✷✮✳

❙❡❥❛ pn=c2/(c2+n)✱ ♦s ❡st✐♠❛❞♦r❡s ❇❛②❡s✐❛♥♦s ❞❡F1∗(·) ❡F(·) sã♦ ❞❛❞♦s ♣♦r

✭✸✳✶✳✻✮ _Fˆ∗

1(t) =pnF1∗,0(t) + (1−pn)F1∗n(t)

❡

✭✸✳✶✳✼✮ _Fˆ₍_t_{) = ˆ}_F∗

1(t) + ˆF2∗(t)

◆♦t❡ q✉❡✱ ♣❛r❛ ❡st✐♠❛çã♦ ❞❡F∗

2 ❜❛st❛ ❛❧t❡r❛r ❛♣r♦♣r✐❛❞❛♠❡♥t❡ ❛ ❊q✉❛çã♦ ✭✸✳✶✳✻✮✳ ◆♦

r❡st❛♥t❡ ❞♦ t❡①t♦ ♥♦s r❡❢❡r✐♠♦s ❛♣❡♥❛s à ❡st✐♠❛çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✶✱ ✉♠❛ ✈❡③ q✉❡ ♣❛r❛ ♦ ❝♦♠♣♦♥❡♥t❡ ✷ ♦s r❡s✉❧t❛❞♦s sã♦ ❛♥á❧♦❣♦s✳

✸✳✷ ❙✐st❡♠❛ ❡♠ ❙ér✐❡

❙❡❥❛♠ m (m 6 n) ❡st❛tíst✐❝❛s ❞❡ ♦r❞❡♠ ❞✐st✐♥t❛s✱ ❞❡ T, T(1) < · · · < T(m)✱ ni = Pn

(21)

✶✾

❉❡✜♥✐♠♦s

✭✸✳✷✳✶✮ i(t) = exp

(

−1

c2+n

Z t

0

dα1(0, s]

1−Fˆ(s) )

❡

✭✸✳✷✳✷✮ π(t) = Y

i:T(i)6t

α1(T(i),∞) +α2(T(i),∞) +ni−di

α1(T(i),∞) +α2(T(i),∞) +ni

,

✸✳✷✳✸ ❚❡♦r❡♠❛✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❢✉♥çã♦ f(s) = (α1(0, s],· · · , αr(0, s]) s❡❥❛ ❝♦♥tí♥✉❛ ❡♠

(0, t)✱ ♣❛r❛ ❝❛❞❛ t >0✱ ❡ F1 ❡ F2 ♥ã♦ t❡♥❤❛♠ ♣♦♥t♦s ❞❡ s❛❧t♦ ❡♠ ❝♦♠✉♠✱ ❡♥tã♦✱ ♣❛r❛ ❝❛❞❛

t≤T(m)✱ t❡♠♦s

✭✸✳✷✳✹✮ _Fˆ₁₍_t_{) = Φ}_s_{( ˆ}_F∗

1(·),Fˆ

∗

2(·), t) = 1−i(t)π(t).

❉❡♠♦♥str❛çã♦✳ ❋❛③❡♥❞♦ ❞❡ ❢♦r♠❛ s✐♠✐❧❛r à P❡t❡rs♦♥ ❬✽❪✱ s✉❜st✐t✉í♠♦s ♦s ❡st✐♠❛❞♦r❡s ❇❛②❡s✐❛♥♦s ❞❡F∗

1 ❡ F2∗ ♥❛ ❊q✉❛çã♦ ✭✷✳✸✳✶✶✮ ❡ t❡♠♦s

✭✸✳✷✳✺✮ _Fˆ₁₍_t_{) = 1}₋_exp"Z t

0

−dFˆ∗

1(s)

1−Fˆ(s) #

Y

s≤t

1−P2_j₌₁Fˆ∗

j(s+)

1−P2_j₌₁Fˆ∗

j(s−)

, t < T(m),

❡♠ q✉❡ Qs≤t é ♦ ♣r♦❞✉t♦ s♦❜r❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ s❛❧t♦ s ❞❡ Fˆ

∗

1 ❡ ❛ ✐♥t❡❣r❛❧ é s♦❜ ♦s

✐♥t❡r✈❛❧♦s ❞✐s❥✉♥t♦s q✉❡ ♥ã♦ ✐♥❝❧✉❡♠ ♦s ♣♦♥t♦s ❞❡ s❛❧t♦ ❞❡_Fˆ∗

1(s)✳

❉❛ ❊q✉❛çã♦ ✭✸✳✶✳✻✮ t❡♠♦s

ˆ

F∗

1 =

c2F1∗,0(t)

c2+n

+ (1− c2

c2+n

)F∗

1n(t)

= c2F

∗

1,0(t)

c2+n

+nF

∗

1n(t)

c2+n

= α1(0, t]

c2+n

+ Pn

i=1I(Ti ≤t, δi = 1)

c2+n

.

❆ ❢✉♥çã♦Pn

i=1I(Ti ≤t, δi = 1)é ❝♦♥st❛♥t❡✱ ❡①❝❡t♦ ♥♦s ♣♦♥t♦s ❞❡ s❛❧t♦✱ ❛ss✐♠ s✉❛ ❞❡r✐✈❛❞❛

é ③❡r♦ ♣❛r❛ ✐♥t❡r✈❛❧♦s ❝♦♥tí♥✉♦s✳ ❊♥tã♦✱

dFˆ∗

1(t) =

dα1(0, t]

c2+n

❡ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❞❛ ❊q✉❛çã♦ ✭✸✳✷✳✺✮ t♦r♥❛✲s❡i(t)✳ P❛r❛ ❝❛❞❛ t >0✜①❛❞♦✱ αj(0,·], j =

1,2 sã♦ ❢✉♥çõ❡s ♠♦♥ót♦♥❛s ❡ ❝♦♥tí♥✉❛s ❡♠ (0, t)✱ ❡ (1/Fˆ(·)) é ♠♦♥ót♦♥❛ ❡♠ (0, t)✳ P♦r✲

t❛♥t♦✱αj(0,·], j = 1, . . . , k✱ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ✉♥✐❝❛♠❡♥t❡ ❝♦♠♦ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❢✉♥çõ❡s

(22)

✷✵

é ❜❡♠ ❞❡✜♥✐❞❛✳ ❆❧é♠ ❞✐ss♦✱ ♦ s❡❣✉♥❞♦ ❢❛t♦r ❞❛ ❊q✉❛çã♦ ✭✸✳✷✳✺✮ é

Y

s≤t

1−P2_j₌₁αj(0, s] +Pn_i₌₁I(Ti ≥s+)

1−P2_j₌₁αj(0, s] +Pn_i₌₁I(Ti ≥s−)

=Y

s≤t P2

j=1αj(s,∞) +Pin=1I(Ti ≥s+) P2

j=1αj(s,∞) +Pin=1I(Ti ≥s−)

=π(t),

❡♠ q✉❡ s é ✉♠ ♣♦♥t♦ ❞❡ s❛❧t♦ ❞❡Fˆ∗

1✱ ♣♦r ✜♠ t < T(m) s❡❣✉❡ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳✽✳

✸✳✸ ❙✐st❡♠❛ ❡♠ P❛r❛❧❡❧♦

❙❡❥❛♠ m (≤ n) ❡st❛tíst✐❝❛s ❞❡ ♦r❞❡♠ ❞✐st✐♥t❛s ❞❡ ❚✱ T(1) < · · · < T(m)✳ Ni = Pn

j=1I(Tj < T(i))✱ ❡Di =Pnj=1I(Tj =T(i), δj = 1), i= 1, . . . , m✳ ❉❡✜♥✐♠♦s

✭✸✳✸✳✶✮ I(t) = exp

"

−1

n+c2

Z ∞

t

dα1(0, s]

ˆ

F(s) #

❡

✭✸✳✸✳✷✮ Π(t) = Y

i:T(i)>t

P2

j=1αj(0, T(i)] +Ni P2

j=1αj(0, T(i)] +Ni+Di

,

✸✳✸✳✸ ❚❡♦r❡♠❛✳ ❙✉♣♦♥❤❛ q✉❡ α1(0,·), α2(0,·) sã♦ ❝♦♥tí♥✉♦s ❡♠ (t,∞)✱ ♣❛r❛ ❝❛❞❛ t > 0✱

❡ F1 ❡ F2 ♥ã♦ tê♠ ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ ❝♦♠✉♠✳ ❊♥tã♦✱ ♣❛r❛ t < T(m)✱

✭✸✳✸✳✹✮ _Fˆ₁₍_t_{) = Φ}_p_{( ˆ}_F∗

1,Fˆ2∗, t) = I(t)Π(t),

é ♦ ❡st✐♠❛❞♦r ❇❛②❡s✐❛♥♦ ♥ã♦ ♣❛r❛♠étr✐❝♦ ❞❡F1(t) ❜❛s❡❛❞♦ ♥❛ ♠é❞✐❛ ♣♦st❡r✐♦r✐✳

❉❡♠♦♥str❛çã♦✳ ❋❛③❡♥❞♦ ❞❡ ❢♦r♠❛ s✐♠✐❧❛r à P♦❧♣♦ ❡ P❡r❡✐r❛ ❬✶✵❪✱ s✉❜st✐t✉í♠♦s ♦s ❡st✐♠❛✲ ❞♦r❡s ❇❛②❡s✐❛♥♦s ❞❡F∗

1 ❡F2∗ ♥❛ ❊q✉❛çã♦ ✭✷✳✸✳✶✷✮ ❡ t❡♠♦s

✭✸✳✸✳✺✮ _Fˆ₁₍_t_{) = exp}"Z ∞

t

−dFˆ∗

1(s)

ˆ

F(s) #

Y

s>t P2

j=1Fˆ

∗

j(s−) P2

j=1Fˆ

∗

j(s+)

, t < T(m),

❡♠ q✉❡ Qs>t é ♦ ♣r♦❞✉t♦ s♦❜r❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ s❛❧t♦ s ❞❡ Fˆ1∗ ❡ ❛ ✐♥t❡❣r❛❧ é s♦❜ ♦s

✐♥t❡r✈❛❧♦s ❞✐s❥✉♥t♦s q✉❡ ♥ã♦ ✐♥❝❧✉❡♠ ♦s ♣♦♥t♦s ❞❡ s❛❧t♦ ❞❡_Fˆ∗

1(s)✳

◆♦t❡ q✉❡✱ ❝♦♠♦ ✈✐st♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✷✳✸✱ dFˆ∗

1(s) =dα1(0, s]/(c2+n)

❡ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❞❛ ❊q✉❛çã♦ ✭✸✳✸✳✺✮ t♦r♥❛✲s❡I(t)✳ P❛r❛ ❝❛❞❛t >0✜①❛❞♦✱ αj(0,·), j =

(23)

✷✶

t❛♥t♦✱αj(0,·), j = 1, . . . , k✱ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ✉♥✐❝❛♠❡♥t❡ ❝♦♠♦ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❢✉♥çõ❡s

♠♦♥ót♦♥❛ ❝♦♥tí♥✉❛ ❡ (1/Fˆ(.))✱ ❢✉♥çã♦ ♠♦♥ót♦♥❛✳ ❆ss✐♠✱ ❛ ✐♥t❡❣r❛❧ R_t∞[dα1(0, s]/Fˆ(s)] é

❜❡♠ ❞❡✜♥✐❞❛✳ ❆❧é♠ ❞✐ss♦✱ ♦ s❡❣✉♥❞♦ ❢❛t♦r ❞❛ ❊q✉❛çã♦ ✭✸✳✸✳✺✮ é

Y

s>t P2

j=1αj(0, s] +Pin=1I(Ti ≤s−) P2

j=1αj(0, s] +Pin=1I(Ti ≤s+)

= Π(t).

❡♠ q✉❡ s é ✉♠ ♣♦♥t♦ ❞❡ s❛❧t♦ ❞❡Fˆ∗

(24)

✷✷

✹ ●❡♥❡r❛❧✐③❛çã♦

✹✳✶ ■♥tr♦❞✉çã♦

❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ s✐st❡♠❛s ❡♠ sér✐❡ ❡ ❡♠ ♣❛r❛❧❡❧♦ ❝♦♠ k ❝♦♠✲

♣♦♥❡♥t❡s✱ k ≥ 2✳ P❛r❛ ✐ss♦ ❞✐✈✐❞✐♠♦s ♦s k ❝♦♠♣♦♥❡♥t❡s ❡♠ ❞♦✐s ❣r✉♣♦s✱ ∆ ❡ ∆c_{✱ ❞❡ t❛❧}

❢♦r♠❛ q✉❡ n(∆) +n(∆c_{) =}_k_{✳ ❉❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡✱ s✉❜st✐t✉✐r❡♠♦s ❛s ❢✉♥çõ❡s} _F

1 ♣♦r F∆

❡F2 ♣♦rF∆c✳

❖❜s❡r✈❛r ♦s ❞❛❞♦s (T1, δ1), . . . ,(Tn, δn) ❡q✉✐✈❛❧❡ ❛ ♦❜s❡r✈❛r✱ ♣❛r❛ ❝❛❞❛ t >0✱ ♦ ✈❡t♦r

❞❡ ❝♦♥t❛❣❡♠ ❛❧❡❛tór✐❛ nF∗

n(t) = (nF

∗

∆n(t), nF∆∗c_n(t), n(1−F_n(t))) q✉❡ t❡♠ ❞✐str✐❜✉✐çã♦ tr✐♥♦♠✐❛❧ ❝♦♠ ❛♠♦str❛ ❞❡ t❛♠❛♥❤♦ n ❡ ♣❛râ♠❡tr♦s(F∗

∆(t), F∆∗c(t); 1−F(t))✱ ❡♠ q✉❡

F∗

n(t) =

1

n

n X

i=1

I(Ti ≤t),

✭✹✳✶✳✶✮

F∗

jn(t) =

1

n

n X

i=1

I(Ti ≤t, δi =j), j = 1,· · · , r,

✭✹✳✶✳✷✮

❡♠ q✉❡F∗

n(t)❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❡♠♣ír✐❝❛ ❞♦ s✐st❡♠❛✱ Fjn∗ (t)é ❛ ❢✉♥çã♦ s✉❜✲❞✐str✐❜✉✐çã♦

❡♠♣ír✐❝❛ ❞♦ j✲és✐♠♦ ❝♦♠♣♦♥❡♥t❡✱ I(A) é ❛ ❢✉♥çã♦ ✐♥❞✐❝❛❞♦r❛ ❞♦ ❝♦♥❥✉♥t♦ A ❡ ♣❡❧❛ Pr♦✲

♣r✐❡❞❛❞❡ ✷✳✸✳✻✱ F∗

∆(t) +F∆∗c(t) + (1−F(t)) = 1✳ ◆♦t❡ q✉❡✱ nF∗

jn(t) é ❛ q✉❛♥t✐❞❛❞❡ ❞❡

❢❛❧❤❛s ❞♦ s✐st❡♠❛ ❛té ♦ t❡♠♣♦ t ❞❡✈✐❞♦ ❛♦ ❝♦♠♣♦♥❡♥t❡ j ❡ n(1−Fn(t)) é ♦ t♦t❛❧ ❞❡

❛♠♦str❛s ❞♦ s✐st❡♠❛ q✉❡ ♥ã♦ ❢❛❧❤❛r❛♠ ❛té ♦ t❡♠♣♦ t✳ ❉❡st❛ ❢♦r♠❛✱ ♣❛r❛ t♦❞♦s ♦s t✱ ❛

✐♥❢♦r♠❛çã♦ r❡❢❡r❡♥t❡ ❛♦s t❡♠♣♦s ❞❡ ❢❛❧❤❛ ✭❞♦ s✐st❡♠❛ ❡ ❝♦♠♣♦♥❡♥t❡s✮ ❝♦♥t✐❞❛ ❡♠n_F∗

n(t)

é ❡q✉✐✈❛❧❡♥t❡ ❛ ✐♥❢♦r♠❛çã♦ ❝♦♥t✐❞❛ ❡♠ (T1, δ1), . . . ,(Tn, δn)✳ ❆❧é♠ ❞✐ss♦✱ nFjn∗ (t) ♦❝♦rr❡

❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛❞❛ ♣❡❧❛ ✈❡r❞❛❞❡✐r❛ s✉❜✲❞✐str✐❜✉✐çã♦F∗

j(t)✱ ❞❡s❝♦♥❤❡❝✐❞❛✳

✹✳✶✳✸ ❉❡✜♥✐çã♦✳ ❙❡❥❛ (X;A) ✉♠ ❡s♣❛ç♦ ♠❡♥s✉rá✈❡❧ ❡ α1,· · ·, αr ♠❡❞✐❞❛s ✜♥✐t❛s✱ ♥ã♦✲

♥❡❣❛t✐✈❛s ❡ ♥ã♦✲♥✉❧❛s ❡♠ (X;A)✳ ❙❡❥❛ ρ = (ρ1,· · · , ρr)❀ P1,· · · , Pr ❡❧❡♠❡♥t♦s ❛❧❡❛tór✐♦s

♠✉t✉❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡✜♥✐❞♦s ❡♠(Ω;F;Q)✳ ❙✉♣♦♥❤❛ q✉❡ρt❡♥❤❛ ✉♠❛ ❞✐str✐❜✉✐çã♦

❞❡ ❉✐r✐❝❤❧❡t D(α1(X),· · · , αr(X))❡ Pj é ✉♠ ♣r♦❝❡ss♦ ❞❡ ❉✐r✐❝❤❧❡t ❝♦♠ ♣❛râ♠❡tr♦ αj✱ ♦✉

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✷✸

❊♥tã♦ P∗ _{é ✉♠ ♣r♦❝❡ss♦ ❞❡ ❉✐r✐❝❤❧❡t ♠✉❧t✐✈❛r✐❛❞♦ ❝♦♠ ♣❛râ♠❡tr♦}

α = (α1,· · · , αr)✱ ♦✉

s❡❥❛✱ P∗ _∼_DM

r(α) ✭❝❢✳✱ P♦❧♣♦ ❡ P❡r❡✐r❛ ❬✶✵❪ ❡ ❙❛❧✐♥❛s ❬✶✸❪✮✳

◆♦ ❝♦♥t❡①t♦ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✱ s❡❥❛♠ α1,· · · , αr ♠❡❞✐❞❛s ✜♥✐t❛s✱ ♥ã♦✲♥❡❣❛t✐✈❛s ❡ ♥ã♦✲

♥✉❧❛s ❡♠ ((0,∞),B(0,∞))✳ ❙❡❥❛♠ t❛♠❜é♠ ρ = (ρ1,· · · , ρr) = (Pr(δ = 1),· · · ,Pr(δ =

r)) ∼ Ds(α1(0,∞),· · ·, αr(0,∞))✱Pj∗ = Pr(T ≤ t|δ = j)✱ P∗ ∼ D(αj)✱ j = 1,· · · , r✳

❙✉♣♦♥❤❛ q✉❡ ρ,_P∗

1,· · · ,P

∗

r sã♦ ♠✉t✉❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❊♥tã♦✱ ❛ ♣r✐♦r✐ ♣❛r❛ ❋

∗ _é

❞❛❞❛ ♣♦r ❋∗

= (ρ1T1∗,· · · , ρrTr∗)∼DM(α1,· · · , αr)✳

❆ ♣r✐♦r✐ ✐♥❞✉③✐❞❛ ♣❛r❛ F∗

∆ é ❞❛❞❛ ♣♦r✿

✭✹✳✶✳✹✮ F∗

∆(t)∼Beta(crF∆∗,0(t), cr(1−F∆∗,0(t))), t >0,

❡♠ q✉❡ cr =Pr_j₌₁αj(0,∞) ❡ F∆∗,0(t) =

P

i∈∆αi(0, t)/cr, i = 1,· · ·, r é ❛ ♠é❞✐❛ ❛ ♣r✐♦r✐

❞❡F∗

∆✳ ❚❛♠❜é♠✱ F0(t) =F∆,0(t) +F∆c_,₀(t)é ❛ ♠é❞✐❛ ♣r✐♦r✐ ❞❡ F(·)✳

❈♦♥s✐❞❡r❛♥❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ❉✐r✐❝❤❧❡t ❝♦♠♦ ♣r✐♦r✐ ♣❛r❛ F∗₍_t)✱ ❛ ❞✐str✐❜✉✐çã♦ ❛ ♣♦st❡✲

r✐♦r✐ ❞❡F∗₍_t₎ _{é ✉♠ ♣r♦❝❡ss♦ ❞❡ ❉✐r✐❝❤❧❡t ♠✉❧t✐✈❛r✐❛❞♦ ❞❛❞♦ ♣♦r✿}

F∗(t)|n_F∗

n(t)∼P D(α1(t,∞) +nF1∗n(t),· · · , αr(t,∞) +nFkn∗ (t);

α1(0, t] +· · ·+αr(0, t] +n(1−Fn(t))),

✭✹✳✶✳✺✮

❡♠ q✉❡ F∗

jn é ❝♦♠♦ ❞❡✜♥✐❞♦ ♥❛ ❊q✉❛çã♦ ✭✹✳✶✳✷✮✳

❙❡❥❛ pn=cr/(cr+n)✱ ♦s ❡st✐♠❛❞♦r❡s ❇❛②❡s✐❛♥♦s ❞❡ F∆∗(·) ❡ F(·) sã♦ ❞❛❞♦s ♣♦r

✭✹✳✶✳✻✮ _Fˆ∗

∆=pnF∆∗,0(t) + (1−pn)F∆∗n(t)

❡

✭✹✳✶✳✼✮ _Fˆ₍_t_{) = ˆ}_F∗

∆(t) + ˆF∆∗c(t) ◆♦t❡ q✉❡✱ ♣❛r❛ ❡st✐♠❛çã♦ ❞❡F∗

∆ ❜❛st❛ ❛❧t❡r❛r ❛♣r♦♣r✐❛❞❛♠❡♥t❡ ❛ ❊q✉❛çã♦ ✭✹✳✶✳✻✮✳ ◆♦

r❡st❛♥t❡ ❞♦ t❡①t♦ ♥♦s r❡❢❡r✐♠♦s ❛♣❡♥❛s ❛ ❡st✐♠❛çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ∆✱ ✉♠❛ ✈❡③ q✉❡ ♣❛r❛ ♦ ❝♦♠♣♦♥❡♥t❡ ∆c _{♦s r❡s✉❧t❛❞♦s sã♦ ❛♥á❧♦❣♦s✳}

✹✳✷ ❙✐st❡♠❛ ❡♠ ❙ér✐❡

❙❡❥❛♠ m (m 6 n) ❡st❛tíst✐❝❛s ❞❡ ♦r❞❡♠ ❞✐st✐♥t❛s✱ ❞❡ T, T(1) < · · · < T(m)✱ ni = Pn

(26)

✷✹

❉❡✜♥✐♠♦s

✭✹✳✷✳✶✮ i∆(t) = exp

(

−1

cr+n X

j∈∆

Z t

0

dαj(0, s]

1−Fˆ(s) )

❡

✭✹✳✷✳✷✮ π∆(t) =

Y

i:T(i)6t Pr

j=1αj(T(i),∞) +ni−d∆i Pr

j=1αj(T(i),∞) +ni

,

✹✳✷✳✸ ❚❡♦r❡♠❛✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❢✉♥çã♦ f(s) = (α1(0, s],· · · , αr(0, s]) s❡❥❛ ❝♦♥tí♥✉❛ ❡♠

(0, t)✱ ♣❛r❛ ❝❛❞❛ t > 0✱ ❡ F∆ ❡ F∆c ♥ã♦ t❡♥❤❛♠ ♣♦♥t♦s ❞❡ s❛❧t♦ ❡♠ ❝♦♠✉♠✱ ❡♥tã♦✱ ♣❛r❛ ❝❛❞❛ t≤T(m)✱ t❡♠♦s

✭✹✳✷✳✹✮ _Fˆ_∆₍_t_{) = Φ}_s_{( ˆ}_F∗

∆(·),Fˆ

∗

∆c(·), t) = 1−i_∆(t)π_∆(t).

❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ é ❛♥á❧♦❣❛ ❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✷✳✸✱ s✉❜st✐t✉✐♥❞♦F1 ♣♦r F∆c ❡F2 ♣♦rF∆c✱ ❡ ❢❛③❡♥❞♦ ❛s ♠♦❞✐✜❝❛çõ❡s ♥❡❝❡ssár✐❛s✳

✹✳✷✳✺ ❊①❡♠♣❧♦✳ ❚❡♠♦s ✉♠ s✐st❡♠❛ ❝♦♠ três ❝♦♠♣♦♥❡♥t❡s ❧✐❣❛❞♦s ❡♠ sér✐❡✱ X1 t❡♠

❞✐str✐❜✉✐çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❝♦♠ ♠é❞✐❛ 1.4 ❡ ✈❛r✐â♥❝✐❛ 0.5✱ X2 t❡♠ ❞✐str✐❜✉✐çã♦ ❣❛♠❛ ❝♦♠

♠é❞✐❛1.4❡ ✈❛r✐â♥❝✐❛ 3.2❡X3 t❡♠ ❞✐str✐❜✉✐çã♦ ❧♦❣✲♥♦r♠❛❧ ❝♦♠ ♠é❞✐❛ 1.5❡ ✈❛r✐â♥❝✐❛ 1.3✳

❙✐♠✉❧❛♠♦s ♦❜s❡r✈❛çõ❡s ❞♦ s✐st❡♠❛ ❝♦♥s✐❞❡r❛♥❞♦ três ❝❛s♦s ❞✐st✐♥t♦s ❡♠ q✉❡ ♦s t❛♠❛✲ ♥❤♦s ❛♠♦str❛✐s ❢♦r❛♠n = 1000✱ n= 100❡n = 30✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♠♦ ♦ s✐st❡♠❛ ❡stá ❧✐❣❛❞♦ ❡♠ sér✐❡✱ ♦ ♣r✐♠❡✐r♦ ❝♦♠♣♦♥❡♥t❡ ❛ ❢❛❧❤❛r é ♦ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛✱ ❡ ♦s ♦✉tr♦s sã♦ ❝❡♥s✉r❛❞♦s✳ ❈♦♠ ✐ss♦ ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

P❛r❛ n= 30

• ❖ ❝♦♠♣♦♥❡♥t❡ 1 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r15 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 2 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r11 ✈❡③❡s✳

• ❖ ❈♦♠♣♦♥❡♥t❡ 3 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r4 ✈❡③❡s✳

❈❛❞❛ ✉♠ ❞❡ss❡s ❝♦♠♣♦♥❡♥t❡s t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ α ❛ ♣r✐♦r✐ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦✲

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✷✺

0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

✹✳✷✳✻ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛ ❡♠ sér✐❡ ✭❝✉r✈❛ ❝♦♥tí♥✉❛ ❡s❝✉r❛✿ ❡st✐✲ ♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛✳✮

❖❜s❡r✈❛♥❞♦ ♦ ❣rá✜❝♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❡st✐♠❛❞❛ ❡stá ♠✉✐t♦ ♣ró①✐♠❛ ❞❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❣❡r❛❞♦r❛ ❞♦s ❞❛❞♦s✳

❆❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ ❛ ❊q✉❛çã♦ ✭✹✳✷✳✹✮✱ ♦❜t❡♠♦s ❛ ❡st✐♠❛t✐✈❛ ❞❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦

(28)

✷✻

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F(t)

(29)

✷✼

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tempo

F(t)

(30)

✷✽

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tempo

F(t)

✹✳✷✳✾ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸ ♥♦ s✐st❡♠❛ ❡♠ sér✐❡ ✭❝✉r✈❛ ❝♦♥tí✲ ♥✉❛ ❡s❝✉r❛✿ ❡st✐♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸✳✮

❈♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♣❡❧❛s ✜❣✉r❛s✱ ❛s ❡st✐♠❛t✐✈❛s ✜❝❛r❛♠ ♣ró①✐♠❛s ❞❛s ✈❡r❞❛❞❡✐r❛s ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦s ❞❛❞♦s✳

P❛r❛ n= 100

• ❖ ❝♦♠♣♦♥❡♥t❡ 1 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r49 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 2 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r34 ✈❡③❡s✳

• ❖ ❈♦♠♣♦♥❡♥t❡ 3 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r17 ✈❡③❡s✳

❈❛❞❛ ✉♠ ❞❡ss❡s ❝♦♠♣♦♥❡♥t❡s t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ α ❛ ♣r✐♦r✐ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦✲

(31)

✷✾

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tempo

F(t)

✹✳✷✳✶✵ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛ ❡♠ sér✐❡ ✭❝✉r✈❛ ❝♦♥tí♥✉❛ ❡s❝✉r❛✿ ❡st✐✲ ♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛✳✮

(32)

✸✵

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F(t)

(33)

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F(t)

(34)

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tempo

F(t)

✹✳✷✳✶✸ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸ ♥♦ s✐st❡♠❛ ❡♠ sér✐❡ ✭❝✉r✈❛ ❝♦♥✲ tí♥✉❛ ❡s❝✉r❛✿ ❡st✐♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸✳✮

❈♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♣❡❧❛s ✜❣✉r❛s✱ ❛s ❡st✐♠❛t✐✈❛s ✜❝❛r❛♠ ♣ró①✐♠❛s ❞❛s ✈❡r❞❛❞❡✐r❛s ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦s ❞❛❞♦s✳

P❛r❛ n= 1000

• ❖ ❝♦♠♣♦♥❡♥t❡ 1 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r527 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 2 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r328 ✈❡③❡s✳

• ❖ ❈♦♠♣♦♥❡♥t❡ 3 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r145 ✈❡③❡s✳

❈❛❞❛ ✉♠ ❞❡ss❡s ❝♦♠♣♦♥❡♥t❡s t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ α ❛ ♣r✐♦r✐ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦✲

(35)

✸✸

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tempo

F(t)

✹✳✷✳✶✹ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛ ❡♠ sér✐❡ ✭❝✉r✈❛ ❝♦♥tí♥✉❛ ❡s❝✉r❛✿ ❡st✐✲ ♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛✳✮

(36)

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F(t)

(37)

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(38)

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tempo

F(t)

✹✳✷✳✶✼ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸ ♥♦ s✐st❡♠❛ ❡♠ sér✐❡ ✭❝✉r✈❛ ❝♦♥✲ tí♥✉❛ ❡s❝✉r❛✿ ❡st✐♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸✳✮

❈♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♣❡❧❛s ✜❣✉r❛s✱ ❛s ❡st✐♠❛t✐✈❛s ✜❝❛r❛♠ ♣ró①✐♠❛s ❞❛s ✈❡r❞❛❞❡✐r❛s ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦s ❞❛❞♦s✱ s❡♥❞♦ ♣✐♦r ❝♦♥❢♦r♠❡ s❡ ❛❢❛st❛ ❞❛ ♦r✐❣❡♠✳

✹✳✸ ❙✐st❡♠❛ ❡♠ P❛r❛❧❡❧♦

❙❡❥❛♠ m (≤ n) ❡st❛tíst✐❝❛s ❞❡ ♦r❞❡♠ ❞✐st✐♥t❛s ❞❡ ❚✱ T(1) < · · · < T(m)✳ Ni = Pn

j=1I(Tj < T(i))✱ ❡D∆i =Pnj=1I(Tj =T(i), δj ∈∆), i= 1, . . . , m✳ ❉❡✜♥✐♠♦s

✭✹✳✸✳✶✮ I∆(t) = exp

"

−1

n+cr X

j∈∆

Z ∞

t

dαj(0, s]

ˆ

F(s) #

❡

✭✹✳✸✳✷✮ Π∆(t) =

Y

i:T(i)>t

Pr

j=1αj(0, T(i)] +Ni

Pr

j=1αj(0, T(i)] +Ni+D∆i

,

(39)

✸✼

t >0✱ ❡ F∆ ❡ F∆c ♥ã♦ tê♠ ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ ❝♦♠✉♠✳ ❊♥tã♦✱ ♣❛r❛ t < T

(m)✱

✭✹✳✸✳✹✮ _Fˆ_∆₍_t_{) = Φ}_p_{( ˆ}_F∗

∆,Fˆ∆∗c, t) = I_∆(t)Π_∆c(t),

é ♦ ❡st✐♠❛❞♦r ❇❛②❡s✐❛♥♦ ♥ã♦ ♣❛r❛♠étr✐❝♦ ❞❡F∆(t) ❜❛s❡❛❞♦ ♥❛ ♠é❞✐❛ ♣♦st❡r✐♦r✐✳

❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ é ❛♥á❧♦❣❛ ❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✸✳✸✱ s✉❜st✐t✉✐♥❞♦F1 ♣♦r F∆ ❡

F2 ♣♦r F∆c✱ ❡ ❢❛③❡♥❞♦ ❛s ♠♦❞✐✜❝❛çõ❡s ♥❡❝❡ssár✐❛s✳

✹✳✸✳✺ ❊①❡♠♣❧♦✳ ❚❡♠♦s ✉♠ s✐st❡♠❛ ❝♦♠ três ❝♦♠♣♦♥❡♥t❡s ❧✐❣❛❞♦s ❡♠ ♣❛r❛❧❡❧♦✱ X1 t❡♠

❞✐str✐❜✉✐çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❝♦♠ ♠é❞✐❛ 1.4 ❡ ✈❛r✐â♥❝✐❛ 0.5✱ X2 t❡♠ ❞✐str✐❜✉✐çã♦ ❣❛♠❛ ❝♦♠

♠é❞✐❛1.4❡ ✈❛r✐â♥❝✐❛ 3.2❡X3 t❡♠ ❞✐str✐❜✉✐çã♦ ❧♦❣✲♥♦r♠❛❧ ❝♦♠ ♠é❞✐❛ 1.5❡ ✈❛r✐â♥❝✐❛ 1.3✳

❙✐♠✉❧❛♠♦s ♦❜s❡r✈❛çõ❡s ❞♦ s✐st❡♠❛ ✉t✐❧✐③❛♥❞♦ três t❛♠❛♥❤♦s ❞❡ ❛♠♦str❛s✱ n = 1000✱

n= 100 ❡n = 30 ♦❜s❡r✈❛çõ❡s ❞♦ t❡♠♣♦ ❞❡ ✈✐❞❛ ❞♦ s✐st❡♠❛✳ ❈♦♠♦ ♦ s✐st❡♠❛ ❡stá ❧✐❣❛❞♦ ❡♠ ♣❛r❛❧❡❧♦✱ ♦ ú❧t✐♠♦ ❝♦♠♣♦♥❡♥t❡ ❛ ❢❛❧❤❛r é ♦ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛✱ ❡ ♦s ♦✉tr♦s sã♦ ❝❡♥s✉r❛❞♦s✳ ❈♦♠ ✐ss♦ ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝❛❞❛ t❛♠❛♥❤♦ ❞❡ ❛♠♦str❛✿

P❛r❛ n= 30

• ❖ ❝♦♠♣♦♥❡♥t❡ 1 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r11 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 2 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r4 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 3 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r15 ✈❡③❡s✳

❈❛❞❛ ✉♠ ❞❡ss❡s ❝♦♠♣♦♥❡♥t❡s t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ α ❛ ♣r✐♦r✐ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦✲

(40)

✸✽

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

✹✳✸✳✻ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦ ✭❝✉r✈❛ ❝♦♥tí♥✉❛ ❡s❝✉r❛✿ ❡st✐♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲ ▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛✳✮

❖❜s❡r✈❛♥❞♦ ♦ ❣rá✜❝♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❡st✐♠❛❞❛ ❡stá ♣ró①✐♠❛ ❞❛ ✈❡r❞❛❞❡✐r❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦s ❞❛❞♦s✳

❆❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ ❛ ❊q✉❛çã♦ ✭✹✳✸✳✹✮✱ ♦❜t❡♠♦s ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ ❡st✐♠❛❞♦r ❞♦

(41)

✸✾

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

(42)

✹✵

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

(43)

✹✶

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

✹✳✸✳✾ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸ ♥♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦ ✭❝✉r✈❛ ❝♦♥✲ tí♥✉❛ ❡s❝✉r❛✿ ❡st✐♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸✳✮

P❛r❛ n= 100

• ❖ ❝♦♠♣♦♥❡♥t❡ 1 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r27 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 2 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r27 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 3 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r46 ✈❡③❡s✳

❈❛❞❛ ✉♠ ❞❡ss❡s ❝♦♠♣♦♥❡♥t❡s t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ α ❛ ♣r✐♦r✐ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦✲

(44)

✹✷

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

✹✳✸✳✶✵ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦ ✭❝✉r✈❛ ❝♦♥tí♥✉❛ ❡s❝✉r❛✿ ❡st✐♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲ ▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ s✐st❡♠❛✳✮

❖❜s❡r✈❛♥❞♦ ♦ ❣rá✜❝♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❡st✐♠❛❞❛ ❡stá ♣ró①✐♠❛ ❞❛ ✈❡r❞❛❞❡✐r❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦s ❞❛❞♦s✳

❆❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ ❛ ❊q✉❛çã♦ ✭✹✳✸✳✹✮✱ ♦❜t❡♠♦s ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ ❡st✐♠❛❞♦r ❞♦

(45)

✹✸

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

(46)

✹✹

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

(47)

✹✺

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

tempo

F(t)

✹✳✸✳✶✸ ❋✐❣✉r❛✳ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸ ♥♦ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦ ✭❝✉r✈❛ ❝♦♥✲ tí♥✉❛ ❡s❝✉r❛✿ ❡st✐♠❛t✐✈❛ ❇❛②❡s✐❛♥❛ ♥ã♦✲♣❛r❛♠étr✐❝❛❀ ❝✉r✈❛ ❝♦♥tí♥✉❛ ❝❧❛r❛✿ ❡st✐♠❛t✐✈❛ ❞❡ ❑❛♣❧❛♥✲▼❡✐❡r❀ ❝✉r✈❛ tr❛❝❡❥❛❞❛✿ ✈❡r❞❛❞❡✐r❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❝♦♠♣♦♥❡♥t❡ ✸✳✮

P❛r❛ n= 1000

• ❖ ❝♦♠♣♦♥❡♥t❡ 1 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r201 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 2 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r232 ✈❡③❡s✳

• ❖ ❝♦♠♣♦♥❡♥t❡ 3 ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❢❛❧❤❛ ❞♦ s✐st❡♠❛ ♣♦r567 ✈❡③❡s✳

❈❛❞❛ ✉♠ ❞❡ss❡s ❝♦♠♣♦♥❡♥t❡s t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ α ❛ ♣r✐♦r✐ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦✲