❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s
♥❛ Pr❡s❡♥ç❛ ❞❡ ❈♦✈❛r✐á✈❡✐s
❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s
♥❛ Pr❡s❡♥ç❛ ❞❡ ❈♦✈❛r✐á✈❡✐s
❆♥❞ré ❨♦s❤✐③✉♠✐ ●♦♠❡s
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❋r❛♥❝✐s❝♦ ▲♦✉③❛❞❛✲◆❡t♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ✲ ❉❊s✴❯❋❙❈❛r✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡✲ q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❊st❛tíst✐❝❛✳
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar
G633fw
Gomes, André Yoshizumi.
Família Weibull de razão de chances na presença de covariáveis / André Yoshizumi Gomes. -- São Carlos : UFSCar, 2012.
115 f.
Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2009.
1. Estatística. 2. Distribuição Weibull. 3. Razão de chances. 4. Estimador de máxima verossimilhança. 5. Bootstrap (Estatística). 6. Cadeias de Markov. I. Título.
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✈
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✈
❆❣r❛❞❡❝✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐①
❘❡s✉♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①
❆❜str❛❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✐
✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
✷ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✷✳✷ ❋♦r♠✉❧❛çã♦ ❞♦ ▼♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✷✳✸ ❆ Pr❡s❡♥ç❛ ❞❡ ❈❡♥s✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷✳✹ ❆ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✷✳✺ ❚❡st❡s ❞❡ ❍✐♣ót❡s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✷✳✻ ●rá✜❝♦s ❚❚❚ ✭❚❡♠♣♦ ❚♦t❛❧ ❡♠ ❚❡st❡✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✼ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ♥❛ Pr❡s❡♥ç❛ ❞❡ ❈♦✈❛r✐á✈❡✐s ✶✸
✷✳✽ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✸ ❊st✉❞♦ ❞❡ ❙✐♠✉❧❛çã♦✿ ■♥❢❡rê♥❝✐❛ ❈❧áss✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸✳✷ ❚❛♠❛♥❤♦ ❡ P♦❞❡r ❞♦ ❚❡st❡ ❞❛ ❘❛③ã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✶✽
✸✳✷✳✶ ❊s♣❡❝✐✜❝❛çõ❡s ❞❛ ❙✐♠✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✸✳✷✳✷ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✸✳✸ Pr♦♣r✐❡❞❛❞❡s ❆ss✐♥tót✐❝❛s ❞♦s ❊st✐♠❛❞♦r❡s ❞❡ ▼á①✐♠❛ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ✷✶
✸✳✸✳✶ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸✳✹ ❊st✐♠❛çã♦ ✈✐❛ ❘❡❛♠♦str❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✹✳✶ ❇♦♦tstr❛♣ ✭❊❢r♦♥ ✫ ❚✐❜s❤✐r❛♥✐✱ ✶✾✾✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✹✳✷ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✺ ❈♦♥s✐❞❡r❛çõ❡s ❙♦❜r❡ ♦ ❘✐s❝♦ ❡♠ ❋♦r♠❛ ❞❡ ❇❛♥❤❡✐r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✹ ■♥❢❡rê♥❝✐❛ ❇❛②❡s✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✹✳✷ Pr♦❝❡❞✐♠❡♥t♦s ❇❛②❡s✐❛♥♦s ❞❡ ❊st✐♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✹✳✷✳✶ ❉❡t❡r♠✐♥❛çã♦ ❞❛s Pr✐♦r✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✹✳✸ ❊st✐♠❛çã♦ ❇❛②❡s✐❛♥❛ ♣❛r❛ ❛ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✸✺
✹✳✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✺ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✺✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✺✳✷ ❊①❡♠♣❧♦ ✶ ✲ ❑✐♠❜❛❧❧ ✭✶✾✻✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✺✳✸ ❊①❡♠♣❧♦ ✷ ✲ ❊❢r♦♥ ✭✶✾✽✽✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✺✳✹ ❊①❡♠♣❧♦ ✸ ✲ ❍❛❧❧❡② ✭✶✻✾✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✺✳✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✻ ❈♦♥❝❧✉sõ❡s ❡ Pr♦♣♦st❛s ❋✉t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
❆ ❘❡s✉❧t❛❞♦s ❆ss✐♥tót✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
❇ ❘❡s✉❧t❛❞♦s ❇♦♦tstr❛♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❈ ❇❛♥❝♦s ❞❡ ❉❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
❉ ❆❧❣♦r✐t♠♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
❉✳✶ ❖ ❆❧❣♦r✐t♠♦ ❇❋●❙ ❞❡ ♦t✐♠✐③❛çã♦ ❣❧♦❜❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
❉✳✷ ❖ ❆❧❣♦r✐t♠♦ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❈✉r✈❛s ❞❡ r✐s❝♦ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✳ ❆ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ✭α❂ ✾✱β ❂ ✵✳✼✱ θ ❂ ✽✺✮✱ ❛ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ✭α ❂ ✵✳✺✱ β ❂ ✵✳✸✱ θ ❂ ✶✵✵✮✱ ❛ ❧✐♥❤❛ ❞❡ tr❛ç♦s ❡ ♣♦♥t♦s ✭α ❂ ✶✱β ❂ ✶✱θ ❂ ✺✵✮✱ ❛ ❧✐♥❤❛ só❧✐❞❛ ❡s❝✉r❛ ✭α ❂ ✽✱β ❂ ✵✳✵✶✱ θ ❂ ✹✺✮ ❡ ❛ ❧✐♥❤❛ só❧✐❞❛ ✭α ❂ ✲✶✳✺✱ β ❂ ✲✵✳✶✱ θ ❂ ✼✺✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ r❡♣r❡s❡♥t❛♠ t❛①❛s ❞❡ r✐s❝♦ ❝r❡s❝❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡✱ ❝♦♥st❛♥t❡✱ ❜❛♥❤❡✐r❛ ❡ ✉♥✐♠♦❞❛❧ ✳ ✳ ✳ ✳ ✼
✷✳✷ ❉❡♥s✐❞❛❞❡s ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦s r✐s❝♦s ❞❛ ❋✐❣✉r❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✷✳✸ ❈✉r✈❛s ❞❛s tr❛♥s❢♦r♠❛❞❛s ❚❚❚ ❡♠♣ír✐❝❛s✳ ❆s ❝✉r✈❛s ❆✱ ❇✱ ❈✱ ❉ ❡ ❊✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✐♥❞✐❝❛♠ q✉❡ ♦s ❞❛❞♦s ♣♦ss✉❡♠ t❛①❛s ❞❡ r✐s❝♦ ❝r❡s❝❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡✱ ❝♦♥st❛♥t❡✱ ❜❛♥❤❡✐r❛ ❡ ✉♥✐♠♦❞❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✸✳✶ ❋✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s q✉❛♥❞♦ α = 8, β = 0.01 ❡θ = 45 ✭❡sq✉❡r❞❛✮✱ ❡ ③♦♦♠ ♥♦
♣♦♥t♦ ❞❡ s❛❧t♦ ✭❞✐r❡✐t❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✺✳✶ ●rá✜❝♦ ❚❚❚ ❞♦s ❞❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ r❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✺✳✷ ❘✐s❝♦ ❡st✐♠❛❞♦ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞♦s r❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✺✳✸ ●rá✜❝♦ ❚❚❚ ❞♦s ❞❛❞♦s ❞❡ ❝â♥❝❡r ❞❡ ❝❛❜❡ç❛ ❡ ♣❡s❝♦ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✺✳✹ ❘✐s❝♦ ❡st✐♠❛❞♦ ♣❡❧❛s ♠❡t♦❞♦❧♦❣✐❛s ❛ss✐♥tót✐❝❛ ✭tr❛ç♦s✮✱ ❜♦♦tstr❛♣ ✭só❧✐❞♦✮ ❡ ❇❛②❡s✐❛♥❛ ✭♣♦♥t♦s✮ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❝â♥❝❡r ❞❡ ❝❛❜❡ç❛ ❡ ♣❡s❝♦ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✺✳✺ ●rá✜❝♦ ❚❚❚ ❞♦s ❞❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ ❲r♦❝❧❛✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✺✳✻ ❘✐s❝♦ ❡st✐♠❛❞♦ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ ❲r♦❝❧❛✇ ✳ ✳ ✳ ✳ ✳ ✹✺
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✸✳✶ P♦❞❡r ❞♦ ❚❡st❡ ♣❛r❛ H01:β = 1 ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✸✳✷ P♦❞❡r ❞♦ ❚❡st❡ ♣❛r❛ H01:β = 1 ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✸✳✸ P♦❞❡r ❞♦ ❚❡st❡ ♣❛r❛ H02:η1 = 0 ♦✉ exp(η1) = 1 ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✷✶
✸✳✹ P♦❞❡r ❞♦ ❚❡st❡ ♣❛r❛ H02:η1 = 0 ♦✉ exp(η1) = 1 ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✷✶
✺✳✶ ■♥❢❡rê♥❝✐❛ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ r❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✺✳✷ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♠♦❞❡❧♦s ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ r❛t♦s ✹✵
✺✳✸ ■♥❢❡rê♥❝✐❛ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❝â♥❝❡r ❞❡ ❝❛❜❡ç❛ ❡ ♣❡s❝♦ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✺✳✹ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♠♦❞❡❧♦s ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❝â♥❝❡r ❞❡ ❝❛❜❡ç❛ ❡ ♣❡s❝♦ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✺✳✺ ■♥❢❡rê♥❝✐❛ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ ❲r♦❝❧❛✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✺✳✻ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♠♦❞❡❧♦s ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ ❲r♦❝❧❛✇ ✹✻
✺✳✼ ■♥❢❡rê♥❝✐❛ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❝â♥❝❡r ❞❡ ♣✉❧♠ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✺✳✽ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♠♦❞❡❧♦s ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❝â♥❝❡r ❞❡ ♣✉❧♠ã♦ ✳ ✳ ✺✵
❆✳✶ ❘✐s❝♦ ❈♦♥st❛♥t❡ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
❆✳✷ ❘✐s❝♦ ❈r❡s❝❡♥t❡ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
❆✳✸ ❘✐s❝♦ ❉❡❝r❡s❝❡♥t❡ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
❆✳✹ ❘✐s❝♦ ❯♥✐♠♦❞❛❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
❆✳✺ ❘✐s❝♦ ❈♦♥st❛♥t❡ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
❆✳✻ ❘✐s❝♦ ❈r❡s❝❡♥t❡ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
❆✳✼ ❘✐s❝♦ ❉❡❝r❡s❝❡♥t❡ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
❆✳✽ ❘✐s❝♦ ❯♥✐♠♦❞❛❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
❆✳✾ ❘✐s❝♦ ❈♦♥st❛♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
❆✳✶✵ ❘✐s❝♦ ❈r❡s❝❡♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
❆✳✶✶ ❘✐s❝♦ ❉❡❝r❡s❝❡♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
❆✳✶✷ ❘✐s❝♦ ❯♥✐♠♦❞❛❧ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
❆✳✶✸ ❘✐s❝♦ ❈♦♥st❛♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
❆✳✶✹ ❘✐s❝♦ ❈r❡s❝❡♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
❆✳✶✺ ❘✐s❝♦ ❉❡❝r❡s❝❡♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
❆✳✶✻ ❘✐s❝♦ ❯♥✐♠♦❞❛❧ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
❇✳✶ ❘✐s❝♦ ❈♦♥st❛♥t❡ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❇✳✷ ❘✐s❝♦ ❈r❡s❝❡♥t❡ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❇✳✸ ❘✐s❝♦ ❉❡❝r❡s❝❡♥t❡ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
❇✳✹ ❘✐s❝♦ ❯♥✐♠♦❞❛❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
❇✳✺ ❘✐s❝♦ ❈♦♥st❛♥t❡ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
❇✳✻ ❘✐s❝♦ ❈r❡s❝❡♥t❡ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
❇✳✼ ❘✐s❝♦ ❉❡❝r❡s❝❡♥t❡ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
❇✳✽ ❘✐s❝♦ ❯♥✐♠♦❞❛❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
❇✳✾ ❘✐s❝♦ ❈♦♥st❛♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
❇✳✶✵ ❘✐s❝♦ ❈r❡s❝❡♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
❇✳✶✶ ❘✐s❝♦ ❉❡❝r❡s❝❡♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
❇✳✶✷ ❘✐s❝♦ ❯♥✐♠♦❞❛❧ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✵✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✈✐✐✐
❇✳✶✹ ❘✐s❝♦ ❈r❡s❝❡♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
❇✳✶✺ ❘✐s❝♦ ❉❡❝r❡s❝❡♥t❡ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
❇✳✶✻ ❘✐s❝♦ ❯♥✐♠♦❞❛❧ ❝♦♠ ❈♦✈❛r✐á✈❡❧ ✭✺✪ ❝❡♥s✉r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
❈✳✶ ❉❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ r❛t♦s ✭❑✐♠❜❛❧❧✱ ✶✾✻✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
❈✳✷ ❉❛❞♦s ❞♦ ❡st✉❞♦ ❞❡ ❝â♥❝❡r ❞❡ ❝❛❜❡ç❛ ❡ ♣❡s❝♦ç♦✱ ❡♠ ♠❡s❡s ✲ ❣r✉♣♦ ❆ ✭❊❢r♦♥✱ ✶✾✽✽✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
❈✳✸ ❉❛❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ ❲r♦❝❧❛✇ ✭❍❛❧❧❡②✱ ✶✻✾✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
❈✳✹ ❉❛❞♦s ❞♦ ❡st✉❞♦ ❞❡ ❝â♥❝❡r ❞❡ ♣✉❧♠ã♦ ✭Pr❡♥t✐❝❡✱ ✶✾✼✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❋r❛♥❝✐s❝♦ ▲♦✉✲ ③❛❞❛✲◆❡t♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❝♦♥❝❡❞✐❞❛ ❛ ❡st❡ ♣r♦❥❡t♦✱ ♣♦r ❝♦♥✜❛r ❡♠ ♠❡✉ tr❛❜❛❧❤♦ ❡ ♣❡❧♦ s✉♣♦rt❡ t♦❞❛ ✈❡③ ❡♠ q✉❡ s❡ ❢❡③ ♥❡❝❡ssár✐♦✳
❚❛♠❜é♠ ❛❣r❛❞❡ç♦ à ❋✉♥❞❛çã♦ ❞❡ ❆♠♣❛r♦ à P❡sq✉✐s❛ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦ ✭❋❆P❊❙P✮ ♣♦r t❡r ✜♥❛♥❝✐❛❞♦ ❡st❡ ♣r♦❥❡t♦ ♣♦r ❜♦❛ ♣❛rt❡ ❞♦ t❡♠♣♦ ❡♠ q✉❡ ❡st✐✈❡ ♥♦ ♣r♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦✳
❆✐♥❞❛✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s✱ ❞❡s❞❡ ❝♦❧❡❣❛s ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ♣r♦❢❡ss♦r❡s ❡ ❛♠✐❣♦s ❞❡ ❧♦♥❣❛ ❞❛t❛✱ ❝✉❥❛ ♣r❡s❡♥ç❛ ❝♦♥st❛♥t❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠♦str❛✲s❡ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ q✉❡ ❡✉ ♣♦ss❛ t❡r ♣❡rs❡✈❡r❛♥ç❛ ❡ s✉♣❡r❛r ♦s ♦❜stá❝✉❧♦s ❞❛ ✈✐❞❛✳ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ à Pr✐s❝✐❧❛✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦✱ ❛♠♦r ❡ ❝❛r✐♥❤♦ ❝♦♠♣❛rt✐❧❤❛❞♦s ♥❡st❡ t❡♠♣♦ ❥✉♥t♦s✳
❋✐♥❛❧♠❡♥t❡✱ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥✲ t✐✈♦✉ ❛ s❡❣✉✐r ❡st❡ ❝❛♠✐♥❤♦✱ ❡ ❡♠ ❡s♣❡❝✐❛❧✱ à ♠✐♥❤❛ ♠ã❡✱ q✉❡ ♠❡ ❛♠♦✉ ❞✉r❛♥t❡ t♦❞❛ ❛ s✉❛ ✈✐❞❛✱ ❞❡s❞❡ ♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ♥❛s❝✐✱ ❡ ❛✐♥❞❛ ❡stá ❛ ♠❡ ✈✐❣✐❛r✳ ❆❧❣✉é♠ q✉❡ ❢♦✐✱ ❡ ❛✐♥❞❛ é✱ t✉❞♦ ♣❛r❛ ♠✐♠✱ ❝✉❥❛ ♣r❡s❡♥ç❛ ❡st❛rá s❡♠♣r❡ ❡♠ ♠❡✉ ❝♦r❛çã♦✳
❘❡s✉♠♦
❆ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ é ✉♠❛ ❡s❝♦❧❤❛ ✐♥✐❝✐❛❧ ❢r❡qü❡♥t❡ ♣❛r❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❞❛❞♦s ❝♦♠ t❛①❛s ❞❡ r✐s❝♦ ♠♦♥ót♦♥❛s✳ ❊♥tr❡t❛♥t♦✱ ❡st❛ ❞✐str✐❜✉✐çã♦ ♥ã♦ ❢♦r♥❡❝❡ ✉♠ ❛❥✉st❡ ♣❛r❛♠étr✐❝♦ r❛③♦á✈❡❧ q✉❛♥❞♦ ❛s ❢✉♥çõ❡s ❞❡ r✐s❝♦ ❛ss✉♠❡♠ ✉♠ ❢♦r♠❛t♦ ✉♥✐♠♦❞❛❧ ♦✉ ❡♠ ❢♦r♠❛ ❞❡ ❜❛♥❤❡✐r❛✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ❈♦♦r❛② ✭✷✵✵✻✮ ♣r♦♣ôs ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❞❛s ❢❛♠í❧✐❛s ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ r❡❢❡r✐❞❛ ❝♦♠♦ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✳ ❊st❛ ❢❛♠í❧✐❛ ♥ã♦ é ❛♣❡♥❛s ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ♠♦❞❡❧❛r t❛①❛s ❞❡ r✐s❝♦ ✉♥✐♠♦❞❛❧ ❡ ❜❛♥❤❡✐r❛✱ ♠❛s t❛♠❜é♠ é ❛❞❡q✉❛❞❛ ♣❛r❛ t❡st❛r ❛ ❛❞❡q✉❛❜✐❧✐❞❛❞❡ ❞♦ ❛❥✉st❡ ❞❛s ❢❛♠í❧✐❛s ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❝♦♠♦ s✉❜♠♦❞❡❧♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s s✐st❡♠❛t✐❝❛♠❡♥t❡ ❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❡ s✉❛s ♣r♦✲ ♣r✐❡❞❛❞❡s✱ ❛♣♦♥t❛♥❞♦ ❛s ♠♦t✐✈❛çõ❡s ♣❛r❛ ♦ s❡✉ ✉s♦✱ ✐♥s❡r✐♥❞♦ ❝♦✈❛r✐á✈❡✐s ♥♦ ♠♦✲ ❞❡❧♦✱ ✈❡r✐✜❝❛♥❞♦ ❛s ❞✐✜❝✉❧❞❛❞❡s r❡❢❡r❡♥t❡s ❛♦ ♣r♦❜❧❡♠❛ ❞❛ ❡st✐♠❛çã♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❡ ♣r♦♣♦♥❞♦ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❡st✐♠❛çã♦ ✐♥t❡r✈❛❧❛r ❡ ❝♦♥str✉çã♦ ❞❡ t❡st❡s ❞❡ ❤✐♣ót❡s❡s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦✳ ❈♦♠♣❛r❛♠♦s ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ♠❡✐♦ ❞♦s ♠ét♦❞♦s ❞❡ r❡❛♠♦str❛❣❡♠ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ✈✐❛ t❡♦r✐❛ ❛ss✐♥tót✐❝❛✳ ❚❛♥t♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦❜❡rt✉r❛ ❞♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♥✜❛♥ç❛ ♣r♦♣♦st♦s q✉❛♥t♦ ♦ t❛♠❛♥❤♦ ❡ ♣♦❞❡r ❞♦s t❡st❡s ❞❡ ❤✐♣ót❡s❡s ❝♦♥s✐❞❡r❛❞♦s ❢♦r❛♠ ❡st✉❞❛❞♦s ✈✐❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ❆❧é♠ ❞✐ss♦✱ ♣r♦♣✉s❡♠♦s ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❇❛②❡s✐❛♥❛ ❞❡ ❡st✐♠❛çã♦ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❜❛s❡❛❞♦s ❡♠ té❝♥✐❝❛s ❞❡ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✈✐❛ ❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧✱ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✱ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛❀ t❡♦r✐❛ ❛ss✐♥tót✐❝❛❀ ❜♦♦tstr❛♣❀ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦❜❡r✲ t✉r❛❀ ✐♥❢❡rê♥❝✐❛ ❇❛②❡s✐❛♥❛❀ ❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✭▼❈▼❈✮✳
❆❜str❛❝t
❚❤❡ ❲❡✐❜✉❧❧ ❞✐str✐❜✉✐t✐♦♥ ✐s ❛ ❝♦♠♠♦♥ ✐♥✐t✐❛❧ ❝❤♦✐❝❡ ❢♦r ♠♦❞❡❧✐♥❣ ❞❛t❛ ✇✐t❤ ♠♦♥♦t♦♥❡ ❤❛③❛r❞ r❛t❡s✳ ❍♦✇❡✈❡r✱ s✉❝❤ ❞✐str✐❜✉t✐♦♥ ❢❛✐❧s t♦ ♣r♦✈✐❞❡ ❛ r❡❛✲ s♦♥❛❜❧❡ ♣❛r❛♠❡tr✐❝ ✜t ✇❤❡♥ t❤❡ ❤❛③❛r❞ ❢✉♥❝t✐♦♥ ✐s ✉♥✐♠♦❞❛❧ ♦r ❜❛t❤t✉❜✲s❤❛♣❡❞✳ ■♥ t❤✐s ❝♦♥t❡①t✱ ❈♦♦r❛② ✭✷✵✵✻✮ ♣r♦♣♦s❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❲❡✐❜✉❧❧ ❢❛♠✐❧② ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦❢ t❤❡ ♦❞❞s ♦❢ ❲❡✐❜✉❧❧ ❛♥❞ ✐♥✈❡rs❡ ❲❡✐❜✉❧❧ ❢❛♠✐❧✐❡s✱ r❡❢❡rr❡❞ ❛s t❤❡ ♦❞❞ ❲❡✐❜✉❧❧ ❢❛♠✐❧② ✇❤✐❝❤ ✐s ♥♦t ❥✉st ✉s❡❢✉❧ ❢♦r ♠♦❞❡❧✐♥❣ ✉♥✐♠♦❞❛❧ ❛♥❞ ❜❛t❤t✉❜✲s❤❛♣❡❞ ❤❛③❛r❞s✱ ❜✉t ✐t ✐s ❛❧s♦ ❝♦♥✈❡♥✐❡♥t ❢♦r t❡st✐♥❣ ❣♦♦❞♥❡ss✲♦❢✲✜t ♦❢ ❲❡✐❜✉❧❧ ❛♥❞ ✐♥✈❡rs❡ ❲❡✐❜✉❧❧ ❛s s✉❜♠♦❞❡❧s✳ ■♥ t❤✐s ♣r♦❥❡❝t ✇❡ ❤❛✈❡ s②st❡♠❛t✐❝❛❧❧② st✉❞✐❡❞ t❤❡ ♦❞❞ ❲❡✐❜✉❧❧ ❢❛♠✐❧② ❛❧♦♥❣ ✇✐t❤ ✐ts ♣r♦♣❡rt✐❡s✱ s❤♦✇✐♥❣ ♠♦t✐✈❛t✐♦♥s ❢♦r ✐ts ✉t✐❧✐③❛t✐♦♥✱ ✐♥s❡rt✐♥❣ ❝♦✈❛r✐❛t❡s ✐♥ t❤❡ ♠♦❞❡❧✱ ♣♦✐♥t✐♥❣ ♦✉t s♦♠❡ tr♦✉❜❧❡s ❛ss♦✲ ❝✐❛t❡❞ ✇✐t❤ t❤❡ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ❛♥❞ ♣r♦♣♦s✐♥❣ ✐♥t❡r✈❛❧ ❡st✐♠❛t✐♦♥ ❛♥❞ ❤②♣♦t❤❡s✐s t❡st ❝♦♥str✉❝t✐♦♥ ♠❡t❤♦❞♦❧♦❣✐❡s ❢♦r t❤❡ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs✳ ❲❡ ❤❛✈❡ ❛❧s♦ ❝♦♠♣❛r❡❞ r❡s❛♠♣❧✐♥❣ r❡s✉❧ts ✇✐t❤ ❛s②♠♣t♦t✐❝ ♦♥❡s✳ ❈♦✈❡r❛❣❡ ♣r♦❜❛❜✐❧✲ ✐t② ❢r♦♠ ♣r♦♣♦s❡❞ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ❛♥❞ s✐③❡ ❛♥❞ ♣♦✇❡r ♦❢ ❝♦♥s✐❞❡r❡❞ ❤②♣♦t❤❡s✐s t❡sts ✇❡r❡ ❜♦t❤ ❛♥❛❧②③❡❞ ❛s ✇❡❧❧ ✈✐❛ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ❇❛②❡s✐❛♥ ❡st✐♠❛t✐♦♥ ♠❡t❤♦❞♦❧♦❣② ❢♦r t❤❡ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs ❜❛s❡❞ ✐♥ ▼♦♥t❡ ❈❛r❧♦ ▼❛r❦♦✈ ❈❤❛✐♥ ✭▼❈▼❈✮ s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s✳
❑❡②✇♦r❞s✿ ❲❡✐❜✉❧❧ ❞✐str✐❜✉t✐♦♥✱ ♦❞❞s r❛t✐♦✱ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡s✲ t✐♠❛t♦r❀ ❛s②♠♣t♦t✐❝ t❤❡♦r②❀ ❜♦♦tstr❛♣❀ ❝♦✈❡r❛❣❡ ♣r♦❜❛❜✐❧✐t②❀ ❇❛②❡s✐❛♥ ✐♥❢❡r❡♥❝❡❀ ▼♦♥t❡ ❈❛r❧♦ ▼❛r❦♦✈ ❈❤❛✐♥s ✭▼❈▼❈✮✳
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
◆❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❝♦♠ t❛①❛s ❞❡ r✐s❝♦ ♠♦♥ót♦♥❛s✱ ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ é ❢r❡qü❡♥t❡♠❡♥t❡ ✉♠❛ ❡s❝♦❧❤❛ ✐♥✐❝✐❛❧✳ P♦ré♠✱ ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ♥ã♦ ❢♦r♥❡❝❡ ✉♠ ❛❥✉st❡ ♣❛r❛♠étr✐❝♦ r❛③♦á✈❡❧ ♣❛r❛ s✐t✉❛çõ❡s ❡♠ q✉❡ ❛s ❢✉♥çõ❡s ❞❡ r✐s❝♦ ❛ss✉♠❡♠ ✉♠ ❢♦r♠❛t♦ ✉♥✐♠♦❞❛❧ ♦✉ ❞❡ ❜❛♥❤❡✐r❛✱ ♦ q✉❡ é ❝♦♠✉♠ ♥❛ ♣rát✐❝❛ ✭▲♦✉③❛❞❛✲◆❡t♦✱ ✶✾✾✾✮✳ P❛r❛ ❞❡s❝r❡✈❡r ❛❞❡q✉❛❞❛♠❡♥t❡ t❛✐s ❝♦♠♣♦r✲ t❛♠❡♥t♦s✱ ❡①t❡♥sõ❡s ❝♦♠ três ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❢♦r❛♠ ♣r♦♣♦st❛s✱ ❝♦♠♦ ❛ ❞✐str✐❜✉✐çã♦ ❣❛♠❛ ❣❡♥❡r❛❧✐③❛❞❛ ✭❙t❛❝②✱ ✶✾✻✷✮ ❡ ❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❡①♣♦✲ ♥❡♥❝✐❛❞❛ ✭▼✉❞❤♦❧❦❛r ❡t ❛❧✳✱ ✶✾✾✺✮✳ P♦ré♠✱ ♠❡s♠♦ t❛✐s ♠♦❞❡❧♦s ♥ã♦ sã♦ ❝❛♣❛③❡s ❞❡ ❛❝♦♠♦❞❛r ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡ r✐s❝♦ ❡♠ ❢♦r♠❛t♦ ❞❡ ❜❛♥❤❡✐r❛✳
◆❡st❡ ❝♦♥t❡①t♦✱ ❈♦♦r❛② ✭✷✵✵✻✮ ♣r♦♣ôs ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❢❛♠í❧✐❛ ❲❡✐✲ ❜✉❧❧ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❞❛s ❢❛♠í❧✐❛s ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ r❡❢❡r✐❞❛ ❝♦♠♦ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✳ ❊st❛ ❢❛♠í❧✐❛ ♥ã♦ é ❛♣❡♥❛s ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ♠♦❞❡❧❛r t❛①❛s ❞❡ r✐s❝♦ ❡♠ ❢♦r♠❛ ❞❡ ❜❛♥❤❡✐r❛✱ ♠❛s t❛♠❜é♠ é ❛❞❡q✉❛❞❛ ♣❛r❛ t❡st❛r ❛ ❛❞❡q✉❛❜✐❧✐❞❛❞❡ ❞♦ ❛❥✉st❡ ❞❛s ❢❛♠í❧✐❛s ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❝♦♠♦ s✉❜♠♦❞❡❧♦s✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❛❝♦♠♦❞❛ ♦s ❝✐♥❝♦ ♣r✐♥❝✐♣❛✐s ❢♦r♠❛t♦s ❞❡ r✐s❝♦✿ ❝♦♥st❛♥t❡✱ ❝r❡s❝❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡✱ ✉♥✐♠♦❞❛❧ ❡ ❜❛♥❤❡✐r❛✳
❆♣❡s❛r ❞❛ ✢❡①✐❜✐❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ♣❛r❛ ❛❝♦♠♦❞❛r ❢✉♥çõ❡s ❞❡ r✐s❝♦ ♥ã♦✲ ♠♦♥ót♦♥❛s✱ ❛ ♣r❡s❡♥ç❛ ❞❡ ❝♦✈❛r✐á✈❡✐s ♥ã♦ é ❝♦♥s✐❞❡r❛❞❛ ♥❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❈♦♦r❛② ✭✷✵✵✻✮✳ ❈♦♠♦ ❝♦♥s❡qüê♥❝✐❛✱ ♠ét♦❞♦s ❢♦r♠❛✐s ❞❡ t❡st❡s ❞❡ ❤✐♣ót❡s❡ ❡ ❡st✐♠❛çã♦
✶✳ ■♥tr♦❞✉çã♦ ✷
♣♦r ✐♥t❡r✈❛❧♦ ♥ã♦ ❢♦r❛♠ ❛✐♥❞❛ ♣❧❡♥❛♠❡♥t❡ ❞❡s❡♥✈♦❧✈✐❞♦s ❡ t❡st❛❞♦s ❝♦♠ r❡❧❛çã♦ às s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❛ss✐♥tót✐❝❛s✳
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡st✉❞❛ s✐st❡♠❛t✐❝❛♠❡♥t❡ ❛ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s✱ ✐♥❞❡①❛❞❛ ♣♦r ❞♦✐s ♣❛râ♠❡tr♦s ❞❡ ❢♦r♠❛ ❡ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛✱ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❛♣♦♥t❛♥❞♦ ❛s ♠♦t✐✈❛çõ❡s ♣❛r❛ s❡✉ ✉s♦❀ ❞❡s❡♥✈♦❧✈❡ ♦ ♠♦❞❡❧♦ ❝♦♠ ❛ ✐♥❝❧✉sã♦ ❞❡ ❝♦✈❛r✐á✈❡✐s❀ ✈❡r✐✜❝❛ ❛s ❞✐✜❝✉❧❞❛❞❡s r❡❢❡r❡♥t❡s ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❡st✐♠❛çã♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ❝♦✈❛r✐á✈❡✐s ❡ ♣r♦♣õ❡ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❡st✐♠❛çã♦ ✐♥t❡r✈❛❧❛r ❡ ❝♦♥str✉çã♦ ❞❡ t❡st❡s ❞❡ ❤✐♣ót❡s❡s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ✈✐❛ ♠ét♦❞♦ ❜♦♦tstr❛♣ ✭❊❢r♦♥ ✫ ❚✐❜s❤✐r❛♥✐✱ ✶✾✾✸❀ ❉❛✈✐s♦♥ ✫ ❍✐♥❦❧❡②✱ ✶✾✾✼✮✳ ❊st❡s ♣r♦❝❡❞✐♠❡♥t♦s t❛♠❜é♠ sã♦ ❝♦♠♣❛r❛❞♦s ❝♦♠ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ♠❡✐♦ ❞❡ t❡♦r✐❛ ❛ss✐♥tót✐❝❛✳ ❆❧é♠ ❞✐ss♦✱ t❛♠❜é♠ é ♣r♦♣♦st❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❡st✐♠❛çã♦ ❇❛②❡s✐❛♥❛ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❜❛s❡❛❞❛ ❡♠ té❝♥✐❝❛s ❞❡ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✈✐❛ ❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ✭●❡♠❛♥ ✫ ●❡♠❛♥✱ ✶✾✽✹❀ ❈❤✐❜ ✫ ●r❡❡♥❜❡r❣✱ ✶✾✾✺✮✳
❈❛♣ít✉❧♦ ✷
❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡
❈❤❛♥❝❡s
✷✳✶ ■♥tr♦❞✉çã♦
❆ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❣❛♥❤♦✉ ♥ít✐❞♦ ❞❡st❛q✉❡ ♥♦s ú❧t✐♠♦s ❛♥♦s ❡ ❤♦❥❡ ♣♦ss✉✐ ✐♥ú♠❡r❛s ❛♣❧✐❝❛çõ❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛ ár❡❛ ♠é❞✐❝❛✳ ❙❡❣✉♥❞♦ ❈♦❧♦s✐♠♦ ✫ ●✐♦❧♦ ✭✷✵✵✻✮✱ é ✉♠❛ ❞❛s ár❡❛s ❞❡ ❛♣❧✐❝❛çã♦ ❞❛ ❡st❛tíst✐❝❛ q✉❡ ♠❛✐s tê♠ ❝r❡s❝✐❞♦ ♥♦s ú❧t✐♠♦s ✷✵ ❛♥♦s✱ ❡♠ r❛③ã♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❞♦ ❛♣r✐♠♦r❛♠❡♥t♦ ❞❡ té❝♥✐❝❛s ❡st❛tíst✐❝❛s✱ ❝♦♠❜✐♥❛❞♦s ❝♦♠ ❝♦♠♣✉t❛❞♦r❡s ❝❛❞❛ ✈❡③ ♠❛✐s rá♣✐❞♦s✳
❊♠ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❛ ✈❛r✐á✈❡❧ ❞❡ ✐♥t❡r❡ss❡ ♥♦ ❡st✉❞♦ é ♦ t❡♠♣♦ ❛té ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❡✈❡♥t♦ ❞❡ ✐♥t❡r❡ss❡✱ ❢r❡qü❡♥t❡♠❡♥t❡ ❝❤❛♠❛❞♦ ❞❡ ❢❛❧❤❛✳ ❊①❡♠♣❧♦s ❞❡ t❡♠♣♦s ❞❡ ❢❛❧❤❛ ✐♥❝❧✉❡♠ ♦s ♣❡rí♦❞♦s ❞❡ ✈✐❞❛ ❞❡ ♠áq✉✐♥❛s ✐♥❞✉str✐❛✐s ❛té ❛ q✉❡❜r❛✱ t❡♠♣♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡ ♣❛❝✐❡♥t❡s ❡♠ ✉♠ ❡st✉❞♦ ❝❧í♥✐❝♦ ❛té ❛ ♠♦rt❡ ❡ ♦ ♣❡rí♦❞♦ ❞❡ ✜❞❡❧✐❞❛❞❡ ❞❡ ❝❧✐❡♥t❡s ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ t✐♣♦ ❞❡ s❡❣✉r❛❞♦r❛✱ ♥❛ ár❡❛ ✜♥❛♥❝❡✐r❛✳ ❉❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❝❛r❛❝t❡r✐③❛♠✲s❡ ♣❡❧❛ ♣r❡s❡♥ç❛ ❞❡ ❝❡♥s✉r❛s✱ q✉❡ r❡♣r❡s❡♥t❛♠ ❛ ♦❜s❡r✈❛çã♦ ✐♥❝♦♠♣❧❡t❛ ❞♦ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥❞✐✈í❞✉♦✳ ❚❡①t♦s ✐♥tr♦❞✉tór✐♦s à ❆♥á❧✐s❡ ❞❡ ❙♦❜r❡✈✐✈ê♥❝✐❛ ❝♦♠✲ ♣r❡❡♥❞❡♠ ❈♦❧♦s✐♠♦ ✫ ●✐♦❧♦ ✭✷✵✵✻✮ ❡ ▲❡❡ ✭✶✾✾✷✮✳
❊♠❜♦r❛ ❡①✐st❛ ✉♠❛ sér✐❡ ❞❡ ♠♦❞❡❧♦s ♣r♦❜❛❜✐❧íst✐❝♦s ✉t✐❧✐③❛❞♦s ❡♠ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❛❧❣✉♥s ❞❡st❡s ♠♦❞❡❧♦s ♦❝✉♣❛♠ ✉♠❛ ♣♦s✐çã♦ ❞❡ ❞❡st❛q✉❡
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✹
♣♦r s✉❛ ❝♦♠♣r♦✈❛❞❛ ❛❞❡q✉❛çã♦ ❛ ✈ár✐❛s s✐t✉❛çõ❡s ♣rát✐❝❛s✱ ❝♦♠♦ ♦ ♠♦❞❡❧♦ ❡①✲ ♣♦♥❡♥❝✐❛❧ ❡ ♦ ❞❡ ❲❡✐❜✉❧❧✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❛♣❛r❡❝❡ ❝♦♠♦ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ q✉❡ s❡ ❞❡st❛❝❛ ♣♦r s✉❛ ✢❡①✐❜✐❧✐❞❛❞❡ ❡♠ ❛❝♦♠♦❞❛r ❛♠❜♦s ♦s ♠♦❞❡❧♦s ❛♥t❡r✐♦r❡s✱ ❛❧é♠ ❞❡ út✐❧ ❡♠ ❞✐✈❡rs❛s ♦✉tr❛s s✐t✉❛çõ❡s✳
❖ ♣r❡s❡♥t❡ ❈❛♣ít✉❧♦ ❛♣r❡s❡♥t❛ ❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✱ ❞❡✜♥✐♥❞♦ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡ ♦s t❡st❡s ❛ss♦❝✐❛❞♦s ❛♦s ❡st✐♠❛❞♦r❡s ❞♦s s❡✉s ♣❛râ♠❡tr♦s✳ ❇r❡✈❡s ❝♦♠❡♥tár✐♦s s♦❜r❡ ♦ ❣rá✜❝♦ ❚❚❚ ✭t❡♠♣♦ t♦t❛❧ ❡♠ t❡st❡✮ ❡ ❛ ♣r❡s❡♥ç❛ ❞❡ ❝❡♥s✉r❛s ♣r❡❝❡❞❡♠ ❛ ❡①t❡♥sã♦ ❞♦ ♠♦❞❡❧♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♣r❡s❡♥ç❛ ❞❡ ❝♦✈❛r✐á✈❡✐s✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❛✉♠❡♥t❛r ❛✐♥❞❛ ♠❛✐s s✉❛ ✢❡①✐❜✐❧✐❞❛❞❡✳
✷✳✷ ❋♦r♠✉❧❛çã♦ ❞♦ ▼♦❞❡❧♦
❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ✭❈♦♦r❛②✱ ✷✵✵✻✮ s✉r❣✐✉ ♣❛r❛ r❡s✲ ♣♦♥❞❡r ❞✉❛s ♣❡r❣✉♥t❛s ❡♥❝♦♥tr❛❞❛s ❡♠ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✿
✶✳ ◗✉❛✐s sã♦ ❛s ❝❤❛♥❝❡s ❞❡ ✉♠ ✐♥❞✐✈í❞✉♦ ♠♦rr❡r ❛♥t❡s ❞♦ t❡♠♣♦ T✱ s❛❜❡♥❞♦ q✉❡T s❡❣✉❡ ✉♠❛ ❝❡rt❛ ❞✐str✐❜✉✐çã♦W❄
✷✳ ❙❡ ❡st❛s ❝❤❛♥❝❡s s❡❣✉❡♠ ❛❧❣✉♠❛ ♦✉tr❛ ❞✐str✐❜✉✐çã♦L✱ ❡♥tã♦ q✉❛❧ ❛ ❝♦rr❡çã♦ ❛ s❡r ❢❡✐t❛ s♦❜r❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ T❄
❊♥q✉❛♥t♦ ❛ r❡s♣♦st❛ ❞❛ ♣r✐♠❡✐r❛ ♣❡r❣✉♥t❛ ♣♦❞❡ s❡r ❞❛❞❛ ❞❡ ♠❛♥❡✐r❛ ✐♠❡❞✐❛t❛ ♣♦r ❞❡♣❡♥❞❡r ❛♣❡♥❛s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ W✱ ❛ r❡s♣♦st❛ ❞❛ s❡❣✉♥❞❛ q✉❡stã♦ ♣♦❞❡ ✈❛r✐❛r ❞❡✈✐❞♦ às ❡s❝♦❧❤❛s ❞❡ L ❡ W✳ ❆ ♣r✐♠❡✐r❛ r❡s♣♦st❛ ♣♦❞❡ s❡r ❞❛❞❛ ❛♦ ❝♦♥s✐❞❡r❛r ❛ ✏r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❞❡ ❢❛❧❤❛✑ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥❞✐✈í❞✉♦✱ ❞❛❞♦ ❡♠ t❡r♠♦s ❞❛ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ♣♦r (1−ST(t))/ST(t)✱ ♦♥❞❡ST(t) = P(T ≥t) = 1−FT(t), t ∈(0,∞)✱ s❡♥❞♦FT(t)❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ ❞❡ T✳ ❊st❛ r❛③ã♦ ♣♦❞❡ s❡r ❞❡♥♦t❛❞❛ ♣♦r y (y ∈ (0,∞)) ❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ✉♠❛
✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳ ❖ ✐♥t❡r❡ss❡ r❡s✐❞❡ ❡♠ ♠♦❞❡❧❛r Y ❛tr❛✈és ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♣❛r❛♠étr✐❝❛ ❛♣r♦♣r✐❛❞❛ FY(y)✱ ✐st♦ é✱
P(Y ≤y) =FY(y) =FY
1−ST(t) ST(t)
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✺
❈♦♥s✐❞❡r❡♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❧♦❣✲❧♦❣íst✐❝❛ ♣❛r❛ ♠♦❞❡❧❛r ❛ ❛❧❡❛t♦r✐❡❞❛❞❡ ❞❛ ✏r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❞❡ ❢❛❧❤❛✑✱ ❝✉❥❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ é ❞❛❞❛ ♣♦r FY(y) = 1−(1 +yγ)−1; 0 < γ < ∞✱ ❡ ❢♦✐ s✉❣❡r✐❞❛ ♣♦r ❈♦♦r❛② ✭✷✵✵✻✮ ♣❡❧❛ ❡①✐stê♥❝✐❛ ❞❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿
1−SY(y) SY(y)
= 1−(1 +y
γ)−1
(1 +yγ)−1 =
= (1 +y
γ−1)(1 +yγ)−1
(1 +yγ)−1 =
= yγ = =
1−ST(t) ST(t)
γ ,
♦♥❞❡ SY(y) = 1−FY(y)✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡ y = 1−ST(t)
ST(t)
=
1−SY(y) SY(y)
1/γ
. ✭✷✳✷✮
❉❡ss❛ ❢♦r♠❛✱ γ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❝♦rr❡çã♦ ❞❛ ❞✐s✲ tr✐❜✉✐çã♦ W✳
❆❣♦r❛ s✉♣♦♥❤❛ q✉❡ ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❞❡ t❡♠♣♦ ❞❡ ✈✐❞❛T s❡❣✉❡ ✉♠❛ ❞✐s✲ tr✐❜✉✐çã♦ ❲❡✐❜✉❧❧✱ ❝♦♠ s✉❛ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❛❞❛ ♣♦rST(t) =❡−(t/θ)
α
; 0< t < ∞, α > 0, θ > 0✳ ❊♥tã♦✱ ❞❡ ✭✷✳✶✮✱ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ ❞❛
❞✐str✐❜✉✐çã♦ ❝♦rr✐❣✐❞❛ ❞❡ T é ❞❛❞❛ ♣♦r
FT(t) = 1−
1 +
1−ST(t) ST(t)
γ−1
= = 1−
1 +
1−❡−(t/θ)α
❡−(t/θ)α
γ−1
= = 1−
1 +
1
❡−(t/θ)α −1
γ−1
=
= 1−1 + ❡(t/θ)α −1γ−1; ✭✷✳✸✮ 0 < t <∞, α >0, γ > 0, θ >0.
❙❡ ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ T s❡❣✉❡ ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ❝♦♠ ❛ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❛❞❛ ♣♦rST(t) = 1−❡−(t/θ)
−α
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✻
❡♥tã♦ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❡ ❡s❝r❡✈❡r ❛ ❞✐str✐❜✉✐çã♦ ❝♦rr✐❣✐❞❛ ❞❡ T ❝♦♠♦
FT(t) = 1−
1 +❡(t/θ)−α−1−γ
−1
; 0< t <∞, α >0, γ >0, θ >0.✭✷✳✹✮ ❆s ❡q✉❛çõ❡s ✭✷✳✸✮ ❡ ✭✷✳✹✮ ♣♦❞❡♠ s❡r ❝♦♠❜✐♥❛❞❛s ❛♦ ❡s❝r❡✈❡r ♦ ♣❛râ♠❡tr♦ ❞❡ ❝♦rr❡çã♦ β = ±γ, γ > 0✱ ♣❛r❛ ♦❜t❡r ❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ ❞❛
❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ♣♦r
F(t;α, β, θ) = 1−1 + ❡(t/θ)α −1β−1; 0< t <∞, θ >0, αβ >0. ✭✷✳✺✮ ❆s ❝♦rr❡s♣♦♥❞❡♥t❡s ❢✉♥çõ❡s ❞❡ ❞❡♥s✐❞❛❞❡✱ r✐s❝♦ ❡ q✉❛♥t✐❧ sã♦ ❞❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r
f(t;α, β, θ) =
αβ t
t θ
α ❡(t/θ)α
❡(t/θ)α
−1β−11 + ❡(t/θ)α −1β−2, ✭✷✳✻✮ h(t;α, β, θ) =
αβ t
t θ
α ❡(t/θ)α
❡(t/θ)α
−1β−11 + ❡(t/θ)α−1β−1, ✭✷✳✼✮ ❡
Q(u) =F−1(u) =θ ❧♥1/α 1 +
u
1−u
1/β!
; 0< u <1. ✭✷✳✽✮ ◆♦t❛çã♦✿ t ∼W eibullRC(α, β, θ)✳
❆ ❞❡♥s✐❞❛❞❡ ❡♥❝♦♥tr❛❞❛ é r❡❞✉③✐❞❛ à ❲❡✐❜✉❧❧ q✉❛♥❞♦β = 1✱ ❡ à ❲❡✐❜✉❧❧
✐♥✈❡rs❛ q✉❛♥❞♦ β = −1✳ ❈❧❛r❛♠❡♥t❡✱ ❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s é
❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ à ❞✐str✐❜✉✐çã♦ ❧♦❣✲❧♦❣íst✐❝❛ ♣❛r❛ ❣r❛♥❞❡s ✈❛❧♦r❡s ❞❡ θ✳ ◗✉❛♥❞♦ ❛♠❜♦s ♦s ♣❛râ♠❡tr♦s ❞❡ ❢♦r♠❛✱ α ❡ β✱ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s sã♦ ✐❣✉❛✐s ❛ ✶✱ ❛ ❢✉♥çã♦ ❞❡ r✐s❝♦ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✼✮ é ❝♦♥st❛♥t❡✱ ✐st♦ é✱ t❡♠♦s ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ◗✉❛♥❞♦ t❡♠♦s (α > 1, αβ > 1)✱ (α < 1, αβ < 1)✱ (α > 1, αβ ≤ 1) ❡ (α < 1, αβ ≥ 1)✱ ❛s ❢♦r♠❛s ❞❛ ❢✉♥çã♦ ❞❡
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✼
❋■●❯❘❆ ✷✳✶✿ ❈✉r✈❛s ❞❡ r✐s❝♦ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✳ ❆ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ✭α ❂ ✾✱ β ❂ ✵✳✼✱ θ ❂ ✽✺✮✱ ❛ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ✭α ❂ ✵✳✺✱ β ❂ ✵✳✸✱ θ ❂ ✶✵✵✮✱ ❛ ❧✐♥❤❛ ❞❡ tr❛ç♦s ❡ ♣♦♥t♦s ✭α❂ ✶✱ β❂ ✶✱ θ❂ ✺✵✮✱ ❛ ❧✐♥❤❛ só❧✐❞❛ ❡s❝✉r❛ ✭α ❂ ✽✱β ❂ ✵✳✵✶✱θ ❂ ✹✺✮ ❡ ❛ ❧✐♥❤❛ só❧✐❞❛ ✭α❂ ✲✶✳✺✱β❂ ✲✵✳✶✱θ❂ ✼✺✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ r❡♣r❡s❡♥t❛♠ t❛①❛s ❞❡ r✐s❝♦ ❝r❡s❝❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡✱ ❝♦♥st❛♥t❡✱ ❜❛♥❤❡✐r❛ ❡ ✉♥✐♠♦❞❛❧
❋■●❯❘❆ ✷✳✷✿ ❉❡♥s✐❞❛❞❡s ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦s r✐s❝♦s ❞❛ ❋✐❣✉r❛ ✶
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✽
E(Tk) = kθ
k
|α|
Z ∞
0
(log(1 +y))αk−1
(1 +y) (1 +yβ)dy. ✭✷✳✾✮ ❖s ♠♦♠❡♥t♦s ♥ã♦ ♣♦ss✉❡♠ ❢♦r♠❛ ❢❡❝❤❛❞❛ ❡✱ ♣♦rt❛♥t♦✱ ❞❡✈❡♠ s❡r ❝❛❧❝✉✲ ❧❛❞♦s ♥✉♠❡r✐❝❛♠❡♥t❡✳ ❈♦♠♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ❡♠ ✭✷✳✾✮✱ s❡ α é ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡β = 1✱ ❡♥tã♦ ♣❛r❛ k=α✱
E(Tα) = θα
Z ∞
0
1
(1 +y2)dy=θ
α.
✷✳✸ ❆ Pr❡s❡♥ç❛ ❞❡ ❈❡♥s✉r❛s
❆ ♣r✐♥❝✐♣❛❧ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ é ❛ ♣r❡s❡♥ç❛ ❞❡ ❝❡♥s✉r❛✱ q✉❡ s❡ r❡s✉♠❡ à ♦❜s❡r✈❛çã♦ ✐♥❝♦♠♣❧❡t❛ ❞❛ ✈❛r✐á✈❡❧ t❡♠♣♦✱ ❝❛✉s❛❞❛ ♣♦r ❛❧❣✉♠ ❢❛t♦r ❡①t❡r♥♦✳ ■st♦ é✱ ♦ ❛❝♦♠♣❛♥❤❛♠❡♥t♦ ❞❡ ❛❧❣✉♠ ❡❧❡♠❡♥t♦ ❢♦✐ ✐♥t❡rr♦♠♣✐❞♦ ♣♦r ✉♠❛ ❝❛✉s❛ ❞✐❢❡r❡♥t❡ ❞❛ ❡s♣❡r❛❞❛✿ s❡ ❡st❛ ❢♦ss❡ ❛ ♠♦rt❡ ❞♦ ♣❛❝✐❡♥t❡ ♣♦r ❝â♥❝❡r✱ ❡❧❡ ♣♦❞❡r✐❛ t❡r ♠✉❞❛❞♦ ❞❡ ❝✐❞❛❞❡✱ ♦ ❡st✉❞♦ ♣♦❞❡r✐❛ t❡r t❡r♠✐♥❛❞♦ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞♦s ❞❛❞♦s ♦✉ ❡❧❡ ❛✐♥❞❛ ♣♦❞❡r✐❛ t❡r ♠♦rr✐❞♦ ❞❡✈✐❞♦ ❛ ♦✉tr❛ ❝❛✉s❛✳
❚♦❞♦s ♦s r❡s✉❧t❛❞♦s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ✉♠ ❡st✉❞♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ♠❡s♠♦ ❝❡♥s✉r❛❞♦s✱ ❞❡✈❡♠ s❡r ✉s❛❞♦s ♥❛ ❛♥á❧✐s❡ ❡st❛tíst✐❝❛ ❡ ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ é ❥✉st✐✜✲ ❝❛❞♦ ♣♦r ❞✉❛s r❛③õ❡s✿
✶✳ ♠❡s♠♦ s❡♥❞♦ ✐♥❝♦♠♣❧❡t❛s✱ ❛s ✐♥❢♦r♠❛çõ❡s ❢♦r♥❡❝✐❞❛s ♣❡❧❛s ♦❜s❡r✈❛çõ❡s ❝❡♥✲ s✉r❛❞❛s sã♦ ✐♠♣♦rt❛♥t❡s❀
✷✳ ❛ ♦♠✐ssã♦ ❞❛s ❝❡♥s✉r❛s ♥♦ ❝á❧❝✉❧♦ ❞❛s ❡st❛tíst✐❝❛s ❞❡ ✐♥t❡r❡ss❡ ❝❡rt❛♠❡♥t❡ ❛❝❛rr❡t❛rá ❡♠ ❝♦♥❝❧✉sõ❡s ✈✐❝✐❛❞❛s✳
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✾
❞✐r❡✐t❛ sã♦ ♠✉✐t♦ ❝♦♠✉♥s ❡♠ ❞❛❞♦s ❞❡ t❡♠♣♦ ❞❡ ✈✐❞❛✱ ❡♥q✉❛♥t♦ q✉❡ ❝❡♥s✉r❛s à ❡sq✉❡r❞❛ sã♦ ❜❛st❛♥t❡ r❛r❛s ✭▲❛✇❧❡ss✱ ✶✾✽✷✮✳
❖s três ♠❡❝❛♥✐s♠♦s ❞❡ ❝❡♥s✉r❛ ♠❛✐s ❝♦♠✉♥s sã♦✿
✶✳ ❝❡♥s✉r❛ ❞♦ t✐♣♦ ■✿ é ❛q✉❡❧❛ ❡♠ q✉❡ s❡ ♣ré✲❡st❛❜❡❧❡❝❡ ✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ♣❛r❛ ♦ tér♠✐♥♦ ❞♦ ❡st✉❞♦❀
✷✳ ❝❡♥s✉r❛ ❞♦ t✐♣♦ ■■✿ é ❛q✉❡❧❛ ♦♥❞❡ ♦ ❡st✉❞♦ s❡rá ❡♥❝❡rr❛❞♦ ❛♣ós ❛ ♦❝♦r✲ rê♥❝✐❛ ❞❛ ❢❛❧❤❛ ❡♠ ✉♠ ♥ú♠❡r♦ ♣ré✲❡st❛❜❡❧❡❝✐❞♦ ❞❡ ✐♥❞✐✈í❞✉♦s❀
✸✳ ❝❡♥s✉r❛ ❛❧❡❛tór✐❛✿ é ❛ q✉❡ ♠❛✐s ♦❝♦rr❡ ♥❛ ♣rát✐❝❛ ♠é❞✐❝❛ ❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ✉♠ ♣❛❝✐❡♥t❡ é r❡t✐r❛❞♦ ❞♦ ❡st✉❞♦ s❡♠ t❡r ♦❝♦rr✐❞♦ ❛ ❢❛❧❤❛✳
❆ ♣r❡s❡♥ç❛ ❞❡ ❝❡♥s✉r❛s ❛❝❛rr❡t❛ ♣r♦❜❧❡♠❛s ♣❛r❛ ❛ ✐♥❢❡rê♥❝✐❛ ❡st❛tíst✐❝❛✱ ❛❧❣✉♥s ✐♠♣♦ssí✈❡✐s ❞❡ s❡r❡♠ ❝♦♠♣❧❡t❛♠❡♥t❡ r❡s♦❧✈✐❞♦s✳ ❊♠❜♦r❛ ♦ ♠♦❞❡❧♦ ❞❡ ❝❡♥s✉r❛s ✐♥❞❡♣❡♥❞❡♥t❡s s❡❥❛ ❢r❡qü❡♥t❡♠❡♥t❡ r❛③♦á✈❡❧✱ ❡①✐st❡♠ s✐t✉❛çõ❡s ♦♥❞❡ ♦ ♠❡❝❛♥✐s♠♦ ❞❡ ❝❡♥s✉r❛ ❡stá ❧✐❣❛❞♦ ❛♦ ♣r♦❝❡ss♦ ❞❡ ❢❛❧❤❛ ❞♦ ❡①♣❡r✐♠❡♥t♦✳ ❊♠ t❛✐s ❝✐r❝✉♥stâ♥❝✐❛s✱ ❛té ♠❡s♠♦ ❞❡s❝r❡✈❡r ✉♠ ♠♦❞❡❧♦ q✉❡ r❡♣r❡s❡♥t❡ ❡st❡ ♣r♦❝❡ss♦ ♣♦❞❡ s❡ ♠♦str❛r ✉♠❛ t❛r❡❢❛ ár❞✉❛✳ ❋❡❧✐③♠❡♥t❡✱ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭❛♣r❡s❡♥✲ t❛❞❛ ❛ s❡❣✉✐r✮ ♠♦str❛✲s❡ ❛♣❧✐❝á✈❡❧ ❛ ✈ár✐❛s s✐t✉❛çõ❡s ❛♣❛r❡♥t❡♠❡♥t❡ ❝♦♠♣❧✐❝❛❞❛s✳
❖s ❡st✉❞♦s ❞❡ s✐♠✉❧❛çã♦ r❡❛❧✐③❛❞♦s ♥♦s ❈❛♣ít✉❧♦s ✸ ❡ ✹ ❝♦♥s✐❞❡r❛♠ ❛ ♣r❡s❡♥ç❛ ❞❡ ❝❡♥s✉r❛s ❛❧❡❛tór✐❛s ♥❛s ❛♠♦str❛s ✉t✐❧✐③❛❞❛s✱ ❛ ✜♠ ❞❡ ❡st✉❞❛r ♦s ❡❢❡✐t♦s ❝❛✉s❛❞♦s ♥❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❡ ❡♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❛ss✐♥tót✐❝❛s✳
✷✳✹ ❆ ❋✉♥çã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛
❖ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❡st✐♠❛çã♦ é ❜❛s❡❛❞♦ ♥❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ ❙✉♣♦♥❤❛ ✉♠❛ ❛♠♦str❛ ❛❧❡❛tór✐❛ ❞❡ ♦❜s❡r✈❛çõ❡st1, . . . , tn❞❡ ✉♠❛ ❝❡rt❛ ♣♦♣✉❧❛çã♦ ❞❡ ✐♥t❡r❡ss❡✳ ❈♦♥s✐❞❡r❡ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ t♦❞❛s ❛s ♦❜s❡r✈❛çõ❡s sã♦ ♥ã♦✲❝❡♥s✉r❛❞❛s✳ ❆ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ✉♠ ♣❛râ♠❡tr♦ ❣❡♥ér✐❝♦ θ é
L(θ) =
n
Y
i=1
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✶✵
❆ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ❝❛❞❛ ♦❜s❡r✈❛çã♦ ♥ã♦✲❝❡♥s✉r❛❞❛ ♣❛r❛ ❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ é ❛ s✉❛ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡✳ ❏á ❛s ♦❜s❡r✈❛çõ❡s ❝❡♥s✉r❛❞❛s✱ q✉❡ s♦♠❡♥t❡ ♥♦s ✐♥❢♦r♠❛♠ q✉❡ ♦ t❡♠♣♦ ❞❡ ❢❛❧❤❛ é ♠❛✐♦r q✉❡ ♦ t❡♠♣♦ ❞❡ ❝❡♥s✉r❛✱ ❝♦♥tr✐❜✉❡♠ ♣❛r❛L(θ)❝♦♠ s✉❛s ❢✉♥çõ❡s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛S(t)✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ à r❛③ã♦ ❡♥tr❡
❛s ❢✉♥çõ❡s ❞❡ ❞❡♥s✐❞❛❞❡ ❡ r✐s❝♦ ✭♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❛ ✶ ♠❡♥♦s ❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ F(t)✮✳
❆♦ s✉♣♦r q✉❡ ❛s ❝❡♥s✉r❛s ❡ ♦s t❡♠♣♦s ♦❜s❡r✈❛❞♦s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ❞❛❞♦s ❝❡♥s✉r❛❞♦s ❛ ❞✐r❡✐t❛ é ❞❛❞❛ ♣♦r
L(θ;t) =
n
Y
i=1
f(t;θ)δiS(t;θ)1−δi ✭✷✳✶✵✮
❖ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛❞rã♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡st✐♠❛r ♦s ♣❛râ♠❡tr♦s ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✱ ❛tr❛✈és ❞❛ ♠❛①✐♠✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ❧♦❣✲✈❡r♦ss✐♠✐❧❤❛♥ç❛ l(t;θ) = lnL(t;θ)✱ ♦♥❞❡ θ = (α, β, θ) ❡ t = (t1, t2, . . . , tn)✱ ✉♠❛ ❛♠♦str❛ ❛❧❡❛tór✐❛ ♦r❞❡♥❛❞❛ ✭✐st♦ é✱ t1 ≤ t2 ≤
. . .≤tn ✮ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✳
❆ ❢✉♥çã♦ ❧♦❣✲✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s é ❞❛❞❛ ♣♦r
l(t1, t2, . . . , tn;θ) = n
X
j=1
δj
ln(αβθ−α) + (α−1) ln tj +
tj θ
α
+ + (β−1) ❧♥ ❡(tj/θ)α −1
−
n
X
j=1
(1 +δj) ❧♥
1 + ❡(tj/θ)α−1β ✭✷✳✶✶✮
♦♥❞❡
δj =
0 s❡ ❛ ❥✲és✐♠❛ ♦❜s❡r✈❛çã♦ é ❝❡♥s✉r❛❞❛ à ❞✐r❡✐t❛, j = 1,2, . . . , n
1 ❝❛s♦ ❝♦♥trár✐♦.
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✶✶
U(θ) = ∂log(L(θ))
∂θ = 0.
❊st❡ s✐st❡♠❛ t❛♠❜é♠ é ❝❤❛♠❛❞♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ ▼ét♦❞♦s ♥✉♠ér✐❝♦s ♣❛r❛ ❧♦❝❛❧✐③❛r ♦ ♣♦♥t♦ q✉❡ ♠❛①✐♠✐③❛ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐✲ ♠✐❧❤❛♥ç❛ ❡♥✈♦❧✈❡♠ ✉♠❛ ❡st✐♠❛t✐✈❛ ✐♥✐❝✐❛❧ θˆ0 = (ˆα0,βˆ0,θˆ0) ❡ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦
✐t❡r❛t✐✈♦ q✉❡ ❝♦♥stró✐ ✉♠❛ s❡qüê♥❝✐❛ ❞❡ ♣♦♥t♦s ❝♦♥✈❡r❣❡♥t❡s ♣❛r❛ θˆ = (ˆα,β,ˆ θˆ)✳
❖ ❛❧❣♦r✐t♠♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞♦ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ❡♥✈♦❧✈❡ ♦ ✉s♦ ❞❛ ♠❛tr✐③ ❞❡ ❞❡r✐✈❛❞❛s s❡❣✉♥❞❛s ✭♦✉ ♠❛tr✐③ ❍❡ss✐❛♥❛✮ ❞❛ ❧♦❣✲✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ♥♦ ❆♣ê♥❞✐❝❡ ❉✳
✷✳✺ ❚❡st❡s ❞❡ ❍✐♣ót❡s❡
❚❡st❛r ❛ ❛❞❡q✉❛❜✐❧✐❞❛❞❡ ❞♦ ❛❥✉st❡ ❞❡ ✉♠ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ é ✉♠❛ t❛r❡❢❛ ❝♦♠♣❧✐❝❛❞❛ ❞❡✈✐❞♦ ❛♦ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ♠♦❞❡❧♦s q✉❡ ♦ tê♠ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r✳ ❆♦ r❡str✐♥❣✐r t❛✐s ❛❧t❡r♥❛t✐✈❛s à ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✱ ❛ ❡st❛tíst✐❝❛ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ♦s ♠♦❞❡❧♦s ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs♦ sã♦ ❛❞❡q✉❛❞♦s✳
❆s ❤✐♣ót❡s❡s ♥✉❧❛s✱ H011 : β = 1✱ H012 : (α = 1, β = 1)✱ H021 : β = −1
❡ H022 : (α = −1, β = −1) ❝♦rr❡s♣♦♥❞❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛♦s s✉❜♠♦❞❡❧♦s
❲❡✐❜✉❧❧✱ ❊①♣♦♥❡♥❝✐❛❧✱ ❲❡✐❜✉❧❧ ✐♥✈❡rs♦ ❡ ❊①♣♦♥❡♥❝✐❛❧ ✐♥✈❡rs♦ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✳ ❆ ❡st❛tíst✐❝❛ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ H0ij ✭i= 1,2;j =
1,2✮ é ✭❘❛♦✱ ✶✾✼✸❀ ▲❛✇❧❡ss✱ ✶✾✽✷✮
Λ = s✉♣R0ijL(α, β, θ;t)
s✉♣U RL(α, β, θ;t),
♦♥❞❡ R0ij r❡♣r❡s❡♥t❛ ♦ ❡s♣❛ç♦ ♣❛r❛♠étr✐❝♦ r❡str✐t♦ ♣❡❧❛ ❤✐♣ót❡s❡ ♥✉❧❛ H0ij✱ i =
1,2✱ j = 1,2✱ ❡ U R r❡♣r❡s❡♥t❛ ♦ ❡s♣❛ç♦ ♣❛r❛♠étr✐❝♦ ✐rr❡str✐t♦✳
❊♠ t❡r♠♦s ❞❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ❛s ❡st❛tíst✐❝❛s r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ♣❛r❛ ❝❛❞❛ ❤✐♣ót❡s❡✱ ♣♦❞❡♠ s❡r ❡s❝r✐t❛s ❝♦♠♦
Λ11 =
L(ˆαw, β = 1,θˆw)
L(ˆα,β,ˆ θˆ) , Λ12 =
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✶✷
Λ21=
L(ˆαiw, β =−1,θˆiw)
L(ˆα,β,ˆ θˆ) , Λ22 =
L(α=−1, β=−1,θˆie) L(ˆα,β,ˆ θˆ) .
❙♦❜ ❛s r❡s♣❡❝t✐✈❛s ❤✐♣ót❡s❡s ♥✉❧❛s✱ −2 ln Λ11✱−2 ln Λ12✱−2 ln Λ21 ❡−2 ln
Λ22 ♣♦ss✉❡♠✱ ❡♠ ♣r✐♥❝í♣✐♦✱ ❞✐str✐❜✉✐çã♦ ❛ss✐♥tót✐❝❛ χ2 ❝♦♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✷✱
✶✱ ✷ ❡ ✶ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳
✷✳✻ ●rá✜❝♦s ❚❚❚ ✭❚❡♠♣♦ ❚♦t❛❧ ❡♠ ❚❡st❡✮
❖s ♣❛râ♠❡tr♦s α ❡ β ❞❡ ✉♠ ♠❡♠❜r♦ ❞❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s ❞❡t❡r♠✐♥❛♠ ♦ ❢♦r♠❛t♦ ❞❛ s✉❛ ❢✉♥çã♦ ❞❡ r✐s❝♦✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ s❡ ♣r❡❞❡t❡r♠✐♥❛r ♣♦ssí✈❡✐s ✈❛❧♦r❡s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❡♠ q✉❡stã♦ é ❛tr❛✈és ❞♦ ❣rá✜❝♦ ❚❚❚ ❡♠♣ír✐❝♦✱ ❝✉❥♦ ❣rá✜❝♦ ♣r♦✈❛✲s❡ út✐❧ ❛♦ ✐♥❞✐❝❛r ❛ ❢♦r♠❛ ❞❛ ❢✉♥çã♦ ❞❡ r✐s❝♦ ❛ s❡r ❛❥✉st❛❞❛ ❛♦s ❞❛❞♦s ❛♥❛❧✐s❛❞♦s✳
❆ tr❛♥s❢♦r♠❛çã♦ ❚❚❚ ❡♠♣ír✐❝❛ é ❞❛❞❛ ♣♦r
φn
r
n
=
Pr
i=1T(Pi)+ (n−r)T(r)
n i=1Ti
,
♦♥❞❡r = 1,2, . . . , n❡ T(i), i= 1,2, . . . , nr❡♣r❡s❡♥t❛♠ ❛s ❡st❛tíst✐❝❛s ❞❡ ♦r❞❡♠ ❞❛
❛♠♦str❛✳
❖ ❣rá✜❝♦ ❞❡r/n♣♦rφn(r/n)❞❡✜♥✐❞♦ ❡♠ ✉♠ q✉❛❞r❛❞♦ ❞❡ ár❡❛ ✶ ♠♦str❛ q✉❛❧ ♦ ❢♦r♠❛t♦ ❞❛ ❢✉♥çã♦ ❞❡ r✐s❝♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦s ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✭❋✐❣✉r❛ ✷✳✸✮✳ ❙❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❚❚❚ ❡♠♣ír✐❝❛ ❢♦r ❝♦♥✈❡①❛✱ ❝ô♥❝❛✈❛✱ ❝♦♥✈❡①❛ ❡ ❝ô♥❝❛✈❛✱ ❡ ❝ô♥❝❛✈❛ ❡ ❝♦♥✈❡①❛ ✭❝♦♠♦ ♠♦str❛❞♦ ♥❛ ❋✐❣✉r❛ ✸✮✱ ❛s ❢✉♥çõ❡s ❞❡ r✐s❝♦ sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡✱ ❝r❡s❝❡♥t❡✱ ❜❛♥❤❡✐r❛ ❡ ✉♥✐♠♦❞❛❧ ✭❇❛r❧♦✇ ✫ ❈❛♠♣♦✱ ✶✾✼✹❀ ❆❛rs❡t✱ ✶✾✽✼❀ ▼✉❞❤♦❧❦❛r ❡t ❛❧✳✱ ✶✾✾✻✮✳
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✶✸
❋■●❯❘❆ ✷✳✸✿ ❈✉r✈❛s ❞❛s tr❛♥s❢♦r♠❛❞❛s ❚❚❚ ❡♠♣ír✐❝❛s✳ ❆s ❝✉r✈❛s ❆✱ ❇✱ ❈✱ ❉ ❡ ❊✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✐♥❞✐❝❛♠ q✉❡ ♦s ❞❛❞♦s ♣♦ss✉❡♠ t❛①❛s ❞❡ r✐s❝♦ ❝r❡s❝❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡✱ ❝♦♥st❛♥t❡✱ ❜❛♥❤❡✐r❛ ❡ ✉♥✐♠♦❞❛❧
✷✳✼ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ♥❛ Pr❡✲
s❡♥ç❛ ❞❡ ❈♦✈❛r✐á✈❡✐s
▼❡t♦❞♦❧♦❣✐❛s ✐♥❢❡r❡♥❝✐❛✐s ♣❛r❛ ❧✐❞❛r ❝♦♠ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❧✐❞❛♠ ❢r❡qü❡♥t❡♠❡♥t❡ ❝♦♠ ❛♠♦str❛s ✉♥✐✈❛r✐❛❞❛s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❞✐s✲ tr✐❜✉✐çã♦✳ ◆❛ ♣rát✐❝❛✱ ♠✉✐t❛s s✐t✉❛çõ❡s ❡♥✈♦❧✈❡♠ ♣♦♣✉❧❛çõ❡s ❤❡t❡r♦❣ê♥❡❛s✱ ❢❛③❡♥❞♦✲ s❡ ✐♠♣♦rt❛♥t❡ ❝♦♥s✐❞❡r❛r ❛ r❡❧❛çã♦ ❞♦ t❡♠♣♦ ❞❡ ✈✐❞❛ ❝♦♠ ♦✉tr♦s ❢❛t♦r❡s✳
✷✳ ❆ ❋❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ ❘❛③ã♦ ❞❡ ❈❤❛♥❝❡s ✶✹
r❡❣r❡ssã♦ ♣❛r❛♠étr✐❝♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♠♦str❛♠✲s❡ ❝♦♠♦ ❡①t❡♥sõ❡s ❞♦s ♠♦✲ ❞❡❧♦s ❞❡ t❡♠♣♦ ❞❡ ✈✐❞❛ ✉♥✐✈❛r✐❛❞♦s✱ ❝✉❥♦s ♣❛râ♠❡tr♦s ✭♦✉ ❛♣❡♥❛s ✉♠ s✉❜❝♦♥✲ ❥✉♥t♦ ❞❡st❡s✮ ❞❡♣❡♥❞❡♠ ❞❛s ✈❛r✐á✈❡✐s r❡❣r❡ss♦r❛s✳ ❆ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s✱ ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r ❞♦✐s ♣❛râ♠❡tr♦s ❞❡ ❢♦r♠❛ ❡ ✉♠ ❞❡ ❡s❝❛❧❛✱ ♣♦❞❡r✐❛ tê✲❧♦s ❞❡♣❡♥❞❡♥t❡s ❞❡ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❝♦✈❛r✐á✈❡✐s x= (1, x1, x2, . . . , xp)′✳
❙❡❣✉♥❞♦ ▲❛✇❧❡ss ✭✶✾✽✷✮✱ ✉♠ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ q✉❡ s❡ ♠♦str❛ ❛❞❡q✉❛❞♦ ❡♠ ❞✐✈❡rs❛s s✐t✉❛çõ❡s ❝♦♥s✐❞❡r❛ ❛♣❡♥❛s ♦ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛θ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❛s ❝♦✈❛r✐á✈❡✐s✳ ❯♠❛ ❝♦♥s✐❞❡r❛çã♦ ❞❡ss❛ ♥❛t✉r❡③❛ s✐❣♥✐✜❝❛ q✉❡ ♦ ❢♦r♠❛t♦ ❞❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ✭❡✱ ♣♦r ❝♦♥s❡qüê♥❝✐❛✱ ❞❛ ❢✉♥çã♦ ❞❡ r✐s❝♦✮ é ♦ ♠❡s♠♦ ♣❛r❛ q✉❛❧q✉❡r ♣♦ssí✈❡❧ ❝♦♥✜❣✉r❛çã♦ ❞❛s ❝♦✈❛r✐á✈❡✐s ❛♥❛❧✐s❛❞❛s✳ ❈♦♠♦ ♦ ♣❛râ♠❡tr♦ θé ♣♦s✐t✐✈♦✱ ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ❡s♣❡❝✐✜❝❛r ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ♣❛râ♠❡tr♦ ❡ ❛s ❝♦✈❛r✐á✈❡✐s é ❞❛❞❛ ♣♦r
θ(x) = exp(η′x),
♦♥❞❡ η′ = (η0, η1, η2, . . . , ηp)✳ ❉❡ss❛ ❢♦r♠❛✱ é ❣❛r❛♥t✐❞♦ q✉❡ θ(x) > 0 s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✐♠♣♦r q✉❛❧q✉❡r r❡str✐çã♦ s♦❜r❡ η✳
❙♦❜ ♦ ❝♦♥t❡①t♦ ❞❡ p ❝♦✈❛r✐á✈❡✐s ♥♦ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛✱ ❛s ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛✱ ❞❡♥s✐❞❛❞❡✱ r✐s❝♦ ❡ q✉❛♥t✐❧ ♣❛r❛ ❛ ❢❛♠í❧✐❛ ❲❡✐❜✉❧❧ ❞❡ r❛③ã♦ ❞❡ ❝❤❛♥❝❡s sã♦ ❞❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r
F(t;α, β,η,x) = 1−
1 +❡(t/exp(η′x))α −1β
−1
; 0< t <∞ ✭✷✳✶✷✮, αβ >0, η∈ ℜp+1
f(t;α, β,η,x) =
αβ t
t
exp(η′x)
α
❡(t/exp(η′x))α
❡(t/exp(η′x))α
−1β−1✭✷✳✶✸✮
×
1 +❡(t/exp(η′x))α−1β
−2
; 0 < t <∞, αβ >0, η ∈ ℜp+1 h(t;α, β,η,x) =
αβ t
t
exp(η′x)
α
❡(t/exp(η′x))α
❡(t/exp(η′x))α
−1β−1✭✷✳✶✹✮
×
1 +❡(t/exp(η′x))α −1β
−1
