Modelos Envolvendo Duas Distribuições Weibull Inversa e Apresentação da Weibull Inversa Generalizada

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

▼♦❞❡❧♦s ❊♥✈♦❧✈❡♥❞♦ ❉✉❛s ❉✐str✐❜✉✐çõ❡s

❲❡✐❜✉❧❧ ■♥✈❡rs❛ ❡ ❆♣r❡s❡♥t❛çã♦ ❞❛ ❲❡✐❜✉❧❧

■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛

❱❛❧❞✐❡❣♦ ❙✐q✉❡✐r❛ ▼❡❧♦

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

▼♦❞❡❧♦s ❊♥✈♦❧✈❡♥❞♦ ❉✉❛s ❉✐str✐❜✉✐çõ❡s

❲❡✐❜✉❧❧ ■♥✈❡rs❛ ❡ ❆♣r❡s❡♥t❛çã♦ ❞❛ ❲❡✐❜✉❧❧

■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛

❱❛❧❞✐❡❣♦ ❙✐q✉❡✐r❛ ▼❡❧♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r❛

Pr♦❢✳❛ _{❉❛♥✐❡❧❡ ❞❛ ❙✐❧✈❛ ❇❛r❛t❡❧❛ ▼❛rt✐♥s ◆❡t♦}

Ficha catalográfica elaborada pela Biblioteca Central da Universidade de Brasília. Acervo 1016196.

Me l o , Va l d i ego S i que i r a .

M528m Mode l os envo l vendo duas d i s t r i bu i ções We i bu l l i nve r sa e ap r esen t ação da We i bu l l i nve r sa gene r a l i zada / Va l d i ego S i que i r a Me l o . - - 2014 .

v i i i , 70 f . : i l . ; 30 cm.

Di sse r t ação (mes t r ado ) - Un i ve r s i dade de Br as í l i a , I ns t i t u t o de Ci ênc i as Exa t as e da Te r r a , Depa r t amen t o de Ma t emá t i ca , 2014 .

I nc l u i b i b l i og r a f i a .

Or i en t ação : Dan i e l e da S i l va Ba r a t e l a Ma r t i ns Ne t o . 1 . Di s t r i bu i ção de We i bu l l . I . Ma r t i ns Ne t o , Dan i e l e da S i l va Ba r a t e l a . I I . Tí t u l o .

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

▼♦❞❡❧♦s ❊♥✈♦❧✈❡♥❞♦ ❉✉❛s ❉✐str✐❜✉✐çõ❡s

❲❡✐❜✉❧❧ ■♥✈❡rs❛ ❡ ❆♣r❡s❡♥t❛çã♦ ❞❛ ❲❡✐❜✉❧❧

■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛

❱❛❧❞✐❡❣♦ ❙✐q✉❡✐r❛ ▼❡❧♦✶

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❈♦♠✐ssã♦ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳❛ _{❉❛♥✐❡❧❡ ❙✳ ❇❛r❛t❡❧❛ ▼❛rt✐♥s ◆❡t♦ ✲ ❖r✐❡♥t❛❞♦r❛ ✭▼❆❚✴❯♥❇✮}

Pr♦❢✳❛ _{❈át✐❛ ❘❡❣✐♥❛ ●♦♥ç❛❧✈❡s ✲ ▼❡♠❜r♦ ✭▼❆❚✴❯♥❇✮}

Pr♦❢✳ ❆r② ❱❛s❝♦♥❝❡❧♦s ▼❡❞✐♥♦ ✲ ▼❡♠❜r♦ ✭▼❆❚✴❯♥❇✮

❇r❛sí❧✐❛✱ ✵✶ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✹

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦s ♠❡✉s ♣❛✐s✱ ▲♦✉r❞❡s ❡ ❱❛❧❞♦✱ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠✐♠ ❡ ♠❡ ❛♣♦✐❛r❡♠ ♥❡st❛ ✐♠♣♦rt❛♥t❡ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ➚ ♠✐♥❤❛ ✐r♠ã ❡ ❛♠✐❣❛✱ ❱❛♥❡ss❛✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ♣❛❧❛✈r❛s ❞❡ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♠❛✐s ♣r❡❝✐s❡✐✳

➚ ♣r♦❢❡ss♦r❛ ❉❛♥✐❡❧❡✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ❞❡❞✐❝❛çã♦ ❡ ✐♥❝❡♥t✐✈♦✳ ➚s ♣r♦❢❡ss♦r❛s ❈át✐❛ ❡ ❉é❜♦r❛✱ ❛❣r❛❞❡ç♦ ♣❡❧❛s ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s✳ ❆♦ ♣r♦❢❡ss♦r ❆r② ♣♦r ❛❝❡✐t❛r ❝♦♠♣♦r ❛ ❜❛♥❝❛ ❞❡ ú❧t✐♠❛ ❤♦r❛ ♥♦ ❧✉❣❛r ❞❛ ♣r♦❢❛ ❉é❜♦r❛✳

❆♦ ♠❡✉ ❛♠✐❣♦✴✐r♠ã♦ ❱✐♥í❝✐✉s ♣❡❧❛ ♣❛r❝❡r✐❛ ❡ ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❊❧✐❛♥❛✱ ❏❤♦♥❛tt❛♥✱ ▼❛①✱ ▲❛ís ❡ ➱❞❡r✳ ❆ ❛♠✐③❛❞❡ ❡ ❝♦♠♣❛♥❤❡✐✲ r✐s♠♦ ❞❡ ✈♦❝ês ❢♦✐ ❡ss❡♥❝✐❛❧✳

❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ♣♦r s✉❛s ❝♦♥tr✐✲ ❜✉✐çõ❡s à ♠✐♥❤❛ ❢♦r♠❛çã♦✳

➚ ❈❛♣❡s ❡ ❛♦ ❈◆P◗ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❡♠ ❢♦r♠❛ ❞❡ ❜♦❧s❛ ❞❡ ❡st✉❞♦s✳

❘❡s✉♠♦

▼♦❞❡❧♦s ❲❡✐❜✉❧❧ tê♠ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❞✐s❝✉t✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛✳ ❊ss❡ ❣r❛♥❞❡ ✐♥t❡✲ r❡ss❡ t❡♠ r❡✢❡t✐❞♦ ♥♦ s✉r❣✐♠❡♥t♦ ❞❡ ♠♦❞✐✜❝❛çõ❡s ❡ ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ ♠♦❞❡❧♦s ❲❡✐❜✉❧❧✳ Pr✐♠❡✐r❛♠❡♥t❡ ♥❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s três ♠♦❞❡❧♦s ✭r✐s❝♦ ❝♦♠♣❡t✐t✐✈♦✱ ♠✉❧t✐♣❧✐✲ ❝❛t✐✈♦ ❡ ♠✐st✉r❛✮ q✉❡ ❡♥✈♦❧✈❡♠ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ❜❛s❡❛❞♦s ♥♦ ❛rt✐❣♦ ❞❡ ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪✳ ❆s ❢♦r♠❛s ❞❛s ❢✉♥çõ❡s ❞❡♥s✐❞❛❞❡ ❡ t❛①❛ ❞❡ ❢❛❧❤❛ sã♦ ❛♣r❡s❡♥t❛✲ ❞❛s ❡ ♠ét♦❞♦s ❣rá✜❝♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♦❜s❡r✈❛çõ❡s ♣♦❞❡ s❡r ❛❥✉st❛❞♦ ♣♦r ✉♠ ❞❡ss❡s ♠♦❞❡❧♦s sã♦ ❞✐s❝✉t✐❞♦s✳ ◆✉♠ s❡❣✉♥❞♦ ♠♦♠❡♥t♦✱ ♥ós ❛♣r❡s❡♥✲ t❛♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛ ❞❡ ●✉s♠ã♦ ❡t ❛❧✳ ❬✽❪ ❡ ❞✐s❝✉t✐♠♦s ❛ ❡st✐♠❛çã♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠ ❞❛❞♦s ❝❡♥s✉r❛❞♦s✳ ❖ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ❞❡ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛ é ✐♥✈❡st✐❣❛❞♦ ❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ é ❞❡♠♦♥str❛❞❛✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❲❡✐❜✉❧❧ ■♥✈❡rs❛✱ ▼♦❞❡❧♦ ❞❡ ❘✐s❝♦✱ ▼♦❞❡❧♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✱ ▼♦❞❡❧♦ ❞❡ ▼✐st✉r❛✱ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡✱ ●rá✜❝♦✲■❲PP✳

❆❜str❛❝t

❲❡✐❜✉❧❧ ♠♦❞❡❧s ❤❛✈❡ ❜❡❡♥ ❡①t❡♥s✐✈❡❧② ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❚❤✐s ❣r❡❛t ✐♥✲ t❡r❡st ✐s r❡✢❡❝t❡❞ ✐♥ t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ ♠♦❞✐✜❝❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ❲❡✐❜✉❧❧ ♠♦❞❡❧s✳ ❋✐rst❧② ✐♥ t❤✐s ✇♦r❦✱ ✇❡ ♣r❡s❡♥t t❤r❡❡ ♠♦❞❡❧s ✭❝♦♠♣❡t✐♥❣ r✐s❦✱ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❛♥❞ ♠✐①t✉r❡✮ ✐♥✈♦❧✈✐♥❣ t✇♦ ✐♥✈❡rs❡ ❲❡✐❜✉❧❧ ❞✐str✐❜✉t✐♦♥s✱ ❜❛s❡❞ ♦♥ ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪ ♣❛♣❡r✳ ❚❤❡ s❤❛♣❡s ♦❢ t❤❡ ❞❡♥s✐t② ❛♥❞ ❢❛✐❧✉r❡ r❛t❡ ❢✉♥❝t✐♦♥s ❛r❡ s❤♦✇♥ ❛♥❞ ❣r❛♣❤✐❝❛❧ ♠❡t❤♦❞s t♦ ❞❡t❡r♠✐♥❡ ✐❢ ❛ ❣✐✈❡♥ ❞❛t❛ s❡t ❝❛♥ ❜❡ ❛❞❥✉st❡❞ ❜② ♦♥❡ ♦❢ t❤❡s❡ ♠♦❞❡❧s ❛r❡ ❞✐s❝✉ss❡❞✳ ❙❡❝♦♥❞❧②✱ ✇❡ ♣r❡s❡♥t ❛ ❣❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ❲❡✐❜✉❧❧ ❞✐str✐❜✉t✐♦♥ ❜② ●✉s♠ã♦ ❡t ❛❧✳ ❬✽❪ ❛♥❞ ❞✐s❝✉ss t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ✇✐t❤ ❝❡♥s♦r❡❞ ❞❛t❛✳ ❚❤❡ ♠✐①t✉r❡ ♠♦❞❡❧ ♦❢ t✇♦ ❣❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ❲❡✐❜✉❧❧ ❞✐str✐❜✉t✐♦♥s ✐s ✐♥✈❡st✐❣❛t❡❞✳ ❚❤❡ ✐❞❡♥t✐✜❛❜✐❧✐t② ♣r♦♣❡rt② ♦❢ t❤❡ ♠✐①❡❞ ♠♦❞❡❧ ✐s ❞❡♠♦♥str❛t❡❞✳

❑❡②✇♦r❞s✿ ■♥✈❡rs❡ ❲❡✐❜✉❧❧✱ ❈♦♠♣❡t✐♥❣ ❘✐s❦ ▼♦❞❡❧✱ ▼✉❧t✐♣❧✐❝❛t✐✈❡ ▼♦❞❡❧✱ ▼✐①t✉r❡ ▼♦❞❡❧✱ ■❞❡♥t✐✜❛❜✐❧✐t②✱ ■❲PP✲♣❧♦t✳

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❋✐❣✉r❛s ✈✐✐✐

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✺

✶✳✶ ❆❥✉st❡ ❞❡ ❘❡t❛ ♣♦r ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸✳✸ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳✹ ●rá✜❝♦s ▼♦❞✐✜❝❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✹✳✶ ❖ ●rá✜❝♦✲❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✹✳✷ ❖ ●rá✜❝♦✲■❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸✳✹✳✸ ❈♦♠♣❛r❛çã♦ ❊♥tr❡ ♦s ●rá✜❝♦s ❲PP ❡ ■❲PP ❡ ❊st✐✲

♠❛çã♦ ❞❡ P❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✹ ▼✐st✉r❛ ❞❡ ❉✐str✐❜✉✐çõ❡s ❡ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷ ❆♥á❧✐s❡ ❞❡ ❉✉❛s ❉✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✷✾

✷✳✶ ▼♦❞❡❧♦ ❞❡ ❘✐s❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✶✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❡ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✳✷ ●rá✜❝♦ ▼♦❞✐✜❝❛❞♦ ✲ ■❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷ ▼♦❞❡❧♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❡ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷✳✷ ●rá✜❝♦ ▼♦❞✐✜❝❛❞♦ ✲ ■❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✸ ▼♦❞❡❧♦ ❞❡ ▼✐st✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✸✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❡ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✷✳✸✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✸✳✸ ●rá✜❝♦ ▼♦❞✐✜❝❛❞♦ ✲ ■❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✸✳✹ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✸ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛ ✺✻

✸✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✸ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✹ ❊st✐♠❛çã♦ ❞❡ ▼á①✐♠❛ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠ ❉❛❞♦s ❈❡♥s✉r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✺ ▼✐st✉r❛ ❞❡ ❉✉❛s ❉✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✺✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❡ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✺✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✺✳✸ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❆❥✉st❡s q✉❡ ❝♦♥s✐❞❡r❛♠ ♦ ♠í♥✐♠♦ ❞❛ s♦♠❛ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞♦s ❡rr♦s✳ ✻ ✶✳✷ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡β ❡

♣❛r❛ λ= 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✶✳✸ ●rá✜❝♦s ❲PP ♣❛r❛ ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ♣❛r❛ α= 5 ❡β = 3,5✳ ✳ ✳ ✶✽

✶✳✹ ●rá✜❝♦s ■❲PP ♣❛r❛ ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ♣❛r❛α = 5 ❡ β= 3,5✳ ✳ ✷✵

✷✳✶ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✺ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✻ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✼ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✽ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✾ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ✲ ❝❛s♦ ✭❛✮ ❡α1 < α2✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✷✳✶✵ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ✲ ❝❛s♦ ✭❛✮ ❡α1 > α2✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✷✳✶✶ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ✲ ❝❛s♦ ✭❜✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✸✳✶ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ❛❧✲ ❣✉♥s ✈❛❧♦r❡s ❞❡ α ❝♦♠ β = 5 ❡ γ = 0,5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✸✳✷ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ α ❝♦♠ β = 5 ❡ γ = 0,5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✸✳✸ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✹ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

■♥tr♦❞✉çã♦

❆ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ t❡♠ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❡st✉❞❛❞❛ ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s✳ ❆ ♣♦♣✉❧❛r✐❞❛❞❡ ❞❡st❛ ❞✐str✐❜✉✐çã♦ é ❡✈✐❞❡♥❝✐❛❞❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠ ❡st✉❞♦s ❞❡ ❝♦♥✜❛❜✐✲ ❧✐❞❛❞❡ ❡ ♥❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ♥♦ ❝♦♥t❡①t♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❢✉♥çõ❡s ❞❡ r✐s❝♦✳ ❙✉❛ ❤❛❜✐❧✐❞❛❞❡ ❞❡ ♠♦❞❡❧❛r t❛①❛s ❞❡ ❢❛❧❤❛s r❡s✐❞❡ ♥❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✳

◆❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛♠♦s ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ❞❡ ❞♦✐s ♣❛râ♠❡tr♦s q✉❡ t❡♠ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❛❞❛ ♣♦r

F(t) = 1₋e−(_αt)β

, t _≥0, ✭✶✮

♦♥❞❡α✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛✱ ❡β✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✱ sã♦ ♣♦s✐t✐✈♦s✳ ❯♠❛ ✈❛r✐❡❞❛❞❡

❞❡ ♠♦❞❡❧♦s t❡♠ s✐❞♦ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ ✭✶✮ ♥❛ t❡♥t❛t✐✈❛ ❞❡ ♠♦❞❡❧❛r ♦✉tr❛s ❢♦r♠❛s ❞❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛✳ ❯♠ ❞❡ss❡s ♠♦❞❡❧♦s ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪ ❛ ♣❛rt✐r ❞❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦✿ ❝♦♥s✐❞❡r❡Y ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❞❡✜♥✐❞❛ ♣♦r

Y = α

2

X,

♦♥❞❡X é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ ❲❡✐❜✉❧❧✳ ❆ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦

❞❡Y é ❞❛❞❛ ♣♦r

F(t) =P(Y _≤t) = e−(α t)

β

, t >0, ✭✷✮

♦♥❞❡ α, β > 0✳ ◆❡st❡ ❝❛s♦✱ s❡ Y é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐✲

çã♦ ❞❛❞❛ ♣♦r ✭✷✮✱ ❡♥tã♦ é ❞✐t❛ t❡r ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✳ ❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ✐♠♣♦rt❛♥t❡ ❞❡st❛ ❞✐str✐❜✉✐çã♦ é q✉❡ ❛ s✉❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ✉♥✐♠♦❞❛❧✳

◆❛ ❧✐t❡r❛t✉r❛ ❛❧❣✉♥s ♠♦❞❡❧♦s ❡♥✈♦❧✈❡♥❞♦ n ❞✐str✐❜✉✐çõ❡s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s sã♦ ♦❜✲

s❡r✈❛❞♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✱ ♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡ ♦ ❞❡ ♠✐st✉r❛✳ P❛r❛ ❡♥t❡♥❞ê✲❧♦s✱ ❛♣r❡s❡♥t❛♠♦s✿

• ❙❡❥❛♠T1, . . . , Tn ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s✱ s❡♥❞♦F1(t), . . . , Fn(t)❛s r❡s✲

■♥tr♦❞✉çã♦ ✷

❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ é ❞❛❞❛ ♣♦r

F(t) = 1₋

n Y

i=1

[1₋Fi(t)].

❊st❡ ♠♦❞❡❧♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✳ ❖✉tr❛s t❡r♠✐♥♦❧♦❣✐❛s t❛♠❜é♠ sã♦ ✉t✐❧✐③❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♠♦❞❡❧♦ ❝♦♠♣♦st♦✱ ♠♦❞❡❧♦ ❞❡ s✐st❡♠❛ ❡♠ sér✐❡ ❡ ♠♦❞❡❧♦ ♠✉❧t✐rr✐s❝♦✳ ❊❧❡ t❡♠ r❡❝❡❜✐❞♦ ❝♦♥s✐❞❡rá✈❡❧ ❛t❡♥çã♦ ♥❛ ❧✐t❡r❛t✉r❛ ❡ t❡♠ s✐❞♦ ❡st✉❞❛❞♦ ❤á ❜❛st❛♥t❡ t❡♠♣♦✳ ◆♦ ❝♦♥t❡①t♦ ❞❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✱ ♦ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ s✐st❡♠❛ ❞❡n❝♦♠♣♦♥❡♥t❡s

❡♠ sér✐❡✱ s❡♥❞♦ Ti ♦ t❡♠♣♦ ❞❡ ✈✐❞❛ ❞❛ ❝♦♠♣♦♥❡♥t❡i❡Z ♦ ❞♦ s✐st❡♠❛✳ ❖ s✐st❡♠❛

❢❛❧❤❛ ♥♦ ♣r✐♠❡✐r♦ ✐♥st❛♥t❡ q✉❡ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❢❛❧❤❛✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✱ ✈❡❥❛ ◆❡❧s♦♥ ❬✶✺❪✳

• ❙❡❥❛♠T1, . . . , Tn ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s✱ s❡♥❞♦F1(t), . . . , Fn(t)❛s r❡s✲

♣❡❝t✐✈❛s ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❛ss♦❝✐❛❞❛s✳ ❙❡ W = max_{T1, . . . , Tn}✱ ❡♥tã♦ ❛

s✉❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ é ❞❛❞❛ ♣♦r

F(t) = P(W _≤t) =P

n \

i=1

(Ti ≤t) !

=

n Y

i=1 Fi(t).

❊st❡ ♠♦❞❡❧♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡✱ ❛♣❡s❛r ❞❡❧❡ ♥ã♦ t❡r s❡ ❞❡st❛❝❛❞♦ t❛♥t♦ q✉❛♥t♦ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✱ ❝♦♠♦ ♦❜s❡r✈❛❞♦ ♣♦r ▼✉rt❤② ❡t ❛❧✳ ❬✶✹❪✱ t❡♠ s✐❞♦ ❛♣❧✐❝❛❞♦ ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ♥♦ ❝♦♥t❡①t♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ✉♠ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦ ❝♦♠ ❝♦♠♣♦♥❡♥t❡s ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❯♠ s✐st❡♠❛ ♣♦❞❡ s❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ✉♠ ♦r❣❛♥✐s♠♦ ♦✉ ✉♠ ❞✐s♣♦s✐t✐✈♦ ♠❡❝â♥✐❝♦✱ ❡ ♦ t❡r♠♦ ♣❛r❛❧❡❧♦ s✐❣♥✐✜❝❛ q✉❡ ❛ s✉❛ ❢❛❧❤❛ ❡stá ❝♦♥❞✐❝✐♦♥❛❞❛ à ❢❛❧❤❛ ❞❡ t♦❞❛s ❛s s✉❛s ♣❛rt❡s✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✱ ✈❡❥❛ ❊❧❛♥❞t✲❏♦❤♥s♦♥ ❡ ❏♦❤♥s♦♥ ❬✼❪✳

• ▼♦❞❡❧♦s ❞❡ ♠✐st✉r❛s ✜♥✐t❛s tê♠ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❞✐s❝✉t✐❞♦s ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s✳

❊ss❡ ✐♥t❡r❡ss❡ é ❞❡❝♦rr❡♥t❡ ❞❡ s✉❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ ár❡❛s ❞✐✈❡rs❛s ❝♦♠♦ ❇✐♦❧♦✲ ❣✐❛✱ ❋ís✐❝❛✱ ❊❝♦♥♦♠✐❛✱ ▼❡❞✐❝✐♥❛ ❡ ❊♥❣❡♥❤❛r✐❛✳ ❊♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s✱ ♦s ❞❛❞♦s ❞✐s♣♦♥í✈❡✐s ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ❝♦♠♦ s❡♥❞♦ ❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ❞❡ ♠✐st✉r❛ ❞❡ ❞✉❛s ♦✉ ♠❛✐s ❞✐str✐❜✉✐çõ❡s✳ ❚✐tt❡r✐♥❣t♦♥ ❡t ❛❧✳ ❬✶✽❪ ❡ ▼❝▲❛❝❤❧❛♥ ❡ P❡❡❧ ❬✶✻❪ sã♦ ❛❧❣✉♠❛s ❞❛s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ♠✐st✉r❛s ✜♥✐t❛s✳

■♥tr♦❞✉çã♦ ✸

❱ár✐♦s ❝❛s♦s ❡s♣❡❝✐❛✐s ❞♦s ♠♦❞❡❧♦s ♠❡♥❝✐♦♥❛❞♦s ❛❝✐♠❛ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s✱ ❞❡♣❡♥✲ ❞❡♥❞♦ ❞♦ ♥ú♠❡r♦ ❞❡ s✉❜♣♦♣✉❧❛çõ❡s ❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❛ss♦❝✐❛❞❛ ❛ ❝❛❞❛ ✉♠❛ ❞❡ss❛s s✉❜♣♦♣✉❧❛çõ❡s✳ ❯♠ ❞♦s ♦❜❥❡t✐✈♦s ❞❡st❡ ❡st✉❞♦ é ❛♥❛❧✐s❛r ❡ss❡s ♠♦❞❡❧♦s ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡n= 2 ❡ ❛♠❜❛s ❛s s✉❜♣♦♣✉❧❛çõ❡s s❡❣✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✳ ▼✉rt❤②

❡t ❛❧✳ ❬✶✹❪ r❡ú♥❡♠ ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ❡st✉❞♦s r❡❧❛❝✐♦♥❛❞♦s às ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ♣✉❜❧✐❝❛❞♦s ❛té ♦ ✐♥í❝✐♦ ❞♦s ❛♥♦s ✷✵✵✵ ❡♥✈♦❧✈❡♥❞♦ ❡ss❡s ♠♦❞❡❧♦s✳ ❈♦♠♦ ❡①❡♠♣❧♦ ♣♦❞❡♠♦s ❝✐t❛r ❏✐❛♥❣ ❡ ▼✉rt❤② ❬✾❪ ❡ ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪✱ ❞✉❛s ♣r✐♥❝✐♣❛✐s r❡❢❡✲ rê♥❝✐❛s ❞♦ ♥♦ss♦ ❡st✉❞♦✳ ❖ ♣r✐♠❡✐r♦ ❛♣r❡s❡♥t❛ ✉♠ ♠ét♦❞♦ ❣rá✜❝♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ❞❡ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧❀ ♦ s❡❣✉♥❞♦ ❛♣r❡s❡♥t❛ ❛ ❢♦r♠❛ ❞❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❡ ❞❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❞❡ ♠✐st✉r❛✱ ❞❡ r✐s❝♦ ❡ ♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ❛❧é♠ ❞❡ ✉♠❛ té❝♥✐❝❛ ❣rá✜❝❛ ♣❛r❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ss❡s ♠♦❞❡❧♦s✳

❚❛♠❜é♠ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ ♠❛✐s r❡❝❡♥t❡✱ ♦♥❞❡ ●✉s♠ã♦ ❡t ❛❧✳ ❬✽❪ ♣r♦♣õ❡♠ ✉♠❛ ♥♦✈❛ ❞✐str✐❜✉✐çã♦ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✳ ❆ s✉❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦✱

G(t)✱ ♣r♦♣♦st❛ ♣❡❧♦s ❛✉t♦r❡s ❡♠ ❬✽❪✱ é ❞❛❞❛ ♣♦r

G(t) = [F(t)]γ =e−γ(α_t)β

, t >0. ✭✸✮

❆ ❞✐str✐❜✉✐çã♦ ❛ss♦❝✐❛❞❛ à ✭✸✮ é ❝❤❛♠❛❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛✳ ❊st❛ ❞✐str✐❜✉✐çã♦ t❡♠ ✉♠❛ ✢❡①✐❜✐❧✐❞❛❞❡ ❛✐♥❞❛ ♠❛✐♦r✱ ❞❡✈✐❞♦ ❛ ✐♥❝❧✉sã♦ ❞♦ ♣❛râ♠❡tr♦γ✱

q✉❡ ♣❡r♠✐t❡ ♦❜t❡r ❝♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ❡①♣♦♥❡♥❝✐❛❧ ✐♥✈❡rs❛ ❡ ❘❛②❧❡✐❣❤ ✐♥✈❡rs❛✳ ❖✉tr❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡st❛ ❞✐str✐❜✉✐çã♦ é q✉❡ ❛ s✉❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ t❡♠ ❢♦r♠❛ ✉♥✐♠♦❞❛❧✳

❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ♦s ♣r✐♥❝✐♣❛✐s tó♣✐❝♦s ❛❜♦r❞❛❞♦s ❡♠ ❝❛❞❛ ❝❛♣ít✉❧♦✳

❖ ❈❛♣ít✉❧♦ ✶ é ❞❡❞✐❝❛❞♦ à ❛♣r❡s❡♥t❛çã♦ ❞❡ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ r❡st❛♥t❡ ❞♦ tr❛❜❛❧❤♦✳ ▼♦str❛♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ é ✉♥✐♠♦❞❛❧✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♠ét♦❞♦ ❣rá✜❝♦✱ ❝❤❛♠❛❞♦ ❣rá✜❝♦✲■❲PP✱ ♣❛r❛ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ❡ ❞✉❛s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s✱ ❞❡ ❈❤❛♥❞r❛ ❬✹❪ ❡ ❆t✐❡♥③❛ ❡t ❛❧✳ ❬✶❪✱ ♣❛r❛ ❣❛r❛♥t✐r ❛ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠✐st✉r❛s ✜♥✐t❛s ❞❡ ❞✐str✐❜✉✐çõ❡s✳

■♥tr♦❞✉çã♦ ✹

Cap´ıtulo

1

Pr❡❧✐♠✐♥❛r❡s

❊st❡ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ à ❛♣r❡s❡♥t❛çã♦ ❞❡ r❡s✉❧t❛❞♦s✱ ❝♦♥❝❡✐t♦s ❡ ❞❡✜♥✐çõ❡s ✐♠✲ ♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳ ■♥✐❝✐❛♠♦s ♥❛ ❙❡çã♦ ✶✳✶ ❝♦♠ ♦ ♠ét♦❞♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♣❛r❛ ❛❥✉st❡ ❞❡ r❡t❛s✳ ◆❛ ❙❡çã♦ ✶✳✷✱ ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦ ❞❡ s♦✲ ❜r❡✈✐✈ê♥❝✐❛ ❡ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛✱ ✐♠♣♦rt❛♥t❡s ❡♠ ❡st✉❞♦s ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❡ ❡①❡♠♣❧✐✜❝❛♠♦✲❛s ♣♦r ♠❡✐♦ ❞❛s ❞✐str✐❜✉✐çõ❡s ❡①♣♦♥❡♥❝✐❛❧ ❡ ❲❡✐❜✉❧❧✳ ❆s r❡❢❡rê♥❝✐❛s ❜ás✐❝❛s ♣❛r❛ ❡ss❛ s❡çã♦ sã♦✿ ❇♦❧❢❛r✐♥❡ ❡t ❛❧✳ ❬✷❪✱ ❈♦❧♦s✐♠♦ ❡ ●✐♦❧♦ ❬✺❪ ❡ ▲❛✇❧❡ss ❬✶✷❪✳ ◆❛ ❙❡çã♦ ✶✳✸✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❝♦♠♦ ✉♠❛ ♠♦❞✐✜❝❛çã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧✱ s❡❣✉✐♥❞♦ ♦ tr❛❜❛❧❤♦ ❞❡ ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪✳ ❆❧é♠ ❞✐ss♦✱ ❡st✉❞❛♠♦s ❛s ❢♦r♠❛s ❞❛s ❢✉♥çõ❡s ❞❡♥s✐❞❛❞❡ ❡ t❛①❛ ❞❡ ❢❛❧❤❛✱ ❝❛❧❝✉❧❛♠♦s ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❡ ❞✐s❝✉t✐♠♦s ❞✉❛s té❝♥✐❝❛s ❣rá✜❝❛s✱ ❛♣r❡s❡♥t❛♥❞♦ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ❜❛s❡❛❞♦s ♥❡ss❡s ♠ét♦❞♦s✳ ◆❛ ❙❡çã♦ ✶✳✹ ✜♥❛❧✐③❛♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠ ❡st✉❞♦ s♦❜r❡ ♠✐st✉r❛s ✜♥✐t❛s ❞❡ ❞✐str✐❜✉✐çõ❡s ❡ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡✱ ♦♥❞❡ ❞❡♠♦♥str❛♠♦s ❞♦✐s r❡s✉❧t❛❞♦s✱ ❞❡ ❈❤❛♥❞r❛ ❬✹❪ ❡ ❆t✐❡♥③❛ ❡t ❛❧✳ ❬✶❪✱ q✉❡ ❣❛r❛♥t❡♠ ❛ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠✐st✉r❛s ✜♥✐t❛s ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ ✉♠❛ ♠❡s♠❛ ❢❛♠í❧✐❛ ♣❛r❛♠étr✐❝❛✳ ❆ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠✐st✉r❛s ❞❡ ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ é t❛♠❜é♠ ❞❡♠♦♥str❛❞❛✳

✶✳✶ ✲ ❆❥✉st❡ ❞❡ ❘❡t❛ ♣♦r ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s

✶✳✶✳ ❆❥✉st❡ ❞❡ ❘❡t❛ ♣♦r ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ✻

✉♠ ❜♦♠ ❛❥✉st❡✳ ■st♦ ♦❝♦rr❡ ♣♦rq✉❡ ♦ ❡rr♦ ♣♦❞❡ s❡r t❛♥t♦ ♣♦s✐t✐✈♦ ✭q✉❛♥❞♦ ♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ❡st✐✈❡r ❛❝✐♠❛ ❞❛ r❡t❛✮ q✉❛♥t♦ ♥❡❣❛t✐✈♦ ✭q✉❛♥❞♦ ♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ❡st✐✈❡r ❛❜❛✐①♦ ❞❛ r❡t❛✮✱ ♦ q✉❡ ♣♦❞❡ ❧❡✈❛r ❛ ✉♠❛ s♦♠❛ ♥✉❧❛ ❞♦s ❡rr♦s ♠❡s♠♦ ❝♦♠ ♦s ✈❛❧♦r❡s ♠✉✐t♦ ❞✐st❛♥t❡s ❞❛ r❡t❛✳

❯♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ s✉♣❡r❛r ❡st❡ ♣r♦❜❧❡♠❛ é ♠✐♥✐♠✐③❛r ❛ s♦♠❛ ❞♦s ✈❛❧♦r❡s ❛❜s♦✲ ❧✉t♦s ❞♦s ❡rr♦s✱ ✉♠❛ ✈❡③ q✉❡ ✐st♦ ❡✈✐t❛ q✉❡ ❣r❛♥❞❡s ❡rr♦s ♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s s❡❥❛♠ ❝♦♠♣❡♥s❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱ t❛❧ ♠ét♦❞♦ ❛✐♥❞❛ ❛♣r❡s❡♥t❛ ✉♠❛ ❞❡s✈❛♥t❛❣❡♠✳ P❡❧❛ ❋✐❣✉r❛ ✶✳✶✱ ♥♦t❡ q✉❡ ♦ ❛❥✉st❡ ❡♠ (b) s❛t✐s❢❛③ ❡st❡ ❝r✐tér✐♦ ♠❡❧❤♦r q✉❡ ♦ ❛❥✉st❡ ❡♠ (a)✳ ▼❛s

❡st❛ ♥ã♦ é ✉♠❛ ❜♦❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✱ ♣♦✐s ♥❡♥❤✉♠❛ ❛t❡♥çã♦ é ❞❛❞❛ ❛♦ ♣♦♥t♦ ❞♦ ♠❡✐♦✳ ❖ ❛❥✉st❡ ❡♠ (a)é ♣r❡❢❡rí✈❡❧ ♣♦rq✉❡ ❧❡✈❛ ❡♠ ❝♦♥s✐❞❡r❛çã♦ t♦❞♦s ♦s ♣♦♥t♦s✳

❯♠❛ ♦✉tr❛ ♦♣çã♦ ♣❛r❛ t❡♥t❛r♠♦s s✉♣❡r❛r ♦ ♥♦ss♦ ♣r♦❜❧❡♠❛ ✐♥✐❝✐❛❧ é ♠✐♥✐♠✐③❛r ❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞♦s ❡rr♦s✳ ❊st❡ ❝r✐tér✐♦ é ❝❤❛♠❛❞♦ ❞❡ ♠ét♦❞♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s✳ ❊❧❡ é ❜❡♠ út✐❧ ❡ ♣rát✐❝♦✱ ✉♠❛ ✈❡③ q✉❡ ♦s ❝á❧❝✉❧♦s ❡♥✈♦❧✈✐❞♦s sã♦ ❜❡♠ s✐♠♣❧❡s✳ ❆♦ ❝♦♥s✐❞❡r❛r♠♦s ♦ q✉❛❞r❛❞♦ ❞♦s ❡rr♦s✱ ❡st❡s s❡ t♦r♥❛♠ ♣♦s✐t✐✈♦s✱ ♦ q✉❡ r❡s♦❧✈❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ s✐♥❛❧✱ ❡✱ ❛❧é♠ ❞✐ss♦✱ ♦s ❣r❛♥❞❡s ❡rr♦s sã♦ ❡✈✐t❛❞♦s✳

❋✐❣✉r❛ ✶✳✶✿ ❆❥✉st❡s q✉❡ ❝♦♥s✐❞❡r❛♠ ♦ ♠í♥✐♠♦ ❞❛ s♦♠❛ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞♦s ❡rr♦s✳

P❛r❛ ❞❡✜♥✐r♠♦s ♦ ♠ét♦❞♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♣❛r❛ ❛❥✉st❡ ❞❡ r❡t❛s✱ ❝♦♥s✐❞❡r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ X ❡Y s❡❥❛♠ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s r❡❧❛❝✐♦♥❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

Y =a0+bX, ✭✶✳✶✮

♦♥❞❡ a0 ❡ b sã♦ ❝♦♥st❛♥t❡s ❞❡s❝♦♥❤❡❝✐❞❛s✳ ❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡st✐♠❛r a0 ❡ b s❡♠ ♦s

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

(x1, y1),(x2, y2), . . . ,(xn, yn)

❞♦ ✈❡t♦r ❛❧❡❛tór✐♦(X, Y)❡ ❝♦♥s✐❞❡r❛♥❞♦ ❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦✿

✶✳✶✳ ❆❥✉st❡ ❞❡ ❘❡t❛ ♣♦r ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ✼

♣❛r❛ i= 1,2, . . . , n❡ ♦♥❞❡ x¯= 1

n

n X

i=1 xi✳

❆ tr❛♥s❢♦r♠❛çã♦ ♣r♦♣♦st❛ ❝♦♥❞✉③ ❛ ❞♦✐s ❢❛t♦s ✐♠♣♦rt❛♥t❡s✳ ❙ã♦ ❡❧❡s✿

✶✳ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ♦ ❡✐①♦y é tr❛♥s❧❛❞❛❞♦ ❞❡ 0 ♣❛r❛ x¯❡ ❛ ❡q✉❛çã♦ ✭✶✳✶✮ ♣♦❞❡ s❡r

r❡❡s❝r✐t❛ ♥❛ ❢♦r♠❛✿

Y =a+bZ,

♦♥❞❡ Z = X ₋X¯ ❝♦♠ X¯ = 1

n

n X

i=1

Xi ❡ X1, . . . , Xn ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛r✐á✈❡✐s

❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛s ❡ ❝♦♠ ❛ ♠❡s♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ X✳

✷✳ ❆❧❣✉♥s ❝á❧❝✉❧♦s sã♦ s✐♠♣❧✐✜❝❛❞♦s✳ ◆♦t❡ q✉❡✿

n X

i=1 zi =

n X

i=1

(xi−x¯) = n X

i=1

xi−nx¯= n X

i=1

xi −n·

1

n

n X

i=1

xi = 0. ✭✶✳✸✮

P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✱ ❞❛q✉✐ ❛té ♦ ✜♥❛❧ ❞❡st❛ s❡çã♦✱ ♦♠✐t✐♠♦s ♦s í♥❞✐❝❡s ❞♦ s♦♠❛tór✐♦✱ ✉t✐❧✐③❛♥❞♦ ♦ sí♠❜♦❧♦P ❛♦ ✐♥✈és ❞❡

n X

i=1

✳

❖ ♦❜❥❡t✐✈♦ ❞♦ ♠ét♦❞♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s é ❡♥❝♦♥tr❛r ˆa ❡ ˆb✱ ❡st✐♠❛❞♦r❡s ❞❡

♠í♥✐♠♦s q✉❛❞r❛❞♦s ❞❡ a ❡ b✱ ♣❡❧♦s ✈❛❧♦r❡s ❞❡ a ❡ b q✉❡ ♠✐♥✐♠✐③❛♠ ❛ ❢✉♥çã♦ S(a, b)✱

❞❛❞❛ ♣♦r

S(a, b) =X(yi−a−bzi)2, ✭✶✳✹✮

❊ss❡s ❡st✐♠❛❞♦r❡s sã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✳

❚❡♦r❡♠❛ ✶✳✶✳ ❙❡S(a, b) é ❛ ❢✉♥çã♦ ❞❛❞❛ ❡♠ ✭✶✳✹✮✱ ❡♥tã♦ ♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠í♥✐♠♦s

q✉❛❞r❛❞♦s ♣❛r❛ a ❡ b sã♦ ❞❛❞♦s ♣♦r

ˆ

a = ¯y= 1

n

X

yi ❡ ˆb = P

yizi P

z2

i

.

❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦✭s✮ ♣♦♥t♦✭s✮ ❝rít✐❝♦✭s✮ ❞❡ ❙✳ ❆ ❞❡r✐✈❛❞❛ ❞❡ ✭✶✳✹✮ ❡♠ r❡❧❛çã♦ ❛a é

S1(a, b) = ∂

∂aS(a, b) = −2

X

(yi−a−bzi). ✭✶✳✺✮

■❣✉❛❧❛♥❞♦ ❛ ③❡r♦ ❡ ✉t✐❧✐③❛♥❞♦ ✭✶✳✸✮✱ s❡❣✉❡ q✉❡

ˆ

a=

P

yi

n = ¯y. ✭✶✳✻✮

❆ ❞❡r✐✈❛❞❛ ❞❡ ✭✶✳✹✮ ❡♠ r❡❧❛çã♦ ❛b é ❞❛❞❛ ♣♦r

S2(a, b) = ∂

∂bS(a, b) = −2

X

✶✳✷✳ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✽

■❣✉❛❧❛♥❞♦ ❛ ③❡r♦ ❡ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ✭✶✳✸✮✱ ♦❜t❡♠♦s

ˆ_b= Pyizi P

z2

i

. ✭✶✳✽✮

▲♦❣♦✱ ♦ ♣♦♥t♦(a, b)✱ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ✭✶✳✻✮ ❡ ✭✶✳✽✮✱ é ♦ ú♥✐❝♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡S✳ ❘❡st❛

♠♦str❛r q✉❡ ❡❧❡ é ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❞❡ S✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞❡ s✉❛s ❞❡r✐✈❛❞❛s

♣❛r❝✐❛✐s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ ❉❡ ✭✶✳✺✮ ❡ ✭✶✳✼✮✱ ✉s❛♥❞♦ ✭✶✳✸✮✱ s❡❣✉❡ q✉❡

S11(a, b) = 2n, S12(a, b) = 0 =S21(a, b) ❡ S22(a, b) = 2Xz2_i.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❤❡ss✐❛♥❛ é ❞❛❞♦ ♣♦r

H(a, b) =

2n 0 0 2Pz2

i = 4n

X

z_i2 >0,

❡ ❝♦♠♦ ❛✐♥❞❛ S11(a, b) = 2n >0✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

✶✳✷ ✲ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛

◆❛s ❞❡✜♥✐çõ❡s s❡❣✉✐♥t❡s ❞❡st❛ s❡çã♦✱ T ❞❡♥♦t❛ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ♥ã♦ ♥❡❣❛t✐✈❛

❡ ❝♦♥tí♥✉❛ q✉❡ r❡♣r❡s❡♥t❛ ♦ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ❞❡ ✉♠ ❡❧❡♠❡♥t♦✱ ♦✉ s❡❥❛✱ ♦ t❡♠♣♦ ❛té ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❡✈❡♥t♦ ❞❡ ✐♥t❡r❡ss❡✳ ❖ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ♣♦❞❡ s❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ t❡♠♣♦ ❞❡ ✈✐❞❛ ❞❡ ✉♠ ♣❛❝✐❡♥t❡ ❞♦ ♠♦♠❡♥t♦ q✉❡ ❢♦✐ ❞✐❛❣♥♦st✐❝❛❞❛ ❛ ❞♦❡♥ç❛ ❛té ❛ s✉❛ ♠♦rt❡ ♦✉ ❛té ❛ ❝✉r❛✱ ♣♦❞❡ ❛✐♥❞❛ s❡r ♦ t❡♠♣♦ út✐❧ ❞❡ ✉♠ ❝❡rt♦ ❡q✉✐♣❛♠❡♥t♦ ❛té ❛ s✉❛ q✉❡✐♠❛✳ ❙❡❥❛♠ f(t) ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ T ❡ F(t) ❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦

❝♦rr❡s♣♦♥❞❡♥t❡✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❆ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡ T é ❞❡✜♥✐❞❛ ♣♦r

S(t) = P(T _≥t) =

Z ∞

t

f(x)dx.

◆♦t❡ q✉❡ ❛ ❢✉♥çã♦ S(t) r❡♣r❡s❡♥t❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♦❜s❡r✈❛çã♦ ♥ã♦ ❢❛❧❤❛r

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

❜r❡✈✐✈❡r ❛♦ t❡♠♣♦t✳ ◗✉❛♥❞♦ tr❛t❛❞❛ ❡♠ ❝♦♥t❡①t♦s q✉❡ ❡♥✈♦❧✈❡♠ s✐st❡♠❛s ♦✉ t❡♠♣♦s

❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡ ✐t❡♥s ♠❛♥✉❢❛t✉r❛❞♦s✱ é t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ❞❡ ❝♦♥✜❛❜✐✲ ❧✐❞❛❞❡✳ ❉❛ ❉❡✜♥✐çã♦ ✶✳✷✱ ✉t✐❧✐③❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡✱ s❡❣✉❡ q✉❡

S(t) é ✉♠❛ ❢✉♥çã♦ ♠♦♥ót♦♥❛ ❞❡❝r❡s❝❡♥t❡✱ ❝♦♥tí♥✉❛✱ ❝♦♠ S(0) = 1 ❡ lim

t→∞S(t) = 0✳

❆❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❛ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡T ♣♦r

✶✳✷✳ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✾

❉❡r✐✈❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡st❛ ✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s✿

f(t) = ₋S′

(t), ✭✶✳✾✮

♦♥❞❡′ _{❞❡♥♦t❛ ❛ ❞❡r✐✈❛❞❛ ❡♠ r❡❧❛çã♦ ❛t✳}

❉❡✜♥✐çã♦ ✶✳✸✳ ❆ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ✭♦✉ ❢✉♥çã♦ r✐s❝♦✮ é ❞❡✜♥✐❞❛ ♣♦r

h(t) = lim

∆t→0

P(t _≤T < t+ ∆t_|T _≥t)

∆t .

◆♦t❡ q✉❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❡s♣❡❝✐✜❝❛ ❛ t❛①❛ ✐♥st❛♥tâ♥❡❛ ❞❡ ❢❛❧❤❛ ♦✉ ♠♦rt❡ ♥♦ t❡♠♣♦t✱ ❞❛❞♦ q✉❡ ♦ ❡❧❡♠❡♥t♦ s♦❜r❡✈✐✈❡✉ ❛té ♦ t❡♠♣♦t ✭▲❛✇❧❡ss ❬✶✷❪✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱

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

h(t) = lim

∆t→0

P(t_≤T < t+ ∆t_|T _≥t)

∆t = lim∆t→0

P(t _≤T < t+ ∆t, T _≥t) ∆t P(T _≥t)

= lim

∆t→0

P(t_≤T < t+ ∆t) ∆t P(T _≥t) ,

♣♦✐s[t, t+ ∆t)_⊂[t,+_∞)✳ ❆ss✐♠✱

h(t) = lim

∆t→0

F(t+ ∆t)₋F(t) ∆t S(t) =

F′_(t)

S(t) =

f(t)

S(t).

◆♦s ❡①❡♠♣❧♦s s❡❣✉✐♥t❡s✱ ❝❛❧❝✉❧❛♠♦s ❛s ❢✉♥çõ❡s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❡ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ❛s ❞✐str✐❜✉✐çõ❡s ❡①♣♦♥❡♥❝✐❛❧ ❡ ❲❡✐❜✉❧❧ ❞❡ ❞♦✐s ♣❛râ♠❡tr♦s✳ ◆♦ ❝♦♥t❡①t♦ ❞❛ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ♦✉ ❞❛ t❡♦r✐❛ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✱ ❛♠❜❛s ♦❝✉♣❛♠ ♣♦s✐çã♦ ❞❡ ❞❡st❛q✉❡ ♣♦r s❡ ❛❞❡q✉❛r❡♠ ❛ ♠✉✐t❛s s✐t✉❛çõ❡s ♣rát✐❝❛s✳

❊①❡♠♣❧♦ ✶✳✹✳ ❆ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ T

❝♦♠ ❞✐str✐❜✉✐çã♦ ❡①♣♦♥❡♥❝✐❛❧ é ❞❛❞❛ ♣♦r

f(t) = λe−λt_{, λ, t >0.}

❆ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡T é ♦❜t✐❞❛ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿

S(t) =

Z ∞

t

λe−λx_dx=e−λt_.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❞❛❞❛ ♣♦r

h(t) = f(t)

S(t) =λ.

✶✳✷✳ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✶✵

❊①❡♠♣❧♦ ✶✳✺✳ ❆ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ T

❝♦♠ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ é ❞❛❞❛ ♣♦r

f(t) = λβ(λt)β−1_e−(λt)β

, t >0,

♦♥❞❡ β✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✱ ❡ λ✱ ♦ ❞❡ ❡s❝❛❧❛✱ sã♦ ❛♠❜♦s ♣♦s✐t✐✈♦s✳ ❆ ❢✉♥çã♦ ❞❡

s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡ T é ❡①♣r❡ss❛ ♣♦r

S(t) =

Z ∞

t

λβ(λx)β−1_e−(λx)β

dx=

Z ∞

(λt)β

e−u_du=e−(λt)β

❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❞❛❞❛ ♣♦r

h(t) = f(t)

S(t) =λβ(λt)

β−1_{, t >0.}

❆ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ é ♠✉✐t♦ ✉t✐❧✐③❛❞❛ ❡♠ ❡st✉❞♦s ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡ ♥❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❞❡✈✐❞♦ ❛ s✉❛ ✢❡①✐❜✐❧✐❞❛❞❡ ❡♠ ❛❥✉st❛r ❞✐❢❡r❡♥t❡s ❢♦r♠❛s ❞❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛✳ P❛r❛β = 1✱ t❡♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r

❞❛ ❲❡✐❧❜✉❧❧ ❡✱ ❝♦♠♦ ✈✐♠♦s ♥♦ ❊①❡♠♣❧♦ ✶✳✹✱ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❝♦♥st❛♥t❡✳ ❙❡β >1✱

❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❝r❡s❝❡♥t❡ ❡✱ ♥♦ ❝❛s♦ ❡♠ q✉❡β <1✱ ❡❧❛ é ❞❡❝r❡s❝❡♥t❡✳ ❉❡ ❢❛t♦✱

❝♦♠♦

h′

(t) = λ2β(β₋1)(λt)β−2

❡ β, λ, t > 0✱ s❡❣✉❡ q✉❡ h′_{(t)> 0 ♣❛r❛ β > 1 ❡✱ ❛ss✐♠✱ h é ❝r❡s❝❡♥t❡ ❡ q✉❛♥❞♦ β < 1✱}

t❡♠♦sh′_{(t)<0✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ h é ❞❡❝r❡s❝❡♥t❡✳ ❱❡❥❛ ❋✐❣✉r❛ ✶✳✷✳}

❋✐❣✉r❛ ✶✳✷✿ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ β ❡

✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✶

✶✳✸ ✲ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛

❆ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ❣❛♥❤♦✉ ❞❡st❛q✉❡ ❛ ♣❛rt✐r ❞♦s ❡st✉❞♦s ❞❡ ❲❛❧♦❞❞✐ ❲❡✐❜✉❧❧ q✉❡ ❡✈✐❞❡♥❝✐❛r❛♠ ❛ s✉❛ ❛♠♣❧❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ ❞✐✈❡rs❛s ár❡❛s✳ ❙❡✉ ♣r✐♠❡✐r♦ tr❛❜❛❧❤♦ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ♥♦ ❛♥♦ ❞❡ ✶✾✸✾ ❡♠ ✉♠ ❥♦r♥❛❧ ❡s❝❛♥❞✐♥❛✈♦ ❡ tr❛t♦✉ ❞❛ r❡s✐stê♥❝✐❛ ❞❡ ♠❛t❡r✐❛✐s✳ ❊♠ ✉♠ ❡st✉❞♦ s✉❜s❡q✉❡♥t❡ ❡♠ ✐♥❣❧ês✱ ❞❡ ✶✾✺✶✱ ❡❧❡ ♠♦str♦✉ ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❡♠ ❞✐❢❡r❡♥t❡s ár❡❛s✳ ❉❡s❞❡ ❡♥tã♦✱ ♦ ♠♦❞❡❧♦ t❡♠ s✐❞♦ ❡①t❡♥s✐✈❛♠❡♥t❡ ✉t✐❧✐③❛❞♦ ❡♠ ❡st✉❞♦s ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✱ ♥❛ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❡ ❡♠ ♦✉tr❛s ár❡❛s ❞❡✈✐❞♦ ❛ s✉❛ ✈❡rs❛t✐❧✐❞❛❞❡✱ ❛ ♠❡❞✐❞❛ q✉❡ ♣♦❞❡ ♠♦❞❡❧❛r t❛①❛s ❞❡ ❢❛❧❤❛s ❝♦♠ ❢♦r♠❛s ❝♦♥st❛♥t❡s✱ ❝r❡s❝❡♥t❡s ❡ ❞❡❝r❡s❝❡♥t❡s ✭✈❡❥❛ ❊①❡♠♣❧♦ ✶✳✺✮✳

▼✉✐t❛s ♠♦❞✐✜❝❛çõ❡s ❡ ❣❡♥❡r❛❧✐③❛çõ❡s ❢♦r❛♠ ♣r♦♣♦st❛s ❡♠ ❞✐✈❡rs♦s tr❛❜❛❧❤♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❡♠ ▼✉rt❤② ❡t ❛❧✳ ❬✶✹❪ ❡ ▲❛✐ ❡t ❛❧✳ ❬✶✶❪✱ q✉❡ ♣❡r♠✐t❡♠ ♠♦❞❡❧❛r ♦✉tr❛s ❢♦r♠❛s ❞❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛✳ ❯♠❛ ❞❡ss❛s é ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ✉♥✐♠♦❞❛❧✱ q✉❡ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞❛ ♣♦r ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝❤❛♠❛❞❛ ❞❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ♣r♦♣♦st❛ ♣♦r ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪ ❝♦♠♦ ✉♠❛ ✈❛r✐❛çã♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧✳

❈♦♥s✐❞❡r❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ❞❡ ❞♦✐s ♣❛râ♠❡tr♦s q✉❡ t❡♠ ❢✉♥çã♦ ❞❡ ❞✐str✐✲ ❜✉✐çã♦ ❞❛❞❛ ♣♦r

F(t) = 1₋e(_αt)β

, α, β >0; t_≥0, ✭✶✳✶✵✮

♦♥❞❡ α é ♦ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ❡ β é ♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✳ ❱❛♠♦s ♥♦s r❡❢❡r✐r ❛

✭✶✳✶✵✮ ❝♦♠♦ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦✳ P♦❞❡♠♦s ✉t✐❧✐③❛r ♣❛r❛♠❡tr✐③❛çõ❡s ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ r❡♣r❡s❡♥tá✲❧♦✱ ❝♦♠♦ ❛s ❧✐st❛❞❛s ❛ s❡❣✉✐r✿

F(t) = 1₋e−(λt)β

, ❝♦♠ λ= 1

α;

F(t) = 1₋e−tβ

α′, ❝♦♠ α′ =αβ;

F(t) = 1₋e−λ′

tβ

, ❝♦♠ λ′

=

1

α

β

.

❊♠❜♦r❛ s❡❥❛♠ t♦❞❛s ❡q✉✐✈❛❧❡♥t❡s✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ❝♦♥t❡①t♦ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ♣❛r✲ t✐❝✉❧❛r ♣♦❞❡ s❡r ♠❛✐s ❛♣r♦♣r✐❛❞❛✳ ◆♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✱ ❛ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦ é ❛ ❡q✉❛çã♦ ✭✶✳✶✵✮✱ ❛ ♠❡♥♦s q✉❡ s❡❥❛ ✐♥❞✐❝❛❞♦ ♦ ❝♦♥trár✐♦✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ♠♦❞❡❧♦s t❡♠ s✐❞♦ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡st❡ ♠♦❞❡❧♦ ❜ás✐❝♦✳ ❯♠ ❞❡❧❡s é ♦ ❡st✉✲ ❞❛❞♦ ❡♠ ❬✶✵❪✱ ❡♠ q✉❡ é ♣r♦♣♦st❛ ❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦✿ ❝♦♥s✐❞❡r❡Y ✉♠❛ ✈❛r✐á✈❡❧

❛❧❡❛tór✐❛ ❞❡✜♥✐❞❛ ♣♦r

Y = α

2

X,

✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✷

❞❡Y é ♦❜t✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

F(t) =P(Y _≤t) = P(α_X2 _≤t) = P(X _≥ α_t2) = 1₋P(X < α_t2) = 1₋P(X _≤ α_t2) = 1₋F(α_t2)

= e−(α t)

β

,

♦♥❞❡ α, β > 0 ❡ t > 0✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ Y t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧

✐♥✈❡rs❛✱ ❝✉❥❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ é ❞❛❞❛ ♣♦r

F(t) = e−(α_t)β

, α, β >0❡ t >0. ✭✶✳✶✶✮

❆ s❡❣✉✐r✱ ✐♥✐❝✐❛♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛❞❛ ♣♦r ✭✶✳✶✶✮✱ ♦❜t❡♥❞♦ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡✱ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❡ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❡st❛ ❞✐str✐❜✉✐çã♦✳ ❋✐♥❛❧✐③❛♠♦s ❛ s❡çã♦ ❛♣r❡s❡♥t❛♥❞♦ ❞✉❛s té❝✲ ♥✐❝❛s ❣rá✜❝❛s✱ ❝❤❛♠❛❞❛s ❣rá✜❝♦✲❲PP ❡ ❣rá✜❝♦✲■❲PP✱ ❞❡s❝r❡✈❡♥❞♦ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ❜❛s❡❛❞♦s ♥❡ss❡s ♠ét♦❞♦s✳

✶✳✸✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡

❉✐❢❡r❡♥❝✐❛♥❞♦F(t) ♦❜t❡♠♦s ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✿

f(t) =αββt−β−1_e−(α t)

β

, α, β >0 ❡t >0. ✭✶✳✶✷✮

❱❛♠♦s ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❛ ♣❛rt✐r ❞♦ ❡st✉❞♦ ❞❛ ♠♦♥♦✲ t♦♥✐❝✐❞❛❞❡ ❞❡ ✉♠❛ ❢✉♥çã♦✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞❡ s✉❛ ❞❡r✐✈❛❞❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ q✉❡ é ♦❜t✐❞❛ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿

f′

(t) = αββt−β−1_e−(α t)

β

[(₋β₋1)t−1_+t−β−1_αβ_β]

= αβ_βt−β−1_e−(α_t)β

t−β−1_{[−(β+ 1)t}β_+αβ_β]

= f(t)t−β−1_[

−(β+ 1)tβ+αββ]. ✭✶✳✶✸✮

❈♦♠♦f(t)t−β−1 _{>0✱ s❡❣✉❡ q✉❡}

f′

(t) = 0 _⇐⇒ tβ = α

β_β

β+ 1 ⇐⇒ t =α

β+ 1

β

−1

β

⇐⇒ t=α

1 + 1

β

−1

β

.

❉❡♥♦t❛♥❞♦ tm = α(1 + _β1)− 1

β✱ t❡♠♦s q✉❡ f′(t) > 0 ♣❛r❛ t < t

m ❡✱ ❛ss✐♠✱ f(t) é ✉♠❛

❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❡ q✉❛♥❞♦t > tm✱ t❡♠♦sf′(t)<0✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡f(t)é ❞❡❝r❡s❝❡♥t❡✳

▲♦❣♦✱ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ é ✉♥✐♠♦❞❛❧✳

■st♦ ❞✐❢❡r❡ ❞♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦✱ ❡♠ q✉❡ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ é ❞❡❝r❡s❝❡♥t❡ ✭♣❛r❛

✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✸

❉❡r✐✈❛♥❞♦ F(t) ❡♠ ✭✶✳✶✵✮✱ ♦❜t❡♠♦s ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧

❜ás✐❝♦✱

f(t) =βα−β_tβ−1_e−(t α)

β

, t >0.

❆ ❞❡r✐✈❛❞❛ ❞❡ f(t) é ❞❛❞❛ ♣♦r

f′

(t) = f(t)tβ−1_[(β

−1)t−β

−βα−β_].

❈♦♠♦ α, β, t, f(t) > 0✱ s❡❣✉❡ q✉❡ f′_{(t) < 0 s❡ β}

≤ 1 ❡✱ ❛ss✐♠✱ f(t) é ❞❡❝r❡s❝❡♥t❡✳ ◆♦

❝❛s♦ ❡♠ q✉❡ β >1✱ t❡♠♦s q✉❡ f′_(t)>0♣❛r❛ _{t < t}

m ❡ f′(t)<0 ♣❛r❛ t > tm✱ ♦♥❞❡

tm =α

1₋ 1

β

1

β

,

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

✶✳✸✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛

❆ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ♣❛r❛ ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ T ❝♦♠ ❞❡♥s✐❞❛❞❡ ❲❡✐❜✉❧❧

✐♥✈❡rs❛✭α, β✮ é ❞❛❞❛ ♣♦r

S(t) = 1₋e−(α_t)β

,

♦♥❞❡ t > 0 ❡ α, β > 0✳ ❆ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❝♦rr❡s♣♦♥❞❡♥t❡ é ❞❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡

❢♦r♠❛✿

h(t) = α

β_βt−β−1_e−(α t)

β

1₋e−(α t)β

,

♦♥❞❡t >0 ❡ α, β >0✳

▲❡♠❛ ✶✳✻✳ ❆ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ s❛t✐s❢❛③ ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿

(i) lim

t→0h(t) = 0 ❡ (ii) limt→∞h(t) = 0.

❉❡♠♦♥str❛çã♦✳

(i) ◆❛ ❡①♣r❡ssã♦ ♣❛r❛ h(t)✱ t❡♠♦s q✉❡ ♦ ❞❡♥♦♠✐♥❛❞♦r t❡♥❞❡ ❛ 1✱ q✉❛♥❞♦ t t❡♥❞❡

❛ 0✳ ❊♥tã♦ ❜❛st❛ ♠♦str❛r q✉❡ ♦ ♥✉♠❡r❛❞♦r t❡♥❞❡ ❛ 0✱ q✉❛♥❞♦ t t❡♥❞❡ ❛ 0✳ ❯s❛♥❞♦

▲✬❍♦s♣✐t❛❧✱ t❡♠♦s q✉❡✿

lim

t→0 βα

β_t−β−1_e−(α t)

β

= βαβ_lim t→0

1

tβ+1

e(αt)β

=βαβlim

t→0

−(β+ 1)t−β−2

−βαβ_t−β−1_e(α t)β

= (β+ 1) lim

t→0

1

t e(α t)β

= 0,

♣♦✐s

lim

t→0

t e(αt) β

✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✹

P❛r❛ ♣r♦✈❛r ♦ ❧✐♠✐t❡ ❡♠ ✭✶✳✶✹✮✱ ♥♦t❡ ✭❢❛③❡♥❞♦h= _t1β✮ q✉❡

lim

t→0t e

1

tβ = lim

h→∞

1

hβ1

eh = lim

h→∞

eh 1

β h 1

β−1

.

❙❡♥❞♦ ❛ss✐♠✱ ❞♦✐s ❝❛s♦s ♣♦❞❡♠ s❡r ❛♥❛❧✐s❛❞♦s s❡♣❛r❛❞❛♠❡♥t❡✳ ❙ã♦ ❡❧❡s✿

• β >1

❚❡♠♦s q✉❡ 1

β <1 ❡ ❛ss✐♠

lim

t→0t e

1

tβ = lim

h→∞βe h_h1−1

β = +∞.

• β <1

◆❡st❡ ❝❛s♦✱ ❛♣❧✐❝❛♥❞♦ ▲✬❍♦s♣✐t❛❧ ✈ár✐❛s ✈❡③❡s ❡ s❡♥❞♦ 1

β >1✱ ♦❜t❡♠♦s✿

lim

t→0t e

1

tβ = lim

h→∞

eh

1

βh 1

β−1

= lim h→∞ eh 1 β 1

β −1

hβ1−2

=_{· · ·} = lim h→∞ eh 1 β 1

β −1

· · ·1

β − ₁

β

−1h1β−

₁

β

= lim

h→∞Ce h_h1β

−_β1

= +_∞,

♦♥❞❡

C = 1

1

β

1

β −1

· · ·1

β − ₁

β

−1 ❡ [x] := min{n ∈Z;n≥x}.

(ii) ❯s❛♥❞♦ ▲✬❍♦s♣✐t❛❧ ♥♦✈❛♠❡♥t❡ ❡ ❛s ❡①♣r❡ssõ❡s ✭✶✳✶✸✮ ❡ ✭✶✳✾✮✱ t❡♠♦s q✉❡✿

lim

t→∞h(t) = limt→∞

f(t)

S(t) = limt→∞

f(t)t−β−1_{[−(β+ 1)t}β _+αβ_β]

−f(t)

= lim

t→∞

β+ 1

t −

αβ_β

tβ+1

= 0.

✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✺

h′_{(t) =} f

′_{(t)S(t) + [f(t)]}2

[S(t)]2 ♣♦✐s S

′

(t) = ₋f(t)

= f(t)t

−β−1_{[−(β+ 1)t}β_+αβ_{β]S(t) + [f(t)]}2

[S(t)]2

= f(t)

S(t)

t−β−1_{[−(β+ 1)t}β_+αβ_{β]S(t) +f(t)}

S(t)

= h(t)t−β−1 _{−(β+ 1)t}β_+αβ_β+αββ e −(α

t) β

1₋e−(α t)β

!

= h(t)t−β−1



₋(β+ 1)tβ ₊α

β_β1−e−(α t)

β

+αβ_{β e}−(α t)

β

1₋e−(α t)

β

 

= h(t)t−β−1

−(β+ 1)tβ₊ αββ

1₋e−(α t)β

.

❚❡♠♦s q✉❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ t❡♠ ❢♦r♠❛ ✉♥✐♠♦❞❛❧ ❡ t❡♠ ♠á①✐♠♦ ❡♠ t = tM

❞❛❞♦ ♣❡❧❛ s♦❧✉çã♦ ❞❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿

(_tα

M)

β

1₋e(tMα )β

= 1 + 1

β.

■st♦ ❞✐❢❡r❡ ❞♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦✱ ❡♠ q✉❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ✭♣❛r❛ β > 1✮✱ ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ✭♣❛r❛ β < 1✮ ♦✉ ❝♦♥st❛♥t❡ ✭♣❛r❛

β= 1✮✱ ❝♦♥❢♦r♠❡ ❊①❡♠♣❧♦ ✶✳✺ ✭❝♦♥s✐❞❡r❡λ= _α1✮✳

✶✳✸✳✸ ▼♦♠❡♥t♦s

❙❡❥❛ Y ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❞❡ ♣❛râ♠❡tr♦sα ❡ β✱ ❞❛❞❛ ♣♦r ✭✶✳✶✶✮✳ ❖ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ❞❡ Y é ❞❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

E(Yk) =

Z ∞

−∞

tkf(t)dt=

Z ∞

0

αββ t(k−β−1)_e−(α t)

β

dt.

❋❛③❡♥❞♦u =αβ_t−β_{✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿ du=−α}β_{β t}−β−1_{dt; u→ ∞ q✉❛♥❞♦} t_→0 ❡ u_→0q✉❛♥❞♦ t _{→ ∞}; tk_{= (t}−β₎−k_β

= _αuβ

−k_β

✳ ❉❡ss❡ ♠♦❞♦✱ ♦❜t❡♠♦s q✉❡

E(Yk) =

Z ∞ 0 _u αβ −k β

e−u_{du= (α}−β₎−k_β Z ∞

0

u[(1−kβ)−1]e−udu

= αkΓ

1₋ k

β

. ✭✶✳✶✺✮

✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✻

✶✳✸✳✹ ●rá✜❝♦s ▼♦❞✐✜❝❛❞♦s

●rá✜❝♦s ♠♦❞✐✜❝❛❞♦s tê♠ s✐❞♦ ✉t✐❧✐③❛❞♦s ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❛❧t❡r♥❛t✐✈♦ ♥❛ r❡♣r❡s❡♥✲ t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ❞❛❞♦s✱ s❡❣✉♥❞♦ ▼✉rt❤② ❡t ❛❧✳ ❬✶✹❪✳ ❆ ✐❞❡✐❛ é tr❛♥s❢♦r♠❛r ♦s ❞❛❞♦s ♥♦ ✐♥t✉✐t♦ ❞❡ ❧✐♥❡❛r✐③❛r ❛ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✭♦✉ ❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦✮ ❞♦ ♠♦❞❡❧♦ ♣r♦♣♦st♦✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❛♠♦s ❞✉❛s ♠♦❞✐✜❝❛çõ❡s ❞❡ ◆❡❧s♦♥ ❬✶✺❪ ❡ ❉r❛♣❡❧❧❛ ❬✻❪✱ ❝❤❛♠❛❞❛s ❲❡✐❜✉❧❧ Pr♦❜❛❜✐❧✐t② P❛♣❡r ✭❲PP✮ ❡ ■♥✈❡rs❡ ❲❡✐❜✉❧❧ Pr♦❜❛❜✐❧✐t② P❛♣❡r ✭■❲PP✮✱ ❛s q✉❛✐s✱ ♥❡st❡ ❡st✉❞♦✱ ♥♦s r❡❢❡r✐♠♦s ❝♦♠♦ ❣rá✜❝♦✲❲PP ❡ ❣rá✜❝♦✲■❲PP✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ♦❜❥❡t✐✈♦ ❛q✉✐ é ❞✐s❝✉t✐r t❛✐s té❝♥✐❝❛s ❣rá✜❝❛s ❡ ❛♣r❡s❡♥t❛r ✉♠ ♣r♦✲ ❝❡❞✐♠❡♥t♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ❢♦r♠❛ ❡ ❡s❝❛❧❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ✭♦✉ ❲❡✐❜✉❧❧✮ ❜❛s❡❛❞♦ ♥❡ss❡s ♠ét♦❞♦s ❣rá✜❝♦s✳

✶✳✸✳✹✳✶ ❖ ●rá✜❝♦✲❲PP

●rá✜❝♦s✲❲PP ♣♦❞❡♠ s❡r ❝♦♥str✉í❞♦s ❞❡ ✈ár✐❛s ♠❛♥❡✐r❛s✳ ❯♠❛ ❞❡❧❛s é ❛ ❛♣r❡s❡♥✲ t❛❞❛ ♣♦r ◆❡❧s♦♥ ❬✶✺❪✱ q✉❡ ❜❛s❡✐❛✲s❡ ♥❛ ♠♦❞✐✜❝❛çã♦ ❞❛❞❛ ♣♦r

y= ln[₋ln(1₋F(t))] ❡ x= lnt. ✭✶✳✶✻✮

❈♦♥s✐❞❡r❛♥❞♦ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦ ✭✶✳✶✵✮✱ ♦❜t❡♠♦s✿

y= ln[₋ln(1₋F(t))] = ln[₋lne−(_αt)β

] = ln

t α

β

=βlnt₋βlnα.

❆ss✐♠✱ s❡♥❞♦ x= lnt✱ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦ é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ r❡❧❛çã♦ ❧✐♥❡❛r

y =βx₋βlnα.

❖ ❣rá✜❝♦ ❞❡ y ♣♦r x é ❝❤❛♠❛❞♦ ❞❡ ❣rá✜❝♦✲❲PP✳ ◆♦t❡ q✉❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠é✲

tr✐❝❛ ❞♦ ❣rá✜❝♦✲❲PP é ✉♠❛ r❡t❛✱ ❝♦♠ ✐♥❝❧✐♥❛çã♦ ❞❛❞❛ ♣❡❧♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛ β ❡

q✉❡ ✐♥t❡r❝❡♣t❛ ♦ ❡✐①♦ y ❡♠ ₋βlnα✳

❆❣♦r❛ s❡ ❛♣❧✐❝❛r♠♦s ❛ ♠♦❞✐✜❝❛çã♦ ✭✶✳✶✻✮ à ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❞❛❞❛ ♣♦r ✭✶✳✶✶✮✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿

y = ln[₋ln(1₋e−z_{)], ✭✶✳✶✼✮}

♦♥❞❡

z =α

t

β

=e[βlnα−βx]_.

◆♦t❡ q✉❡y é ✉♠❛ ❢✉♥çã♦ ♥ã♦✲❧✐♥❡❛r ❞❡x✳ ❖s t❡♦r❡♠❛s ❛ s❡❣✉✐r ❞ã♦ ❞♦✐s r❡s✉❧t❛❞♦s

❞♦ ❣rá✜❝♦✲❲PP ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛✳