Robustecendo a distribuição normal

❘♦❜✉st❡❝❡♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧

▼❛r❝♦s ❘❛❢❛❡❧ ◆♦❣✉❡✐r❛ ❈❛✈❛❧❝❛♥t❡

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛

❛♦

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

♣❛r❛

♦❜t❡♥çã♦ ❞♦ tít✉❧♦

❞❡

▼❡str❡ ❡♠ ❈✐ê♥❝✐❛s

Pr♦❣r❛♠❛✿ ❊st❛tíst✐❝❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❍❡❧❡♥♦ ❇♦❧❢❛r✐♥❡

❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ♦ ❛✉t♦r r❡❝❡❜❡✉ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞♦ ❈◆Pq

❘♦❜✉st❡❝❡♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧

❊st❛ ✈❡rsã♦ ❞❛ ❞✐ss❡rt❛çã♦ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❞♦ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛❞❛ ❡♠ ✵✻✴✶✶✴✷✵✶✺✳ ❯♠❛ ❝ó♣✐❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❡stá ❞✐s♣♦♥í✈❡❧ ♥♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

• Pr♦❢✳ ❉r✳ ❍❡❧❡♥♦ ❇♦❧❢❛r✐♥❡ ✲ ■▼❊✲❯❙P • Pr♦❢❛✳ ❉r❛✳ ❙✐❧✈✐❛ ◆❛❣✐❜ ❊❧✐❛♥ ✲ ■▼❊✲❯❙P

❍✐♥♦ ❉❡▼♦❧❛②

❆ ❝♦r♦❛ ❞❛ ❥✉✈❡♥t✉❞❡ ✐♥✐❝✐❛ ❆té ♦ ♠❡r✐❞✐❛♥♦ ❛ ♥♦ss❛ ❥♦r♥❛❞❛ ❈♦♥t❡♠♣❧❛ ❡♠ ♥ós✱ ❜r✐❧❤♦ ❞♦ ♠❡✐♦ ❞✐❛ P❡r❛♥t❡ ❡st❡ ❛❧t❛r✱ ❛ ♣r♦♠❡ss❛ s❛❣r❛❞❛✳

◗✉❡ s♦❜❡r❛♥♦s s❡❥❛♠ ♦s ♥♦ss♦s ✐❞❡❛✐s ▲✉③❡s ♥♦ ❝❛♠✐♥❤♦ ❞❡ ✈✐rt✉❞❡s ✐♠♦rt❛✐s ◗✉❡ ❡st❛s s❡t❡ ✈❡❧❛s s❡❥❛♠ ♥♦ss❛ ▲❡✐ ❖ ❇r❛sã♦ ❍❡r♦✐❝♦ ❞❛ ❖r❞❡♠ ❉❡▼♦❧❛②✳

❈♦♥s❛❣r❛❞❛ ❜❛t❛❧❤❛ ❞❛ ✈✐❞❛ ❈♦♥❞✉③ ♦ ❝❛♠✐♥❤♦ ❞❛ r❡t✐❞ã♦ ❊♠ ♥♦ss❛ ❜❛♥❞❡✐r❛ ✐♠♣♦♥❡♥t❡✱ ❡st❡♥❞✐❞❛ ❊stã♦ ♦s ❜❛❧✉❛rt❡s ❞❛ ♥♦ss❛ ◆❛çã♦✳

◗✉❡ s♦❜❡r❛♥♦s s❡❥❛♠ ♦s ♥♦ss♦s ✐❞❡❛✐s ▲✉③❡s ♥♦ ❝❛♠✐♥❤♦ ❞❡ ✈✐rt✉❞❡s ✐♠♦rt❛✐s ◗✉❡ ❡st❛s s❡t❡ ✈❡❧❛s s❡❥❛♠ ♥♦ss❛ ▲❡✐ ❖ ❇r❛sã♦ ❍❡r♦✐❝♦ ❞❛ ❖r❞❡♠ ❉❡▼♦❧❛②✳

❙♦❜ ❛ r❡❣ê♥❝✐❛ ❞♦ P❛✐ ❈❡❧❡st✐❛❧ ◆♦s ❞✐❛s ❞❡ ❛✉r♦r❛ ❛té ♦ ❛♣♦❣❡✉ ◗✉❡ ❡♠ ♥♦ss❛ ❖r❞❡♠ s❡❥❛♠ ✉♠ s✐♥❛❧ ❉❡ ❤♦♥r❛ q✉❡ ♦ ❢♦❣♦ ♥ã♦ ❢❡♥❡❝❡✉✳

◗✉❡ s♦❜❡r❛♥♦s s❡❥❛♠ ♦s ♥♦ss♦s ✐❞❡❛✐s ▲✉③❡s ♥♦ ❝❛♠✐♥❤♦ ❞❡ ✈✐rt✉❞❡s ✐♠♦rt❛✐s ◗✉❡ ❡st❛s s❡t❡ ✈❡❧❛s s❡❥❛♠ ♥♦ss❛ ▲❡✐ ❖ ❇r❛sã♦ ❍❡r♦✐❝♦ ❞❛ ❖r❞❡♠ ❉❡▼♦❧❛②✳

◗✉❡ ❉❡✉s t❡ ❛❜❡♥ç♦❡ ♠ã❡✳ ◗✉❡ ❉❡✉s t❡ ❛❜❡♥ç♦❡ ♣❛✐✳ ◗✉❡ ❉❡✉s ❛❜❡♥ç♦❡ ❛ ❝❛✉s❛ ❞❛ ❖r❞❡♠ ❉❡▼♦❧❛②✳ ❆♠é♠✦

❆❣r❛❞❡❝✐♠❡♥t♦s

●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r✿

Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♥♦ss♦ P❛✐ ❈❡❧❡st✐❛❧✱ ♣♦✐s s❡♠ ❊❧❡ ♥❛❞❛ s❡r✐❛ ♣♦ssí✈❡❧ ❡ é ♣♦r ❝❛✉s❛ ❉❡❧❡ q✉❡ ❝♦♥s❡❣✉✐ s✉♣❡r❛r t♦❞♦s ♦s ♦❜stá❝✉❧♦s ❛♦ ❧♦♥❣♦ ❞♦s ♠❡✉s ❡st✉❞♦s✳

❆♦s ♠❡✉s ♣❛✐s✱ ▲✉③✐❡✈❛ ❡ ▼❛r❝♦s✱ ♣❡❧♦s s❡✉s ❝❛r✐♥❤♦s✱ ❝♦♥s❡❧❤♦s✱ ❝✉✐❞❛❞♦s✱ ❡♥s✐♥❛♠❡♥t♦s ❡ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧✳ ❙❡♠ ❡❧❡s ❡✉ ♥ã♦ t❡r✐❛ ❢♦rç❛ ♣❛r❛ s❡❣✉✐r ❡♠ ❢r❡♥t❡✱ ♣♦✐s ❡❧❡s sã♦ ♦ ♠❡✉ ♣♦rt♦ s❡❣✉r♦✳ ❋♦r❛♠ ❡❧❡s q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ❛ ❧✉t❛r ♣❡❧♦s ♠❡✉s s♦♥❤♦s✳ ❙ã♦ ❡❧❡s q✉❡ ❡✉ s❡♠♣r❡ ♣♦❞❡r❡✐ ❝♦♥t❛r ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡ ♠✐♥❤❛ ✈✐❞❛✳ ❊ ♦s ♠❡✉s ✐r♠ã♦s q✉❡ ❛♣❡s❛r ❞❛s ♥♦ss❛s ❜r✐❣❛s✱ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ❡ t♦r❝❡r❛♠ ♣♦r ♠✐♠✳

➚ ♠✐♥❤❛ ❡s♣♦s❛✱ ❘♦s✐❛♥❡✱ q✉❡ s❡♠♣r❡ ❡st❡✈❡ ❛♦ ♠❡✉ ❧❛❞♦ t♦❞♦s ♦s ❞✐❛s ❡ ❢♦✐ ❝♦♠♣r❡❡♥s✐✈❛ ♥♦s ♠♦♠❡♥t♦s q✉❡ ♥ã♦ ♣✉❞❡ ❧❤❡ ❞❛r ❛ ❛t❡♥çã♦ q✉❡ ❡❧❛ ♠❡r❡❝❡✳

❚❡♥❤♦ ♠✉✐t♦ ❛ ❛❣r❛❞❡❝❡r ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❍❡❧❡♥♦✱ ♣♦✐s s❡♠ s✉❛ ❣r❛♥❞❡ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♠♣r❡✲ ❡♥sã♦ ❡✉ ♥ã♦ ♣♦❞❡r✐❛ t❡r ♦❜t✐❞♦ ❡st❡ tít✉❧♦ q✉❡ ♠❡ ♦r❣✉❧❤♦ t❛♥t♦✳

◆ã♦ ♣♦ss♦ ❞❡✐①❛r ❞❡ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s ♣r♦❢❡ss♦r❡s q✉❡ ♠❡ ❡♥s✐♥❛r❛♠ t♦❞♦ q✉❡ ❡✉ s❡✐✱ s❡♠ s❡✉s ❡♥s✐♥❛♠❡♥t♦s ❡ ❝♦♥s❡❧❤♦s ❡✉ ♥ã♦ ♣♦❞❡r✐❛ t❡r ❝❤❡❣❛❞♦ ❛té ❛q✉✐✳ ❋♦✐ ❣r❛ç❛s ❛♦s ♠❡✉s ♠❡str❡s q✉❡ t✐✈❡ ❛♦ ❧♦♥❣♦ ❞❛ ♠✐♥❤❛ ✈✐❞❛ ❞❡ ❡st✉❞❛♥t❡ q✉❡ ♠❡ t♦r♥❡✐ q✉❡♠ s♦✉ ❤♦❥❡✳

❆♦s ♠❡✉s ❣r❛♥❞❡s ❛♠✐❣♦s q✉❡ ❝♦♥❤❡❝✐ t♦❞♦s ❡st❡s ❛♥♦s✳ ◆❛ ❣r❛❞✉❛çã♦ ❝♦♥❤❡❝✐ ♣❡ss♦❛s ✐♥❝rí✈❡✐s q✉❡ ♠❡ ❡♥s✐♥❛r❛♠ ♠✉✐t♦✳ ◆❛ r❡s✐❞ê♥❝✐❛ ✉♥✐✈❡rs✐tár✐❛ ♣✉❞❡ ❝♦♥✈✐✈❡r ❝♦♠ ♣❡ss♦❛s ♠✉✐t♦ ❞✐❢❡r❡♥t❡s ❞❛s ♠❛✐s ❞✐✈❡rs❛s ♦♣✐♥✐õ❡s✱ ❡❧❛s ♠❡ ❛❥✉❞❛r❛♠ ♠✉✐t♦ ❛ ❝r❡s❝❡r✳ ◆♦ ■▼❊ ❝♦♥❤❡❝✐ ❛♠✐❣♦s q✉❡ ✐r❡✐ ❧❡✈❛r ♣❛r❛ t♦❞❛ ❛ ✈✐❞❛✳ ❋♦✐ ❣r❛ç❛s ❛ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❢❡❧✐③❡s q✉❡ t✐✈❡♠♦s q✉❡ ♣✉❞❡ ❛♣r♦✈❡✐t❛r ❛♦ ♠á①✐♠♦ ❡st❛ ❥♦r♥❛❞❛✳

◆✉♥❝❛ ♣♦❞❡r✐❛ ❞❡✐①❛r ❞❡ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s ✐r♠ã♦s ❞❛ ❖r❞❡♠ ❉❡▼♦❧❛② ❡ ❛♦s ♠❡✉s t✐♦s ▼❛✲ ç♦♥s✳ ❋♦✐ ❣r❛ç❛s ❛ ❡❧❡s q✉❡ ♣✉❞❡ ♠❡❧❤♦r❛r ❛s ✈✐rt✉❞❡s q✉❡ ♠❡ ❢♦r❛♠ ❡♥s✐♥❛❞❛s ♣♦r ♠❡✉s ♣❛✐s✳ ❙ã♦ ❛s ✈✐rt✉❞❡s ❞❡ ✉♠ ❉❡▼♦❧❛② q✉❡ ♠♦❧❞❛r❛♠ ♦ ❤♦♠❡♠ q✉❡ ❡✉ s♦✉✳

❊♥✜♠✱ ❛ t♦❞❛s ❛s ♣❡ss♦❛s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡ ❝❛❞❛ ♠♦♠❡♥t♦ q✉❡ ❧❡✈♦✉ à ♠✐♥❤❛ ❝♦♥❝❧✉sã♦ ❞♦ ♠❡str❛❞♦✳

❘❡s✉♠♦

❈❆❱❆▲❈❆◆❚❊✱ ▼✳ ❘✳ ◆✳ ❘♦❜✉st❡❝❡♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ ✷✵✶✺✳ ✾✸ ❢✳ ❉✐ss❡rt❛çã♦ ✭▼❡s✲ tr❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✺✳

❊st❛ ❞✐ss❡rt❛çã♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♦ ❡st✉❞♦ ❞❛ ❞✐str✐❜✉✐çã♦ ✏s❧❛s❤✑✱ ❝♦♥s✐❞❡r❛♥❞♦ s❡✉s ❝❛s♦s s✐♠étr✐❝♦ ❡ ❛ss✐♠étr✐❝♦ ✉♥✐✈❛r✐❛❞♦s✳ ❙❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♣r♦♣r✐❡❞❛❞❡s ♣r♦❜❛❜✐❧íst✐❝❛s ❡ ✐♥❢❡r❡♥❝✐❛✐s ❞❡ss❛ ❞✐str✐❜✉✐çã♦✱ ❛ss✐♠ ❝♦♠♦ ♣❡❝✉❧✐❛r✐❞❛❞❡s ❡ ♣r♦❜❧❡♠❛s✳ P❛r❛ s❡r❡♠ ❢❡✐t❛s ✐♥❢❡rê♥❝✐❛s s❡rá ❝♦♥✲ s✐❞❡r❛❞♦ ♦ ❡♥❢♦q✉❡ ❝❧áss✐❝♦ ❛tr❛✈és ❞♦ ✉s♦ ❞♦s ♠ét♦❞♦s ❞♦s ♠♦♠❡♥t♦s ❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ ❙ã♦ ❛♣r❡s❡♥t❛❞♦s t❛♠❜é♠ ♦s ❝á❧❝✉❧♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡st❡s ❡st✐♠❛❞♦r❡s✳ ◆♦s ❝❛s♦s ♦♥❞❡ ❡st❡s ❡s✲ t✐♠❛❞♦r❡s ♥ã♦ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s✱ ❛tr❛✈és ❞❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❊▼✳ P❛r❛ ✐st♦✱ ❢♦✐ ✉t✐❧✐③❛❞♦ ♦ s♦❢t✇❛r❡ ❘ ❡ ♦s ❝♦♠❛♥❞♦s ❡stã♦ ♥♦ ❆♣ê♥❞✐❝❡❆✳ ◆♦ ❝❛s♦ ❞♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ s❡rá ✐♠♣❧❡♠❡♥t❛❞♦ ♦ ♠ét♦❞♦ ❞❡ ▲♦✉✐s ♣❛r❛ ❡st✐♠❛r ♦s ❡❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r✳ ❋♦r❛♠ r❡❛❧✐③❛❞♦s ❡st✉❞♦s ❞❡ s✐♠✉❧❛çã♦ ❡ ❛♣❧✐❝❛çõ❡s ♣❛r❛ ❞❛❞♦s r❡❛✐s✳ ◆❛s ❛♣❧✐❝❛çõ❡s ❢♦✐ ❛♥❛❧✐s❛❞♦ ♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r s✐♠♣❧❡s✱ ♦♥❞❡ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ q✉❡ ♦s ❡rr♦s s❡❣✉❡♠ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❉✐str✐❜✉✐çã♦ s❧❛s❤✱ ❉✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛✳

❆❜str❛❝t

❈❆❱❆▲❈❆◆❚❊✱ ▼✳ ❘✳ ◆✳ ❘♦❜✉st✐❢②✐♥❣ t❤❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✳ ✷✵✶✺✳ ✾✸ ❢✳ ▼❙❝ ❞✐ss❡rt❛t✐♦♥ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✺✳

❚❤✐s ❞✐ss❡rt❛t✐♦♥ ❛✐♠s ❛t st✉❞②✐♥❣ t❤❡ ✏s❧❛s❤✑ ❞✐str✐❜✉t✐♦♥ ❝♦♥s✐❞❡r✐♥❣ ✐ts s②♠♠❡tr✐❝ ❛♥❞ ❛s②♠✲ ♠❡tr✐❝ ✈❡rs✐♦♥s✳ ❲❡ ♣r❡s❡♥t ♣r♦❜❛❜✐❧✐st✐❝ ❛s ✇❡❧❧ ❛s ✐♥❢❡r❡♥t✐❛❧ ❛s♣❡❝ts ♦❢ t❤✐s ❞✐str✐❜✉t✐♦♥✱ ✐♥❝❧✉❞✐♥❣ ♣❡❝✉❧✐❛r✐t✐❡s ❛♥❞ ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ ♠♦❞❡❧ ✜tt✐♥❣✳ ❚❤❡ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ♠❛①✐♠✉♠ ❧✐❦❡✲ ❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✐s ✉s❡❞✳ ▼♦♠❡♥ts ❡st✐♠❛t✐♦♥ ✐s ❛❧s♦ ❝♦♥s✐❞❡r❡❞ ❛s st❛rt✐♥❣ ✈❛❧✉❡s ❢♦r t❤❡ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥✳ ❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ✐s ❞❡✈❡❧♦♣❡❞ ❢♦r t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❛♣♣r♦❛❝❤✳ ❋♦r t❤✐s ✐♠♣❧❡♠❡♥t❛t✐♦♥ s♦❢t✇❛r❡ ❘ ✇❛s ✉s❡❞ ❛♥❞ ❝♦❞❡s r❡q✉✐r❡❞ ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❆♣♣❡♥❞✐①✳ ❆s ❛ ❜②♣r♦❞✉❝t ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠✱ ▲♦✉✐s ♠❡t❤♦❞ ✐s ❝♦♥s✐❞❡r❡❞ ❢♦r ❡st✐♠❛t✐♥❣ t❤❡ ❋✐s❤❡r ✐♥❢♦r♠❛t✐♦♥ ♠❛tr✐① ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ❝♦♠♣✉t✐♥❣ ❧❛r❣❡ s❛♠♣❧❡ ✐♥t❡r✈❛❧s ❢♦r ♠♦❞❡❧ ♣❛r❛♠❡t❡rs✳ ❊①t❡♥s✐♦♥s ❢♦r ❛ s✐♠♣❧❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ✐s ❝♦♥s✐❞❡r❡❞✳ ❙✐♠✉❧❛t✐♦♥ st✉❞✐❡s ❛r❡ ♣r❡s❡♥t❡❞ ✐❧❧✉str❛t✐♥❣ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❡st✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤ ❝♦♥s✐❞❡r❡❞✳ ❘❡s✉❧ts ♦❢ r❡❛❧ ❞❛t❛ ❛♥❛❧②s✐s ✐♥❞✐❝❛t❡ t❤❛t t❤❡ ♠❡t❤♦❞♦❧♦❣② ❝❛♥ ♣❡r❢♦r♠ ✇❡❧❧ ✐♥ ❛♣♣❧✐❡❞ s❝❡♥❛r✐♦s✳

❑❡②✇♦r❞s✿ ❉✐str✐❜✉t✐♦♥ s❧❛s❤✱ ❞✐str✐❜✉t✐♦♥ s❧❛s❤ ❛s②♠♠❡tr✐❝❛❧✳

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s ①✐

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ①✐✐✐

▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✈

▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✈✐✐

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❖r❣❛♥✐③❛çã♦ ❞❛ ❞✐ss❡rt❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✷ ❉✐str✐❜✉✐çã♦ s❧❛s❤ s✐♠étr✐❝❛ ✸

✷✳✶ ■♥tr♦❞✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✷ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷✳✶ ❊st✐♠❛❞♦r❡s ♣❡❧♦ ♠ét♦❞♦ ❞♦s ♠♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ ❊st✐♠❛çã♦ ♣♦r ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸✳✶ ❆❧❣♦r✐t♠♦ ❊▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸✳✷ ▼❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸✳✸ ▼ét♦❞♦ ❞❡ ▲♦✉✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸✳✹ ❆♣❧✐❝❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✹ ❊st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✹✳✶ ✶◦ _{❈❛s♦ ✿q ❝♦♥❤❡❝✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸} ✷✳✹✳✷ ✷◦ _{❈❛s♦ ✿q ❞❡s❝♦♥❤❡❝✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻} ✷✳✺ ❆♣❧✐❝❛çã♦ ❡♠ ❞❛❞♦s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✸ ❉✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛ ✷✸

✸✳✶ ■♥tr♦❞✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ❉✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✸ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✸✳✶ ❆ss✐♠❡tr✐❛ ❡ ❝✉rt♦s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✸✳✷ ❊st✐♠❛❞♦r❡s ♣❡❧♦ ♠ét♦❞♦ ❞♦s ♠♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✹ ▼á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✹✳✶ ❆❧❣♦r✐t♠♦ ❊▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✹✳✷ ▼❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✹✳✸ ▼ét♦❞♦ ❞❡ ▲♦✉✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✹✳✹ ❆♣❧✐❝❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

① ❙❯▼➪❘■❖

✸✳✺ ❊st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✺✳✶ q ❝♦♥❤❡❝✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✸✳✻ ❆♣❧✐❝❛çã♦ ❡♠ ❞❛❞♦s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✹ ❘❡❣r❡ssã♦ ❧✐♥❡❛r ✸✾

✹✳✶ ■♥tr♦❞✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✷ ❘❡❣r❡ssã♦ ❧✐♥❡❛r s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷✳✶ ❆❧❣♦r✐t♠♦ ❊▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷✳✷ ▼❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✸ ▼ét♦❞♦ ❞❡ ▲♦✉✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✹ ❆♣❧✐❝❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✸ ❆♣❧✐❝❛çã♦ ❡♠ ❞❛❞♦s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✺ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✺✸

✺✳✶ ❚r❛❜❛❧❤♦s ❢✉t✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

❆ ❈♦♠❛♥❞♦s ❞♦ ❘ ✺✺

❆✳✶ ❉✐str✐❜✉✐çã♦ s❧❛s❤ ❝♦♠ q ❝♦♥❤❡❝✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

❆✳✶✳✶ ❙✐♠✉❧❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ❆✳✶✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ❆✳✷ ❉✐str✐❜✉✐çã♦ s❧❛s❤ ❝♦♠ q ❞❡s❝♦♥❤❡❝✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

❆✳✷✳✶ ❙✐♠✉❧❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ❆✳✷✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ❆✳✸ ❉✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛ ❝♦♠q ❝♦♥❤❡❝✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸

❆✳✸✳✶ ❙✐♠✉❧❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ❆✳✸✳✷ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ❆✳✹ ❘❡❣r❡ssã♦ ❧✐♥❡❛r s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s

❆❙❙ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❛ss✐♠❡tr✐❛ ❊❈ ❊①❝❡ss♦ ❞❡ ❝✉rt♦s❡

❊▼ ❆❧❣♦r✐t♠♦ ❊▼ ✲ ❊s♣❡r❛♥ç❛ ❡ ▼❛①✐♠✐③❛çã♦ ❊◗▼ ❊rr♦ q✉❛❞rát✐❝♦ ♠é❞✐♦

❙▲ ❉✐str✐❜✉✐çã♦ s❧❛s❤

❙◆ ❉✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ❛ss✐♠étr✐❝❛ ❙❙▲ ❉✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s

γ ❋✉♥çã♦ ❣❛♠❛ ✐♥❝♦♠♣❧❡t❛

Γ ❋✉♥çã♦ ❣❛♠❛

Ψ ❋✉♥çã♦ ❛❝✉♠✉❧❛❞❛ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❣❛♠❛ ♥♦ ♣♦♥t♦ ✶

φ ❉❡♥s✐❞❛❞❡ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♣❛❞rã♦

Φ ❆❝✉♠✉❧❛❞❛ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♣❛❞rã♦ ˆ

µm,ˆσ2m,qˆm,ˆλm ❊st✐♠❛❞♦r❡s ♣❡❧♦ ♠ét♦❞♦ ❞♦s ♠♦♠❡♥t♦s

ˆ

µmv,σˆmv2 ,qˆmv,ηˆmv,ˆτmv ❊st✐♠❛❞♦r❡s ♣❡❧♦ ♠ét♦❞♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ DGI ❋✉♥çã♦ ❞✐❣❛♠❛ ✐♥❝♦♠♣❧❡t❛

IF(θ) ▼❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r IO(θ) ▼❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ♦❜s❡r✈❛❞❛

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❉❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❇❡t❛✭q✱✶✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❉❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❙▲✭✵✱✶✱q✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✸ ❇♦①♣❧♦t ❞❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦

❙❧❛s❤✭✶✵✱✾✱✺✮ ❝♦♥s✐❞❡r❛♥❞♦q ❝♦♥❤❡❝✐❞♦✳ ✭❛✮ ❡st✐♠❛t✐✈❛s ❞❡µ❡ ✭❜✮ ❡st✐♠❛t✐✈❛s ❞❡σ2✳ ✶✹

✷✳✹ ❇♦①♣❧♦t ❞❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ ❙❧❛s❤✭✶✵✱✾✱✺✮ ❝♦♥s✐❞❡r❛♥❞♦ q ❝♦♥❤❡❝✐❞♦✳ ✭❛✮ ❡st✐♠❛t✐✈❛s ❞❡ µ✱ ✭❜✮ ❡st✐♠❛t✐✈❛s ❞❡σ2

❡ ✭❝✮ ❡st✐♠❛t✐✈❛s ❞❡q✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷✳✺ ❍✐st♦❣r❛♠❛ ❞♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❣♦r❞✉r❛ ❞♦s ✷✵✷ ❛t❧❡t❛s ❛✉str❛❧✐❛♥♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶ ❉❡♥s✐❞❛❞❡ ❞❛ ♥♦r♠❛❧ ❛ss✐♠étr✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ❉❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ SSL(µ, σ2_{, λ, q)✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺} ✸✳✸ ❇♦①♣❧♦t ❞❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦

❙❙▲✭✶✵✱✾✱✺✱ ✲✷✮ ❝♦♥s✐❞❡r❛♥❞♦q ❝♦♥❤❡❝✐❞♦✳ ✭❛✮ ❡st✐♠❛t✐✈❛s ❞❡µ✱ ✭❜✮ ❡st✐♠❛t✐✈❛s ❞❡σ2

❡ ✭❝✮ ❡st✐♠❛t✐✈❛s ❞❡λ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✹✳✶ ❇♦①♣❧♦t ❞♦s ❛t❧❡t❛s ❛✉str❛❧✐❛♥♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✷ ❉✐s♣❡rsã♦ ❞♦s ❛t❧❡t❛s ❛✉str❛❧✐❛♥♦s ♣♦r ♣❡r❝❡♥t✉❛❧ ❞❡ ❣♦r❞✉r❛ ❡ ♣❡s♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✷✳✶ ❊st✐♠❛t✐✈❛s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ ❙▲✭✶✵✱✾✱✺✮✱ ❝♦♠ ✺✵✵ ré♣❧✐❝❛s ❞❡ t❛✲ ♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❱í❝✐♦ ❡ ❊rr♦ ◗✉❛❞rát✐❝♦ ▼é❞✐♦ ✭❊◗▼✮ ♣❛r❛ ❛s ❡st✐♠❛t✐✈❛s ❞♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐s✲

tr✐❜✉✐çã♦ ❙▲✭✶✵✱✾✱✺✮✱ ❝♦♠ ✺✵✵ ré♣❧✐❝❛s ❞❡ t❛♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ ❘❡s✉♠♦ ❞♦ ♥ú♠❡r♦ ❞❡ ✐t❡r❛çõ❡s ♣❛r❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛

✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡♠ ✺✵✵ ré♣❧✐❝❛s ❞❡ t❛♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✹ ❊st✐♠❛t✐✈❛s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ ❙▲✭✶✵✱✾✱✺✮✱ ❝♦♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

✷✷✽✱ ✸✹✼ ❡ ✹✷✵ ré♣❧✐❝❛s ❞❡ t❛♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✺ ❱í❝✐♦ ❡ ❊rr♦ ◗✉❛❞rát✐❝♦ ▼é❞✐♦ ✭❊◗▼✮ ♣❛r❛ ❛s ❡st✐♠❛t✐✈❛s ❞♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐✲

❜✉✐çã♦ ❙▲✭✶✵✱✾✱✺✮✱ ❝♦♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✷✷✽✱ ✸✹✼ ❡ ✹✷✵ ré♣❧✐❝❛s ❞❡ t❛♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✻ ❘❡s✉♠♦ ❞♦ ♥ú♠❡r♦ ❞❡ ✐t❡r❛çõ❡s ♣❛r❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛

✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✷✷✽✱ ✸✹✼ ❡ ✹✷✵ ré♣❧✐❝❛s ❞❡ t❛♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✼ ❊st❛tíst✐❝❛s ♣❛r❛ ❛ ✈❛r✐á✈❡❧Bf at✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷✳✽ ❊st✐♠❛t✐✈❛s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛ ✈❛r✐á✈❡❧Bf at✱ s✉♣♦♥❞♦ q✉❡ s❡❣✉❡

✉♠❛ ❞✐str✐❜✉✐çã♦ SL(µ, σ2, q)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷✳✾ ❊st✐♠❛t✐✈❛s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛ ✈❛r✐á✈❡❧Bf at✱ s✉♣♦♥❞♦ q✉❡ s❡❣✉❡

✉♠❛ ❞✐str✐❜✉✐çã♦ SL(µ, σ2_{, q)✱ ♦♥❞❡ q∈[2,1; 19]✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶} ✷✳✶✵ ❊st✐♠❛t✐✈❛s ❞♦s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛s ❞♦s ❡st✐♠❛❞♦r❡s ❞♦s ♣❛râ✲

♠❡tr♦sµ ❡σ2 ♣❛r❛q_∈[2,1; 19]✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✸✳✶ ❊st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ♦s ❡st✐♠❛❞♦r❡s ❞♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐s✲ tr✐❜✉✐çã♦ ❙❙▲✭✶✵✱✾✱✺✱✲✷✮✱ ❝♦♠ ✺✵✵ ré♣❧✐❝❛s ❞❡ t❛♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ❱í❝✐♦ ❡ ❊rr♦ ◗✉❛❞rát✐❝♦ ▼é❞✐♦ ✭❊◗▼✮ ❞❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛

♣❛r❛ ♦s ❡st✐♠❛❞♦r❡s ❞♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ ❙❙▲✭✶✵✱✾✱✺✱✲✷✮✱ ❝♦♠ ✺✵✵ ré♣❧✐❝❛s ❞❡ t❛♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✸ ❘❡s✉♠♦ ❞♦ ♥ú♠❡r♦ ❞❡ ✐t❡r❛çõ❡s ♣❛r❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛

✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡♠ ✺✵✵ ré♣❧✐❝❛s ❞❡ t❛♠❛♥❤♦ ✐❣✉❛❧ ❛ ✷✵✱ ✺✵ ❡ ✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✹ ❊st✐♠❛t✐✈❛s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛ ✈❛r✐á✈❡❧Bf at✱ s✉♣♦♥❞♦ q✉❡ s❡❣✉❡

✉♠❛ ❞✐str✐❜✉✐çã♦ SSL(µ, σ2, q, λ)✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✸✳✺ ❊st✐♠❛t✐✈❛s ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛s ❞♦s ❡st✐♠❛❞♦r❡s ❞♦s ♣❛râ♠❡tr♦s

µ✱η ❡τ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✹✳✶ ❊st❛tíst✐❝❛s ❞♦s r❡sí❞✉♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❊①✐st❡♠ ♠✉✐t❛s s✐t✉❛çõ❡s ♣rát✐❝❛s ♦♥❞❡ ❛ ✉s✉❛❧ s✉♣♦s✐çã♦ ❞❡ ♥♦r♠❛❧✐❞❛❞❡ ❞♦s ❞❛❞♦s ♥ã♦ é ❛ ✐❞❡❛❧✳ ■ss♦ ♦❝♦rr❡ ❞❡✈✐❞♦ ❛ ✈ár✐♦s ♣r♦❜❧❡♠❛s✱ ✉♠ ❞❡st❡s é ❛ ❢❛❧t❛ ❞❡ s✐♠❡tr✐❛ ❞♦s ❞❛❞♦s✳ ❯♠❛ ❛❧✲ t❡r♥❛t✐✈❛ é ❛ ✐♥❝❧✉sã♦ ❞❡ ✉♠ ♣❛râ♠❡tr♦✱ λ✱ ♣❛r❛ ♠♦❞❡❧❛r ❛ ❛ss✐♠❡tr✐❛✱ ❛ss✐♠ t❡♠✲s❡ ❛ ❞✐str✐❜✉✐çã♦

♥♦r♠❛❧ ❛ss✐♠étr✐❝❛✳ ◗✉❛♥❞♦ ✉t✐❧✐③❛✲s❡ ❡♠ ♠♦❞❡❧♦s ❞❡ r❡❣r❡ssã♦ ❛s ❞✐str✐❜✉✐çõ❡s ♥♦r♠❛❧ ❡ ♥♦r♠❛❧ ❛ss✐♠étr✐❝❛✱ ❡st❡s sã♦ s❡♥sí✈❡✐s ❛ ♣r❡s❡♥ç❛ ❞❡ ♦❜s❡r✈❛çõ❡s ❡①tr❡♠❛s ♦✉ ❛❜❡rr❛♥t❡s ✭✏♦✉t❧✐❡rs✑✮✳ ❆ss✐♠✱ ❡st✉❞❛r❡♠♦s ❝♦♠♦ ❛❧t❡r♥❛t✐✈❛ ✉♠❛ ❢❛♠í❧✐❛ ♠❛✐s ❣❡r❛❧ ❞❡ ❞✐str✐❜✉✐çõ❡s q✉❡ ✐♥❝❧✉✐ ❝♦♠♦ ❝❛s♦s ♣❛r✲ t✐❝✉❧❛r❡s ❛s ❞✐str✐❜✉✐çõ❡s ♥♦r♠❛❧ ❡ ♥♦r♠❛❧ ❛ss✐♠étr✐❝❛✳ ❊st❛ ❢❛♠í❧✐❛ ❞❡ ❞✐str✐❜✉✐çõ❡s é ❞❡♥♦♠✐♥❛❞❛ s❧❛s❤ ❛ss✐♠étr✐❝❛✳

❆ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛ ♣♦ss✉✐ q✉❛tr♦ ♣❛râ♠❡tr♦s✿ ♣♦s✐çã♦✱ ❡s❝❛❧❛✱ ❢♦r♠❛ ❡ ❛ss✐♠❡tr✐❛✳ ❈♦♠ ♦s ♣❛râ♠❡tr♦s ❞❡ ❢♦r♠❛ ❡ ❛ss✐♠❡tr✐❛ ❡st❛ ❞✐str✐❜✉✐çã♦ é ❜❡♠ ♠❛✐s ❣❡r❛❧ ❡ ❜❡♠ ♠❡♥♦s s❡♥sí✈❡❧✱ s❡♥❞♦ ✉♠❛ ❜♦❛ ❛❧t❡r♥❛t✐✈❛ à ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳

❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❞✐s❝✉t✐r ♦s ❛s♣❡❝t♦s ✐♥❢❡r❡♥❝✐❛✐s ♥❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡✲ tr♦s ❞❛s ❞✐str✐❜✉✐çõ❡s s❧❛s❤ ❡ s❧❛s❤ ❛ss✐♠étr✐❝❛✳ P❛r❛ t❛❧ s❡rá ✉t✐❧✐③❛❞♦ ♦ ❡♥❢♦q✉❡ ❝❧áss✐❝♦✳

P❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦s ❡st✐♠❛❞♦r❡s ❞♦s ♣❛râ♠❡tr♦s s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s ♦s ♠ét♦❞♦s ❞♦s ♠♦♠❡♥t♦s ❡ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ ◆♦ ♠ét♦❞♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦s ❡st✐♠❛❞♦r❡s ❢♦✐ ✉t✐❧✐③❛❞♦ ♦ ❛❧❣♦r✐t♠♦ ❊▼✳ ❈♦♠ ❛ ✐♥❝❧✉sã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✱q✱ ♦ ❛❧❣♦r✐t♠♦ ✜❝❛ ♠❛✐s ❧❡♥t♦✱

✉♠❛ ❢♦r♠❛ ❞❡ ❛❝❡❧❡r❛r ♦ ❛❧❣♦r✐t♠♦ é ❝♦♥s✐❞❡r❛r q ❝♦♥❤❡❝✐❞♦✳ P❛r❛ ✉♠❛ ❡s❝♦❧❤❛ ♠❛✐s ❡✜❝❛③ ❞❡q ❢♦✐

✉t✐❧✐③❛❞♦ ♦ ♠ét♦❞♦ ❞❡ ▲♦✉✐s✳

P❛r❛ ❛✈❛❧✐❛r ♦s ❡st✐♠❛❞♦r❡s ❢♦r❛♠ r❡❛❧✐③❛❞❛s s✐♠✉❧❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s ❡♠ ❞❛❞♦s r❡❛✐s✳

❋♦✐ r❡❛❧✐③❛❞♦ t❛♠❜é♠ ✉♠ ❡st✉❞♦ s♦❜r❡ ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r s✐♠♣❧❡s✱ ♦♥❞❡ ❢♦✐ s✉♣♦st♦ q✉❡ ♦s ❡rr♦s s❡❣✉❡♠ ❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛✳

✶✳✶ ❖r❣❛♥✐③❛çã♦ ❞❛ ❞✐ss❡rt❛çã♦

❆ ♣r❡s❡♥t❡ ❞✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦ ❡stá ❞✐✈✐❞❛ ❡♠ ❝✐♥❝♦ ❝❛♣ít✉❧♦s✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥✲ t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❞❡♥tr❡ ❡❧❛s✱ ♦s ♠♦♠❡♥t♦s✱ ❛ss✐♠❡tr✐❛ ❡ ❝✉rt♦s❡✳ ❙ã♦ ❛♣r❡s❡♥t❛❞♦s t❛♠❜é♠ ♦s ❡st✐♠❛❞♦r❡s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ♣❡❧♦ ♠ét♦❞♦ ❞♦s ♠♦♠❡♥t♦s ❡ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ P❛r❛ ❛ ❛✈❛❧✐❛çã♦ ❞♦s ❡st✐♠❛❞♦r❡s sã♦ r❡❛❧✐③❛❞❛s s✐♠✉❧❛çõ❡s ❡ ❛♣❧✐❝❛çã♦ ❡♠ ❞❛❞♦s r❡❛✐s✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❛s ❞✐str✐❜✉✐çõ❡s ♥♦r♠❛❧ ❛ss✐♠étr✐❝❛ ❡ s❧❛s❤ ❛ss✐♠étr✐❝❛ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s✿ ❝♦♠♦ ♦s ♠♦♠❡♥t♦s✱ ❛ss✐♠❡tr✐❛✱ ❝✉rt♦s❡ ❡ ♦✉tr♦s✳ ❙ã♦ ❛♣r❡s❡♥✲ t❛❞♦s t❛♠❜é♠ ♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ P❛r❛ ❛ ❛✈❛❧✐❛çã♦ ❞♦s ❡st✐♠❛❞♦r❡s sã♦ r❡❛❧✐③❛❞❛s s✐♠✉❧❛çõ❡s ❡ ❛♣❧✐❝❛çã♦ ❡♠ ❞❛❞♦s r❡❛✐s✳

◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦ ❡st✉❞❛♠♦s ♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❧✐♥❡❛r s✐♠♣❧❡s✱ ♦♥❞❡ ❛♦ ✐♥✈és ❞❡ s✉♣♦r✲ ♠♦s q✉❡ ♦s ❞❛❞♦s s❡❣✉❡♠ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✱ s✉♣♦r❡♠♦s q✉❡ ♦s ❞❛❞♦s s❡❣✉❡♠ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ❛ss✐♠étr✐❝❛✳

❊ ♥♦ q✉✐♥t♦ ❝❛♣ít✉❧♦✱ sã♦ ❛♣r❡s❡♥t❛❞❛s ❝♦♥❝❧✉sõ❡s ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❡st❡ tr❛❜❛❧❤♦ ❡ ♣❡rs✲ ♣❡❝t✐✈❛s ❞❡ tr❛❜❛❧❤♦s ❢✉t✉r♦s✳

❈❛♣ít✉❧♦ ✷

❉✐str✐❜✉✐çã♦ s❧❛s❤ s✐♠étr✐❝❛

◆❡st❡ ❝❛♣✐t✉❧♦ s❡rá ❞❡✜♥✐❞❛ ❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ s✐♠étr✐❝❛✳ ❙❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♠ét♦❞♦s ❞❡ ❡st✐♠❛çã♦✳ ❖s ❡st✐♠❛❞♦r❡s ❛♣r❡s❡♥t❛❞♦s sã♦ ♦❜t✐❞♦s ♣❡❧♦s ♠ét♦❞♦s ❞♦s ♠♦♠❡♥t♦s ❡ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ ◆♦ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ s❡rá ✉t✐❧✐③❛❞♦ ♦ ❛❧❣♦r✐t♠♦ ❊▼✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ♠ét♦❞♦ ❞❡ ▲♦✉✐s ♣❛r❛ ❡st✐♠❛r ❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r✳ ❙ã♦ ❛♣r❡s❡♥t❛❞♦s t❛♠❜é♠ ✉♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ♣❛r❛ ❞♦✐s ❝❛s♦s✿q ✭♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✮ ❝♦♥❤❡❝✐❞♦ ❡ ❞❡s❝♦♥❤❡❝✐❞♦✳

❙❡rá t❛♠❜é♠ r❡❛❧✐③❛❞❛ ✉♠❛ ❛♥á❧✐s❡ ♣❛r❛ ❞❛❞♦s r❡❛✐s✳

✷✳✶ ■♥tr♦❞✉çã♦

❊♠ s✐t✉❛çõ❡s ♣rát✐❝❛s ♠✉✐t❛s ✈❡③❡s ❛ s✉♣♦s✐çã♦ ❞❡ ♥♦r♠❛❧✐❞❛❞❡ ❞♦s ❞❛❞♦s ♥ã♦ é ❛ ✐❞❡❛❧✱ ❝♦♠♦ q✉❛♥❞♦ ❡①✐st❡♠ ♦✉t❧✐❡rs✳ ❆ ❞✐str✐❜✉✐çã♦ s❧❛s❤ é ✉♠❛ ❛❧t❡r♥❛t✐✈❛ à ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ ■st♦ ♦❝♦rr❡ ♣♦rq✉❡ ❡❧❛ ♣♦ss✉✐ ♣r♦♣r✐❡❞❛❞❡s ✐♥t❡r❡ss❛♥t❡s ❝♦♠♦ ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ ❆❧é♠ ❞✐ss♦ t❡♠ ❛ ❞✐str✐✲ ❜✉✐çã♦ ♥♦r♠❛❧ ❝♦♠♦ ❝❛s♦ ❧✐♠✐t❡ ❡ ❛✐♥❞❛ ♣♦ss✉✐ ❝❛✉❞❛s ♠❛✐s ♣❡s❛❞❛s✱ ♦ q✉❡ ❛ t♦r♥❛ ♠❡♥♦s s❡♥sí✈❡❧ ❛ ♦✉t❧✐❡rs✳

❲❛♥❣ ❡ ●❡♥t♦♥ ✭✷✵✵✻✮ ❛♣r❡s❡♥t❛♠ ❝♦♠♦ ❞❡✜♥✐çã♦ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ♦ s❡❣✉✐♥t❡ ❝♦❝✐❡♥t❡

S= Z

U1/q ∼SL(0,1, q), q >0,

♦♥❞❡✱Z _∼N ormal(0,1)✐♥❞❡♣❡♥❞❡♥t❡ ❞❡U _∼U nif orme(0,1)✳

❈♦♥s✐❞❡r❛♥❞♦ ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ U✱ ✉♥✐❢♦r♠❡♠❡♥t❡ ❞✐str✐❜✉í❞❛ ♥♦ ✐♥t❡r✈❛❧♦ ✭✵✱✶✮✳ ❊ ❛ tr❛♥s✲

❢♦r♠❛çã♦M =U1/q✳ ❈♦♠♦ ❡st❛ tr❛♥s❢♦r♠❛çã♦ é ❜✐✉♥í✈♦❝❛✱ ❡♥tã♦

FM(m) =P(M ≤m) =P(U1/q≤m) =P(U ≤mq) =FU(mq)

❙❛❜❡✲s❡ q✉❡ s❡ U _∼U nif orme(0,1)✱ ❡♥tã♦ FU(u) =uI(0,1)(u) +I[1,∞)(u)✳ ❆ss✐♠✱

FM(m) =mqI(0,1)(m) +I[1,∞)(m) =⇒ fM(m) =qmq−1I(0,1)(m). ❉❡st❛ ❢♦r♠❛✱U1/q _∼Beta(q,1)✳

❆ss✐♠✱ ♣♦❞❡✲s❡ ✉t✐❧✐③❛r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤✳ ❉❡✜♥❡✲s❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✱S✱ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❙❧❛s❤ q✉❛♥❞♦ ❡st❛ é ❞❛❞❛ ♣♦r

S = Z

U ∼SL(0,1, q), q >0

♦♥❞❡✱Z _∼N ormal(0,1)✐♥❞❡♣❡♥❞❡♥t❡ ❞❡U _∼Beta(q,1)✳ ❈♦♠ ❞❡♥s✐❞❛❞❡s ❞❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

♣♦r

fZ(z) = e−z2_/2

√

2π IR(z) ❡ fU(u) =qu q−1_I

(0,1)(u)

✹ ❉■❙❚❘■❇❯■➬➹❖ ❙▲❆❙❍ ❙■▼➱❚❘■❈❆ ✷✳✶

◆❛ ❋✐❣✉r❛✷✳✶ ❡♥❝♦♥tr❛✲s❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❜❡t❛ ♣❛r❛ ✈ár✐♦s ✈❛❧♦r❡s ❞❡q✳

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

x

f(x)

Beta(0,1;1) Beta(0,2;1) Beta(0,5;1) Beta(1;1) Beta(2;1) Beta(5;1) Beta(10;1)

❋✐❣✉r❛ ✷✳✶✿ ❉❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❇❡t❛✭q✱✶✮✳

❯t✐❧✐③❛♥❞♦✲s❡ ♦ ♠ét♦❞♦ ❞♦ ❥❛❝♦❜✐❛♥♦ t❡♠✲s❡ q✉❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ S é ❞❛❞❛ ♣♦r

fS(s) =

Z 1

0

quqφ(su)du= √q

8πγ

q+ 1 2 , s

2

IR(s),

♦♥❞❡✱φ(u) é ❛ ❞❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧ ♣❛❞rã♦ ♥♦ ♣♦♥t♦u✳

❆ ❢✉♥çã♦ ❣❛♠❛ ✐♥❝♦♠♣❧❡t❛ é ❞❛❞❛ ♣♦r

γ(α, β) = Γ (α)β−αΨ (α, β).

❆ ❢✉♥çã♦ Ψ (α, β) é ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❛❝✉♠✉❧❛❞❛ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ Gama(α, β) ♥♦

♣♦♥t♦ ✶✳

❙❡ q = 1✱ ♦❜té♠✲s❡ ❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ♣❛❞rã♦✱ ♦✉ ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ q✉❡ ♣♦ss✉✐ ❞❡♥s✐❞❛❞❡ ♥❛

❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛

fS(s) =

( _φ(0)

−φ(s)

s2 , s❡ s6= 0;

φ(0)

2 , s❡ s= 0.

P❛r❛ ♦❜t❡r ✉♠❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ❝♦♠ ♣❛râ♠❡tr♦s ❞❡ ♣♦s✐çã♦ ❡ ❡s❝❛❧❛✱ q✉❛♥❞♦ ❤á ✐♥t❡r❡ss❡✱ ❜❛st❛ ✉s❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❧✐♥❡❛r✐❞❛❞❡ ✭✈❡r ❲❛♥❣ ❡ ●❡♥t♦♥✱ ✷✵✵✻✮✳ ❖✉ s❡❥❛✱ s❡ ♦ ✐♥t❡r❡ss❡ é ♦❜t❡r ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝♦♠ ♣❛râ♠❡tr♦s ❞❡ ♣♦s✐çã♦ ❡ ❡s❝❛❧❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱µ❡ σ2 é só ✉t✐❧✐③❛r ❛ s❧❛s❤

❝❛♥ô♥✐❝❛✱ ❡ ❡♠ s❡❣✉✐❞❛✱ ❢❛③❡r ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳ ❖ ♠♦❞❡❧♦ ❡stá✱ ♣♦rt❛♥t♦✱ ♥❛ ❝❧❛ss❡ ❞♦s ♠♦❞❡❧♦s ❞❡ ❧♦❝❛❧✐③❛çã♦✲❡s❝❛❧❛✳ ❆ss✐♠ s❡

✷✳✷ ▼❖▼❊◆❚❖❙ ✺

◆❛ ❋✐❣✉r❛✷✳✷ ❡♥❝♦♥tr❛✲s❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ♣❛r❛ ✈ár✐♦s ✈❛❧♦r❡s ❞❡q✳

−10 −5 0 5 10

0.0

0.1

0.2

0.3

0.4

x

f(x)

Normal(0;1) SL(0;1;0,1) SL(0;1;0,2) SL(0;1;0,5) SL(0;1;1) SL(0;1;2) SL(0;1;5) SL(0;1;10)

❋✐❣✉r❛ ✷✳✷✿ ❉❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❙▲✭✵✱✶✱q✮✳

✷✳✷ ▼♦♠❡♥t♦s

❙❡❥❛ S =Z/U _∼SL(0,1, q)✳ P❛r❛ ❡♥❝♦♥tr❛r ♦s ♠♦♠❡♥t♦s ♥ã♦ ❝❡♥tr❛✐s ❜❛st❛ ✉t✐❧✐③❛r ♦ ❢❛t♦ ❞❡

q✉❡Z ❡ U sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❆ss✐♠ ♦ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ♥ã♦ ❝❡♥tr❛❧ é ❞❛❞♦ ♣♦r

E[Sk] =E

Zk Uk

=E[Zk]E

1

Uk

.

❉❡st❛ ❢♦r♠❛✱ ♣r❡❝✐s❛✲s❡ ❡♥❝♦♥tr❛r ♦s ♠♦♠❡♥t♦s ♥ã♦ ❝❡♥tr❛✐s ❞❛s ❞✐str✐❜✉✐çõ❡s ♥♦r♠❛❧ ♣❛❞rã♦ ❡ ❜❡t❛✳

❈♦♥s✐❞❡r❛♥❞♦ Z _∼N ormal(0,1)✱ t❡♠✲s❡

E[Zk] =

(

0, s❡ k é í♠♣❛r;

2k/2_Γ(k+1

2 )

√

π , s❡ k é ♣❛r.

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ U _∼Beta(q,1)

E

1

Uk

= q

q₋k, ♣❛r❛ q > k.

❙❡♥❞♦ ❛ss✐♠✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ ♦ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ♥ã♦ ❝❡♥tr❛❧ ❞❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ é ❞❛❞♦ ♣♦r

E[Sk] =

(

0, s❡ k é í♠♣❛r ❡ q > k;

2k/2_Γ(k+1

2 )

✻ ❉■❙❚❘■❇❯■➬➹❖ ❙▲❆❙❍ ❙■▼➱❚❘■❈❆ ✷✳✸

P♦❞❡✲s❡ ✈❡r✐✜❝❛r t❛♠❜é♠✱ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ só ♣♦ss✉✐ ❡s♣❡r❛♥ç❛ ♣❛r❛q >1✱ s❡♥❞♦E[S] = 0✱ ❡ só ♣♦ss✉✐ ✈❛r✐â♥❝✐❛ ♣❛r❛q >2✱ s❡♥❞♦ V ar[S] = _q−q₂ ✭✈❡r ❲❛♥❣ ❡ ●❡♥t♦♥✱ ✷✵✵✻✮✳

❖ ❡①❝❡ss♦ ❞❡ ❝✉rt♦s❡✱ EC✱ é ❞❛❞♦ ♣♦r

EC= E[(S−E[S])

4_]

E[(S₋E[S])2_]2−3 =

E[S4]

E[S2_]2−3 = 3

q q₋4

q₋2

q

2

−3 = 3

q2₋4q+ 4

q2_−4q −1

= 12

q(q₋4).

◆♦t❛✲s❡ q✉❡ só é ♣♦ssí✈❡❧ ❝❛❧❝✉❧❛r ❛ ❝✉rt♦s❡ ♣❛r❛ q > 4✱ ❡ q✉❡ EC > 0✳ ❆ss✐♠ ❛ ❞✐str✐❜✉✐çã♦

s❧❛s❤ é ❧❡♣t♦❝úrt✐❝❛✱ ♦✉ s❡❥❛✱ ♣♦ss✉✐ ❝❛✉❞❛ ♠❛✐s ♣❡s❛❞❛ ❞♦ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ♥♦r♠❛❧✳ ◗✉❛♥❞♦ q

❛✉♠❡♥t❛ ♦ ❡①❝❡ss♦ ❞❡ ❝✉rt♦s❡ t❡♥❞❡ ❛ ③❡r♦✳

P❛r❛ ❡♥❝♦♥tr❛r ♦ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ♥ã♦ ❝❡♥tr❛❧ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ❣❡r❛❧ é só ✉s❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❧✐♥❡❛r✐❞❛❞❡ ♠❡♥❝✐♦♥❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✳

✷✳✷✳✶ ❊st✐♠❛❞♦r❡s ♣❡❧♦ ♠ét♦❞♦ ❞♦s ♠♦♠❡♥t♦s

❉❡✜♥✐♥❞♦ ✉♠❛ ❛♠♦str❛ ❛❧❡❛tór✐❛ ❞❡ t❛♠❛♥❤♦ n ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ X✳ ❙❛❜❡✲s❡ q✉❡ ♦

❦✲és✐♠♦ ♠♦♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ ❛♠♦str❛❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ µk ❡mk✱ sã♦ ❞❛❞♦s ♣♦r

µk=E[Xk] ❡ mk=

Pn

i=1Xik

n .

❈♦♥s✐❞❡r❛♥❞♦ S_∼SL(0,1, q)✱ t❡♠✲s❡ q✉❡

E[S] = 0, q >1; E[S2] = q

q₋2, q >2;

E[S3] = 0, q >3; E[S4] = 3 q

q₋4, q >4.

❆❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ X=µ+σS _∼SL(µ, σ2_{, q)✱ t❡♠♦s q✉❡}

E[X] =E[µ+σS] =µ, q >1; E[X2] =E[(µ+σS)2] =µ2+σ2 q

q₋2, q >2;

E[X3] =E[(µ+σS)3] =µ3+ 3µσ2 q

q₋2, q >3;

❡

E[X4] =E[(µ+σS)4] =µ4+ 6µ2σ2 q

q₋2 + 3σ

4 q

q₋4, q >4.

■❣✉❛❧❛♥❞♦ ♦s ♠♦♠❡♥t♦s ♣♦♣✉❧❛❝✐♦♥❛✐s ❛♦s ♠♦♠❡♥t♦s ❛♠♦str❛✐s ♦❜té♠✲s❡ ♦s ❡st✐♠❛❞♦r❡s ♣❡❧♦ ♠é✲ t♦❞♦ ❞♦s ♠♦♠❡♥t♦s✳ ❆ ❡q✉❛çã♦ ❡♥❝♦♥tr❛❞❛ ✐❣✉❛❧❛♥❞♦ ♦ t❡r❝❡✐r♦ ♠♦♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❛♦ ❛♠♦str❛❧ ♥ã♦ ♦❜té♠ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ♦ ♣❛râ♠❡tr♦q✱ ❛ss✐♠ ♣r❡❝✐s❛✲s❡ ✉t✐❧✐③❛r ❛ ❡q✉❛çã♦ ❞♦ q✉❛rt♦ ♠♦♠❡♥t♦✳

ˆ

µm = ¯X, q >1; σˆm2 =



 

 

q−2

q σˆ2, s❡ q é ❝♦♥❤❡❝✐❞♦ ❡ ♠❛✐♦r q✉❡ ✷❀

√ 4+k

2+√4+cσˆ

2_{, s❡ q é ❞❡s❝♦♥❤❡❝✐❞♦ ❡ ♠❛✐♦r q✉❡ ✹❀}

ˆ

qm = 2 +√4 +c, s❡ q é ❞❡s❝♦♥❤❡❝✐❞♦ ❡ ♠❛✐♦r q✉❡ ✹❀

♦♥❞❡ _X¯ ₌ Pn i=1Xi

n ✱σˆ2 =

Pn

i=1(Xi−X¯)2

n ❡c= 12ˆσ

4

1/nPn

✷✳✸ ❊❙❚■▼❆➬➹❖ P❖❘ ▼➪❳■▼❆ ❱❊❘❖❙❙■▼■▲❍❆◆➬❆ ✼

✷✳✸ ❊st✐♠❛çã♦ ♣♦r ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛

❖s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ tê♠ ❛ ✈❛♥t❛❣❡♠ ❞❡ q✉❡ s✉❛ ✈❛r✐â♥❝✐❛ ❛ss✐♥tót✐❝❛ é ❞❛❞❛ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞♦ ✐♥✈❡rs♦ ❞❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r ❡ ♣♦rt❛♥t♦ sã♦ ♠❛✐s ❡✜❝✐❡♥t❡s q✉❡ ♦s ❡st✐♠❛❞♦r❡s ♣❡❧♦ ♠ét♦❞♦ ❞♦s ♠♦♠❡♥t♦s✳ P♦r ✐ss♦ sã♦ ♠❛✐s ✉t✐❧✐③❛❞♦s ❞♦ q✉❡ ♦s ❡st✐♠❛❞♦r❡s ❞♦ ♠ét♦❞♦ ❞♦s ♠♦♠❡♥t♦s✳

❖s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❛ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ♥ã♦ ♣♦s✲ s✉❡♠ ❢♦r♠❛ ❢❡❝❤❛❞❛✳ ▲♦❣♦ ♣r❡❝✐s❛✲s❡ ✉t✐❧✐③❛r ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ❡st✐♠❛❞♦r❡s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❡ss❛ ❞✐str✐❜✉✐çã♦✳ ❯♠ ♠ét♦❞♦ ❜❛st❛♥t❡ ✉t✐❧✐③❛❞♦ é ♦ ❛❧❣♦r✐t♠♦ ❊▼ ✭❊s♣❡r❛♥ç❛ ❡ ▼❛①✐♠✐③❛çã♦✮✳

❆ ❞✐str✐❜✉✐çã♦ s❧❛s❤ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❝♦♠♦ ✉♠❛ ♠✐st✉r❛ ❞❡ ♥♦r♠❛✐s ♥♦ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ✭✈❡r ❆❧❜❡r❣❤✐♥✐✱ ✷✵✶✶✮✳ ❙✉❛ ❞❡♥s✐❞❛❞❡ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ♣♦r

fX(x) =

Z 1

0

f_X|U(x_|u)fU(u)du, ✭✷✳✶✮

♦♥❞❡✱X_|U =u_∼N ormal(µ, σ2_u−2_{)✱U ∼Beta(q,1)✱ ❡ X∼SL(µ, σ}2_{, q)✳}

✷✳✸✳✶ ❆❧❣♦r✐t♠♦ ❊▼

◗✉❛♥❞♦ ✉t✐❧✐③❛✲s❡ ♦ ❛❧❣♦r✐t♠♦ ❊▼ tr❛❜❛❧❤❛✲s❡ ❝♦♠ ♦✉tr❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ❞❡♥♦♠✐♥❛❞❛ ✈❡r♦s✲ s✐♠✐❧❤❛♥ç❛ ❝♦♠♣❧❡t❛✳ ❆ss✐♠ ♣r❡❝✐s❛✲s❡ ♠♦❞✐✜❝❛r ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ q✉❡ s❡ ♦❜t❡♥❤❛ ✉♠ ♣r♦❞✉t♦ ❞❡ ❞❡♥s✐❞❛❞❡s✱ ✉♠❛ ❝♦♥❞✐❝✐♦♥❛❧ ♣♦r ✉♠❛ ♠❛r❣✐♥❛❧✱ ❝♦♠♦ ♥♦ ✐♥t❡❣r❛♥❞♦ ❡♠ ✷✳✶✳ ❆ ❞✐s✲ tr✐❜✉✐çã♦ ♠❛r❣✐♥❛❧ é ❝❤❛♠❛❞❛ ❞❡ ❞❛❞♦s ❢❛❧t❛♥t❡s ✭✏♠✐ss✐♥❣ ✈❛❧✉❡s✑✮✱ ♣♦✐s ♥ã♦ sã♦ ♦❜s❡r✈❛❞♦s ❡ ❛ ❞✐str✐❜✉✐çã♦ ❝♦♥❞✐❝✐♦♥❛❧ é ❝❤❛♠❛❞❛ ❞❡ ❞❛❞♦s ♦❜s❡r✈❛❞♦s✳ ❆ss✐♠ ❛♣ós ♦❜s❡r✈❛r ✉♠❛ ❛♠♦str❛ ❛❧❡✲ ❛tór✐❛ ❞❡ t❛♠❛♥❤♦ n ♦❜t❡♠♦s ❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠♣❧❡t❛✱ ❝♦♠♣♦st❛ ♣❡❧♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s ❡ ♦s

❞❛❞♦s ❢❛❧t❛♥t❡s ✭✏♠✐ss✐♥❣ ✈❛❧✉❡s✑✮✳

❆ ❞✐str✐❜✉✐çã♦ s❧❛s❤ s❡♥❞♦ ♦❜s❡r✈❛❞❛ ❝♦♠♦ ♠✐st✉r❛ ❞❡ ♥♦r♠❛✐s ♥❛ ❡s❝❛❧❛ ❥á ❡stá ♥❛ ❢♦r♠❛ ❞❡s❡❥❛❞❛✳ ❈♦♥s✐❞❡r❛✲s❡ ❝♦♠♦ ❞❛❞♦s ❢❛❧t❛♥t❡s ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛U✳ ❆ ❞❡♥s✐❞❛❞❡ ❝♦♥❥✉♥t❛ ❞❡(X, U)✱

♣❛r❛ ♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s ❡ ❢❛❧t❛♥t❡s✱ é ❡①♣r❡ss❛ ♣♦r

fX,U(x, u) = quq

√

2πσ2e −12u2 (

x−µ)2 σ2 _I

R(x)I(0,1)(u).

❈♦♥s✐❞❡r❛♠♦s ❛❣♦r❛ ✉♠❛ ❛♠♦str❛ ❛❧❡❛tór✐❛ ❞❡ t❛♠❛♥❤♦ n❞❛ ❞✐str✐❜✉✐çã♦ ❝♦♥❥✉♥t❛ ❞❡ (X, U)✳

❖❜té♠✲s❡ ❡♥tã♦✱ ❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠♣❧❡t❛✱ ♦✉ s❡❥❛

L(θ) =qn(2πσ2)−n/2

n

Y

i=1

ui

!q

e

−

n

X

i=1

u2_i(xi−µ)2

2σ2

,

♦♥❞❡ θ= (µ, σ2_{, q)}T_✳

➱ ❝♦♠✉♠ ✉t✐❧✐③❛r✲s❡ ♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳ P♦✐s✱ ❝♦♠♦ ❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✱ ❡♥tã♦ ♠❛①✐♠✐③❛r L(θ) é ❡q✉✐✈❛❧❡♥t❡ ❛ ♠❛①✐♠✐③❛r l(θ)✱ ❞❡

♠♦❞♦ q✉❡

l(θ) =log(L(θ)) =nlog(q)₋n

2log(2πσ

2_{) +q}

n

X

i=1

log(ui)− n

X

i=1

u2

i(xi−µ)2

✽ ❉■❙❚❘■❇❯■➬➹❖ ❙▲❆❙❍ ❙■▼➱❚❘■❈❆ ✷✳✸

P❛ss♦ ❊

◆♦ ❛❧❣♦r✐t♠♦ ❊▼✱ ♥❛ ❡t❛♣❛ j✱ ♣r❡❝✐s❛✲s❡ ❡♥❝♦♥tr❛r ❛ ❡s♣❡r❛♥ç❛ ❡♠ r❡❧❛çã♦ ❛ U ❞♦ ❧♦❣❛r✐t♠♦

❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❞❛ ❛♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s ❡ ❛♦s ♣❛râ♠❡tr♦s ❡♥❝♦♥tr❛❞♦s ♥❛ ❡t❛♣❛j₋1✳ ❆ss✐♠

Q(θ,θ(j−1)) =E[l(θ)_|x,θ(j−1)] =nlog(q)₋n

2 log(2πσ

2_{) +q}

n

X

i=1

β₁(j_i)₋ n

X

i=1

β₂(j_i)(xi−µ)2

2σ2 .

P❛r❛ ❢❛❝✐❧✐t❛r ♦s ❝á❧❝✉❧♦s ❢♦✐ ✉t✐❧✐③❛❞♦ ❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦ R=U2✳ ▲♦❣♦ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ R_|X é ❞❛❞❛ ♣♦r

fR|X(r|x) =

fX,R(x, r) fX(x)

= r

q−1

2 e−r

(x−µ)2

2σ2

R1

0 r

q−1

2 e−r

(x−µ)2

2σ2 _dr

.

❉❡st❛ ❢♦r♠❛✱

E[log(U)_|X] =E[log(R1/2)_|X] = 1

2E[log(R)|X] = 1 2

R1

0(logr)r

q−1

2 e−r

(x−µ)2

2σ2 dr

R1

0 r

q−1

2 e−r

(x−µ)2

2σ2 _dr

,

❡

E[U2_|X] =E[R_|X] =

R1

0 r

q+1

2 e−r

(x−µ)2

2σ2 dr

R1

0 r

q−1

2 e−r

(x−µ)2

2σ2 _dr

.

▼❛♥✐♣✉❧❛♥❞♦ ❛s ❡s♣❡r❛♥ç❛s ❛❝✐♠❛ ❡♥❝♦♥tr❛✲s❡ β1 ❡β2✳ ❆ss✐♠✱

β₁(j_i)=E[log(Ui)|x,θ(j−1)] = 1

2

γ′

q(j−1)₊₁

2 ,12

xi−µ(j−1)

σ(j−1)

2 γ q+1 2 , 1 2

xi−µ(j−1)

σ(j−1)

2 =

1 2DGIi





q(j−1)+ 1

2 ,

1 2

xi−µ(j−1) σ(j−1)

!2 ,

❡

β₂(j_i)=E[U_i2_|x,θ(j−1)] =

γ

q(j−1)₊₃

2 ,12

xi−µ(j−1)

σ(j−1)

2

γ

q(j−1)₊₁

2 , 1 2

xi−µ(j−1)

σ(j−1) 2,

s❡♥❞♦ DGI(α, β) = ∂log(_∂αγ(α,β)) = γ_γ′₍(_α,βα,β₎) ❛ ❢✉♥çã♦ ❞✐❣❛♠❛ ✐♥❝♦♠♣❧❡t❛ ❡ γ′(α, β) = ∂γ(_∂αα,β) =

R1

✷✳✸ ❊❙❚■▼❆➬➹❖ P❖❘ ▼➪❳■▼❆ ❱❊❘❖❙❙■▼■▲❍❆◆➬❆ ✾

P❛ss♦ ▼

◆♦ s❡❣✉♥❞♦ ♣❛ss♦✱ ♥❛ ❡t❛♣❛ j✱ ❞♦ ❛❧❣♦r✐t♠♦ ♣r❡❝✐s❛✲s❡ ♠❛①✐♠✐③❛r ❛ ❡s♣❡r❛♥ç❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞❛

✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠♣❧❡t❛ ❡♠ r❡❧❛çã♦ ❛♦s ♣❛râ♠❡tr♦s✳ P❛r❛ t❛❧ ❡♥❝♦♥tr❛✲s❡ ❛s s❡❣✉✐♥t❡s ❞❡r✐✈❛❞❛s ✭❢✉♥çõ❡s ❡s❝♦r❡✮

∂Q(θ,θ(j−1))

∂µ =

n

X

i=1

(xi−µ)β₂(j_i)

σ2 ;

∂Q(θ,θ(j−1))

∂σ2 =−

n

2σ2 +

n

X

i=1

(xi−µ)2β₂(j_i)

2σ4 ;

∂Q(θ,θ(j−1))

∂q =

n

q +

n

X

i=1

β(₁j_i).

■❣✉❛❧❛♥❞♦ ❛s ❞❡r✐✈❛❞❛s ❛ ③❡r♦ ♦❜té♠✲s❡ q✉❡ ♦s ❡st✐♠❛❞♦r❡s ❞♦s ♣❛râ♠❡tr♦s ♥❛ ❡t❛♣❛j sã♦ ❞❛❞♦s

♣♦r

ˆ

µmv =µ(j)=

Pn

i=1β (j) 2i xi

Pn

i=1β (j) 2i

, σˆ2_mv = (σ(j))2 =

Pn

i=1(xi−µˆ(j))2β (j) 2i

n ❡ qˆmv =q

(j)₌₋ n

Pn

i=1β (j) 1i

.

✷✳✸✳✷ ▼❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r

❖s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s ❛ss✐♥tót✐❝❛s ót✐♠❛s✳ ❈♦♥s✐❞❡✲ r❛♥❞♦ ✉♠ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦s θ= (µ, σ2, q)T✱ ❡♥tã♦

ˆ

θ_∼a N3(θ, I_F−1(θ)).

❆ss✐♠ ♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ sã♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ♥♦r♠❛✐s✱ ❛ss✐♥t♦t✐❝❛✲ ♠❡♥t❡ ♥ã♦ ✈✐❡s❛❞♦s✱ E[θˆ] =a θ✱ ❡ ♣♦ss✉❡♠ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛s ❛ss✐♥tót✐❝❛ ✐❣✉❛❧ ❛♦ ✐♥✈❡rs♦ ❞❛

♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r✳ ❙❡♥❞♦ q✉❡✱ ♣❡❧♦ ❝r✐tér✐♦ ❞❛ ✐♥❢♦r♠❛çã♦✱ ❡♥tr❡ ♦s ❡st✐♠❛❞♦r❡s ♥ã♦ ✈✐❡s❛❞♦s ❛ ✈❛r✐â♥❝✐❛ ♠í♥✐♠❛ é ❛ ✈❛r✐â♥❝✐❛ ❡♥❝♦♥tr❛❞❛ ♥♦s ❡❧❡♠❡♥t♦s ❞♦ ✐♥✈❡rs♦ ❞❛ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r✳

❆ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r é ❞❛❞❛ ♣♦r

IF(θ) =E

−∂ 2_l(θ)

∂θ∂θT

.

❊①✐st❡♠ ❝❛s♦s ♦♥❞❡ ❡♥❝♦♥tr❛r ❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r é ♠✉✐t♦ ❝♦♠♣❧✐❝❛❞♦✳ ◆❡st❡s ❝❛s♦s ♣♦❞❡✲s❡ ❡st✐♠❛✲❧❛ ♣❡❧❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ♦❜s❡r✈❛❞❛✱ s❡♥❞♦ ❡st❛ ✉♠ ❡st✐♠❛❞♦r ❝♦♥s✐st❡♥t❡✱ q✉❡ é ❞❛❞❛ ♣♦r

IO(θ) = − ∂2l(θ)

∂θ∂θT

_θ

=θˆ

✶✵ ❉■❙❚❘■❇❯■➬➹❖ ❙▲❆❙❍ ❙■▼➱❚❘■❈❆ ✷✳✸

✷✳✸✳✸ ▼ét♦❞♦ ❞❡ ▲♦✉✐s

◗✉❛♥❞♦ ✉t✐❧✐③❛✲s❡ ♦ ❛❧❣♦r✐t♠♦ ❊▼✱ ❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛s ❛ss✐♥tót✐❝❛ ❞♦s ❡st✐♠❛❞♦r❡s ❞♦s ♣❛râ♠❡tr♦s ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ♦❜s❡r✈❛❞❛ é s✉♣❡r❡st✐♠❛❞❛✱ ♣♦✐s ✉t✐❧✐③❛✲s❡ ♦ ❧♦❣❛r✐t♠♦ ❞❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠♣❧❡t❛✱ ♦♥❞❡ ❡st❛ ♣♦ss✉✐ ♠❛✐s ✐♥❢♦r♠❛çã♦ ❞♦ q✉❡ ❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♦❜s❡r✈❛❞❛✳ ❆ss✐♠ ♣r❡❝✐s❛✲s❡ ❝♦rr✐❣✐r ❡st❛ ❡st✐♠❛t✐✈❛✱ ❡✱ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ é ✉s❛r ♦ ♠ét♦❞♦ ❞❡ ▲♦✉✐s ✭✈❡r ▲✐♠✱ ✷✵✵✼✮✳ ❆ ♣r♦♣♦st❛ ❞❡ ▲♦✉✐s ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦

−∂ 2_l(θ)

∂θ∂θT

_θ

=θˆ ≈ −∂

2_Q(θ,θˆ)

∂θ∂θT

_θ

=θˆ −V ar

∂l(θ)

∂θ

x,θˆ

θ=θˆ

.

P❛r❛ ❡♥❝♦♥tr❛r ❛ ❡st✐♠❛t✐✈❛ ❞❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ✭♦❜s❡r✈❛❞❛ ♦✉ ❡s♣❡r❛❞❛✮ ♣r❡❝✐s❛✲s❡ ❡♥❝♦♥✲ tr❛r ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ♠❛tr✐③ ❞❡ s❡❣✉♥❞❛s ❞❡r✐✈❛❞❛s✳ ❆ss✐♠✱

A= ∂

2_Q(θ,θˆ)

∂θ∂θT

_θ

=θˆ

=





a11 a12 a13

a21 a22 a23

a31 a32 a33



.

a11=−

n

X

i=1

β₂(j_i)

ˆ

σ2 ; a22=

n

2ˆσ4 −

n

X

i=1

(xi−µˆ)2β(2ji)

ˆ

σ6 ; a33=−

n

ˆ

q2;

a12=a21=−

n

X

i=1

(xi−µˆ)β(2ji)

ˆ

σ4 ; a13=a31=a23=a32= 0.

❊♠ s❡❣✉✐❞❛ ♣r❡❝✐s❛✲s❡ ❡♥❝♦♥tr❛r ❛s ❞❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠♣❧❡t❛✳

l(θ) =nlog(q)₋nlog(2πσ

2₎

2 +q

n

X

i=1

log(ui)− n

X

i=1

u2_i(xi−µ)2

2σ2 ;

∂l(θ)

∂µ =

n

X

i=1

u2

i(xi−µ)

σ2 ;

∂l(θ)

∂σ2 =−

n

2σ2 +

n

X

i=1

u2_i(xi−µ)2

2σ4 ;

∂l(θ)

∂q = n q + n X i=1

log(ui).

❆❣♦r❛✱ ❝❛❧❝✉❧❛♥❞♦ ❛ ✈❛r✐â♥❝✐❛ ❞❛s ❞❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠♣❧❡t❛ ❝♦♥❞✐❝✐♦♥❛❞❛ ❛♦s ❞❛❞♦s ❡ ❛s ❡st✐♠❛t✐✈❛s ❞♦s ♣❛râ♠❡tr♦s ❡♥❝♦♥tr❛❞❛s ♥❛ ❡t❛♣❛j✱ t❡♠✲s❡ q✉❡

B =V ar

∂l(θ)

∂θ

x,θ

(j−1)

θ=θˆ

=





b11 b12 b13

b21 b22 b23

b31 b32 b33



,

♦♥❞❡

b11=

n

X

i=1

(xi−µˆ)2

ˆ

σ4 V ar[U 2

i|x,θ(j−1)] = n

X

i=1

(xi−µˆ)2(β₄(j_i)−(β(₂j_i))2)

ˆ

σ4 ;

b22=

n

X

i=1

(xi−µˆ)4

4ˆσ8 V ar[U 2

i|x,θ(j−1)] = n

X

i=1

(xi−µˆ)4(β₄(j_i)−(β(₂j_i))2)

4ˆσ8 ;

b33=

n

X

i=1

V ar[log(Ui)|x,θ(j−1)] = n

X

i=1