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Modelos Série de Potência com Excesso de Zeros Observáveis e Latentes

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▼♦❞❡❧♦s ❙ér✐❡ ❞❡ P♦tê♥❝✐❛ ❝♦♠ ❊①❝❡ss♦

❞❡ ❩❡r♦s ❖❜s❡r✈á✈❡✐s ❡ ▲❛t❡♥t❡s

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❡ ❚❡❝♥♦❧♦❣✐❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛

▼♦❞❡❧♦s ❙ér✐❡ ❞❡ P♦tê♥❝✐❛ ❝♦♠ ❊①❝❡ss♦

❞❡ ❩❡r♦s ❖❜s❡r✈á✈❡✐s ❡ ▲❛t❡♥t❡s

❑❛t❤❡r✐♥❡ ❊❧✐③❛❜❡t❤ ❈♦❛❣✉✐❧❛ ❩❛✈❛❧❡t❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉♦✉t♦r ❱✐❝❡♥t❡ ●❛r✐❜❛② ❈❛♥❝❤♦

❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉♦✉t♦r ❆❞r✐❛♥♦ ❑❛♠✐♠✉r❛ ❙✉③✉❦✐

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊s✲

t❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦

❈❛r❧♦s ✲ ❉❊s✴❯❋❙❈❛r ❝♦♠♦ ♣❛rt❡ ❞♦s r❡✲

q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r

❡♠ ❊st❛tíst✐❝❛✳

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ à ❉❡✉s ♣♦r s❡r ♠❡✉ ♣❛r❝❡✐r♦ ♥❛ ❧✉t❛✱ ♣❡❧❛ ❢♦rç❛ ♣❛r❛ s✉♣❡r❛r ❛s ❞✐✜❝✉❧❞❛❞❡s

❞❛ ✈✐❞❛ ❡ ❛❣♦r❛ ♣❡❧❛ ✈✐tór✐❛ ❞❡st❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❞❡ ❞♦✉t♦r❛❞♦✱ ❱✐❝❡♥t❡ ●❛r✐❜❛② ❈❛♥❝❤♦✱ ♣❡❧❛s s✉❛s ✈❛❧✐♦s❛s ♦r✐✲

❡♥t❛çõ❡s q✉❡ ❢♦r❛♠ ❡ss❡♥❝✐❛✐s ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❞❡ t❡♠♣♦✱ ♣❛✲

❝✐ê♥❝✐❛ ❡ ❛♠✐③❛❞❡✳

❆♦ ♠❡✉ ❝♦✲♦r✐❡♥t❛❞♦r ❞❡ ❞♦✉t♦r❛❞♦✱ ❆❞r✐❛♥♦ ❑❛♠✐♠✉r❛ ❙✉③✉❦✐✱ ♣❡❧♦ ❛❝♦♠♣❛♥❤❛✲

♠❡♥t♦ ❞❡ s✉♣❡r✈✐sã♦ ❡ ❡stí♠✉❧♦ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳

❆♦s ♣r♦❢❡ss♦r❡s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ●❧❡✐❝✐ ❞❛ ❙✐❧✈❛ ❈❛str♦ P❡r❞♦♥á✱ ❏✉❧✐❛♥❛ ❈♦❜r❡✱

❏♦s❡♠❛r ❘♦❞r✐❣✉❡s ❡ ❱❡r❛ ▲✉❝✐❛ ❉❛♠❛s❝❡♥♦ ❚♦♠❛③❡❧❧❛ ♣❡❧❛s s✉❣❡stõ❡s q✉❡ ❛❥✉❞❛r❛♠ ♥♦

♠❡❧❤♦r❛♠❡♥t♦ ❞❡st❛ t❡s❡✳

❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡✲

❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s q✉❡ ♠❡ ♣r♦♣✐❝✐❛r❛♠ ❝♦♥❞✐çõ❡s ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

➚ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ✭❈❆P❊❙✮ ♣❡❧♦

❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❝♦♥❝❡❞✐❞♦ ❞✉r❛♥t❡ ♦ ♣❡rí♦❞♦ ❞❡ ❞♦✉t♦r❛❞♦✳

❆ ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s q✉❡ ❢❡③ ❡♠ ❙ã♦ ❈❛r❧♦s ♣♦r t❡r ♠❡ ❛❝♦♠♣❛♥❤❛❞♦ ♥♦s

♠♦♠❡♥t♦s ❞❡ ❛❧❡❣r✐❛✱ ❝♦♥s❡❧❤♦ ❡ ❡st✉❞♦✳

❊ ❝♦♠ ♠✉✐t♦ ❛♠♦r✱

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❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡✈❡ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧✱ ❡st✉❞❛r ❛ s✐❣♥✐✜❝â♥❝✐❛ ❞❡ ③❡r♦s

♥✉♠❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ♦❜s❡r✈á✈❡✐s ❡ ❧❛t❡♥t❡s✳ ◆♦s ❝♦♥❥✉♥t♦s ❞❡ ❞❛❞♦s ♦❜s❡r✈á✈❡✐s q✉❡ ♦❝♦r✲

r❡♠ ❡①❝❡ss♦s ❞❡ ③❡r♦s✱ é ❝♦♠✉♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❜r❡❞✐s♣❡rsã♦✳ ◆❡st❡ s❡♥t✐❞♦ ♦s ♠♦❞❡❧♦s

❩❡r♦✲■♥✢❛❝✐♦♥❛❞♦s ❙ér✐❡ ❞❡ P♦tê♥❝✐❛ ✭❩■❙P✮ ❢♦r❛♠ ♣r♦♣♦st♦s ♣❛r❛ ❛❝♦♠♦❞❛r ♦ ❡①❝❡ss♦ ❞❡

③❡r♦s✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ♦❜s❡r✈á✈❡✐s ❝♦♠ ❡①❝❡ss♦ ❞❡ ③❡r♦s ❞❡s❡♥✈♦❧✈❡✲

♠♦s ✉♠ ❡st✉❞♦ ❞❛ ❡st❛tíst✐❝❛ ❣r❛❞✐❡♥t❡✱ ♣r♦♣♦st❛ ♣♦r

❚❡rr❡❧❧

✭✷✵✵✷✮✱ ♣❛r❛ t❡st❛r ❛s ❤✐♣ót❡s❡s

❡♠ r❡❧❛çã♦ ❛♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥✢❛çã♦ ❞♦ ♠♦❞❡❧♦ ❩■❙P✱ ❜❛s❡❛❞♦ ♥❛ ❛✈❛❧✐❛çã♦ ❞❛ ♣❡r❢♦r♠❛♥❝❡

❞❛ ❡st❛tíst✐❝❛ ❣r❛❞✐❡♥t❡ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛s ❡st❛tíst✐❝❛s ❝❧áss✐❝❛s ❞❛ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐✲

❧❤❛♥ç❛ ✭❲✐❧❦s✱

✶✾✸✽✮✱ ❡s❝♦r❡ ✭❘❛♦✱

✶✾✹✽✮ ❡ ❲❛❧❞ ✭❲❛❧❞✱

✶✾✹✸✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ r❡❝❡♥t❡♠❡♥t❡

❛ ❢r❛❣✐❧✐❞❛❞❡ é ♠♦❞❡❧❛❞❛ ♣♦r ❞✐str✐❜✉✐çõ❡s ❞✐s❝r❡t❛s s♦❜ ♦s ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s ❡ ♣❡r♠✐t❡

❢r❛❣✐❧✐❞❛❞❡ ③❡r♦✱ ✐st♦ é✱ ✐♥❞✐✈í❞✉♦s q✉❡ ♥ã♦ ❛♣r❡s❡♥t❛♠ ♦ ❡✈❡♥t♦ ❞❡ ✐♥t❡r❡ss❡ ✭❢r❛çã♦ ❞❡ r✐s❝♦

③❡r♦✮✳ P❛r❛ ❡st❡ t✐♣♦ ❞❛❞♦s ❞❡ ❧❛t❡♥t❡s✱ ♣r♦♣✉s❡♠♦s ✉♠ ♥♦✈♦ ♠♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✐♥❞✉✲

③✐❞❛ ♣♦r ❢r❛❣✐❧✐❞❛❞❡ ❞✐s❝r❡t❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❩■❙P✳ ❊ss❛ ♣r♦♣♦st❛ tr❛③ ✉♠❛ ❞❡s❝r✐çã♦ ♠❛✐s

r❡❛❧ ❞♦s ✐♥❞✐✈í❞✉♦s s❡♠ r✐s❝♦✱ ♣♦✐s ✐♥❝❧✉✐ ✐♥❞✐✈í❞✉♦s ❝✉r❛❞♦s ❞❡✈✐❞♦ ❛♦s ❢❛t♦r❡s ❣❡♥ét✐❝♦s

✭✐♠✉♥❡s✮ ♠♦❞❡❧❛❞♦s ❝♦♠♦ ❛ ❢r❛çã♦ ❞❡ r✐s❝♦ ③❡r♦ ❞❡t❡r♠✐♥íst✐❝♦✱ ❡♥q✉❛♥t♦ q✉❡✱ ♦s ✐♥❞✐✈í❞✉♦s

❝✉r❛❞♦s ♣♦r tr❛t❛♠❡♥t♦ sã♦ ♠♦❞❡❧❛❞♦s ♣❡❧❛ ❢r❛çã♦ ❞❡ r✐s❝♦ ③❡r♦ ❛❧❡❛tór✐♦✳ ◆❡st❡ ❝♦♥t❡①t♦

❞❡s❡♥✈♦❧✈❡♠♦s t❛♠❜é♠ ❛ ❡st❛tíst✐❝❛ ❣r❛❞✐❡♥t❡ ♣❛r❛ ✈❡r✐✜❝❛r ❛ s✐❣♥✐✜❝â♥❝✐❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡

r✐s❝♦ ③❡r♦ ♣❛r❛ ❞❛❞♦s ♠♦❞❡❧❛❞♦s ♣❡❧❛ ❢r❛çã♦ ❞❡ r✐s❝♦ ③❡r♦ ❞❡t❡r♠✐♥íst✐❝♦✳ ❊ ♣❛r❛ ❝♦♠♣❧❡✲

t❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ♣r♦♣♦st❛s✱ ❛♣r❡s❡♥t❛♠♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ❡st✉❞♦s ❞❡ s✐♠✉❧❛çã♦ ❡

❡①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çã♦ ❝♦♠ ✉s♦ ❞❡ ❞❛❞♦s r❡❛✐s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❚❡st❡ ❣r❛❞✐❡♥t❡✱ ▼♦❞❡❧♦s ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦s ❙ér✐❡ ❞❡ P♦tê♥❝✐❛✱ ▼♦❞❡❧♦

❙ér✐❡ ❞❡ P♦tê♥❝✐❛✱ ❆♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❋r❛❣✐❧✐❞❛❞❡✱ ❘✐s❝♦ ③❡r♦✳

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❆❜str❛❝t

❚❤❡ ♣r❡s❡♥t ✇♦r❦✬s ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ✐s t♦ st✉❞② t❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ③❡r♦s ✐♥ ❛♥ ♦❜s❡r✲

✈❛❜❧❡ ❛♥❞ ❧❛t❡♥t ❞❛t❛✳ ■♥ ♦❜s❡r✈❛❜❧❡ ❞❛t❛ s❡t t❤❛t ♦❝❝✉r ❡①❝❡ss ♦❢ ③❡r♦s✱ ✐ts ❝♦♠♠♦♥ t♦ ❤❛✈❡

s♦❜r❡❞✐s♣❡rs✐♦♥✳ ■♥ t❤✐s s❡♥s❡✱ t❤❡ ♠♦❞❡❧s ③❡r♦✲✐♥✢❛t❡❞ ♣♦✇❡r s❡r✐❡s ✭❩■❙P✮ ✇❡r❡ ♣r♦♣♦s❡❞

t♦ ❛❝❝♦♠♠♦❞❛t❡ t❤❡s❡ ❡①❝❡ss❡s✳ ❙♣❡❝✐✜❝❛❧❧② ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ ♦❜s❡r✈❡❞ ❞❛t❛✱ ✐t ✇❛s ♠❛❞❡

❛ st✉❞② ♦❢ ❣r❛❞✐❡♥t st❛t✐st✐❝✱ ♣r♦♣♦s❡❞ ❜②

❚❡rr❡❧❧

✭✷✵✵✷✮✱ t♦ t❡st t❤❡ ❤②♣♦t❤❡s❡s ✐♥ r❡❧❛t✐♦♥

t♦ ✐♥✢❛t✐♦♥ ♣❛r❛♠❡t❡r ❩■❙P ♠♦❞❡❧s✳ ❚❤✐s t❡st ✐s ❜❛s❡❞ ♦♥ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ♣❡r❢♦r♠❛♥❝❡

♦❢ ❣r❛❞✐❡♥t st❛t✐st✐❝ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ❝❧❛ss✐❝❛❧ ❧✐❦❡❧✐❤♦♦❞ r❛t✐♦ ✭❲✐❧❦s✱

✶✾✸✽✮✱ s❝♦r❡ ✭❘❛♦✱

✶✾✹✽✮ ❛♥❞ ❲❛❧❞ ✭❲❛❧❞✱

✶✾✹✸✮ st❛t✐st✐❝s✳ ■♥ ❛❞❞✐t✐♦♥✱ r❡❝❡♥t❧②✱ ❢r❛❣✐❧✐t② ❤❛s ❜❡✐♥❣ ♠♦❞❡❧❡❞

❜② ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥s ✉s✐♥❣ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ✈❛❧✉❡s t❤❛t ❛❧❧♦✇s ③❡r♦ ❢r❛❣✐❧✐t②✱ ✇❤✐❝❤

♠❡❛♥s✱ ✐♥❞✐✈✐❞✉❛❧s ✇❤♦ ❞♦ ♥♦t ♣r❡s❡♥t t❤❡ ❡✈❡♥t ♦❢ ✐♥t❡r❡st ✭❢r❛❝t✐♦♥ ♦❢ ③❡r♦ r✐s❦✮✳ ❋♦r t❤✐s

t②♣❡ ♦❢ ❧❛t❡♥t ❞❛t❛✱ ✇❡ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ♥❡✇ s✉r✈✐✈❛❧ ♠♦❞❡❧ ✐♥❞✉❝❡❞ ❜② ❞✐s❝r❡t❡ ❢r❛✐❧t② ✇✐t❤

❩■❙P ❞✐str✐❜✉t✐♦♥✳ ❚❤✐s ♣r♦♣♦s❛❧ ❜r✐♥❣s ❛ r❡❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ ✐♥❞✐✈✐❞✉❛❧s ✇✐t❤♦✉t r✐s❦✱ ❜❡❝❛✉s❡

✐♥❞✐✈✐❞✉❛❧s ❝✉r❡❞ ❞✉❡ t♦ ❣❡♥❡t✐❝ ❢❛❝t♦rs ✭✐♠♠✉♥❡✮ ❛r❡ ♠♦❞❡❧❡❞ ❜② ❢r❛❝t✐♦♥ ♦❢ ❞❡t❡r♠✐♥✐st✐❝

③❡r♦ r✐s❦✱ ✇❤✐❧❡ t❤❡ ❝✉r❡❞ ❜② tr❡❛t♠❡♥t ❛r❡ ♠♦❞❡❧❡❞ ❜② ❢r❛❝t✐♦♥ ♦❢ r❛♥❞♦♠ ③❡r♦ r✐s❦✳ ■♥ t❤✐s

❝♦♥t❡①t✱ ✇❡ ❛❧s♦ ❞❡✈❡❧♦♣❡❞ t❤❡ ❣r❛❞✐❡♥t st❛t✐st✐❝ t♦ ✈❡r✐❢② ♣❛r❛♠❡t❡r s✐❣♥✐✜❝❛♥❝❡ ♦❢ ③❡r♦ r✐s❦

❢♦r ❞❛t❛ ♠♦❞❡❧❡❞ ❜② ❢r❛❝t✐♦♥ ♦❢ ❞❡t❡r♠✐♥✐st✐❝ ③❡r♦ r✐s❦✳ ❚♦ s❤♦✇ ♦✉r ♣r♦♣♦s❛❧s✱ ✇❡ ♣r❡s❡♥t

t❤❡ r❡s✉❧ts ♦❢ s✐♠✉❧❛t✐♦♥ st✉❞✐❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ✉s✐♥❣ r❡❛❧ ❞❛t❛✳

❑❡②✇♦r❞s✿ ●r❛❞✐❡♥t st❛t✐st✐❝✱ ❩❡r♦✲■♥✢❛t❡❞ P♦✇❡r ❙❡r✐❡s ▼♦❞❡❧✱ P♦✇❡r ❙❡r✐❡s ▼♦❞❡❧✱ ❙✉r✲

✈✐✈❛❧ ❆♥❛❧②s✐s✱ ❋r❛❣✐❧✐t②✱ ③❡r♦ r✐s❦✳

(7)

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✈✐

✶ ■♥tr♦❞✉çã♦

✶✳✶ ❊st❛tíst✐❝❛ ❣r❛❞✐❡♥t❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷ ❖❜❥❡t✐✈♦s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞❛ t❡s❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ❚❡st❡ ●r❛❞✐❡♥t❡ ♥♦ ▼♦❞❡❧♦ ❩❡r♦✲■♥✢❛❝✐♦♥❛❞♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛

✷✳✶ ❖s ♠♦❞❡❧♦s ❩❡r♦✲■♥✢❛❝✐♦♥❛❞♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✶✳✶ ❈❛s♦s ♣❛rt✐❝✉❧❛r❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✷ ❚❡st❡ ❣r❛❞✐❡♥t❡ ♥♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛

✳ ✶✺

✷✳✷✳✶ ❈❛s♦s ♣❛rt✐❝✉❧❛r❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷✳✸ ❊st✉❞♦ ❞❡ s✐♠✉❧❛çã♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷✳✸✳✶ ❙✐♠✉❧❛çõ❡s s❡♠ ❝♦✈❛r✐á✈❡✐s ♥♦ ♠♦❞❡❧♦ ❩■✲P♦✐ss♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷✳✸✳✷ ❙✐♠✉❧❛çõ❡s s❡♠ ❝♦✈❛r✐á✈❡✐s ♥♦ ♠♦❞❡❧♦ ❩■✲●❡♦♠étr✐❝❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✷✳✸✳✸

❙✐♠✉❧❛çã♦ ❝♦♠ ❝♦✈❛r✐á✈❡✐s ♥♦ ♠♦❞❡❧♦ ❩■✲P♦✐ss♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✷✳✸✳✹ ❙✐♠✉❧❛çõ❡s ❝♦♠ ❝♦✈❛r✐á✈❡✐s ♥♦ ♠♦❞❡❧♦ ❩■✲●❡♦♠étr✐❝❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✷✳✹ ❆♣❧✐❝❛çã♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✷✳✹✳✶ ❆♣❧✐❝❛çã♦ ✶✿ ▲❡sõ❡s ❡♠ ❡♠♣r❡❣❛❞♦s ❞❛ ❧✐♠♣❡③❛✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✷✳✹✳✷ ❆♣❧✐❝❛çã♦ ✷✿ ❈✉❧t✐✈♦s ❞❡ ♠❛çãs

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

(8)

✷✳✺ ❈♦♥❝❧✉sõ❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✸ ❯♠ ▼♦❞❡❧♦ ❞❡ ❙♦❜r❡✈✐✈ê♥❝✐❛ ■♥❞✉③✐❞❛ ♣♦r ❋r❛❣✐❧✐❞❛❞❡ ❉✐s❝r❡t❛✿ ❊♥❢♦q✉❡

❈❧áss✐❝♦

✹✼

✸✳✶ ■♥tr♦❞✉çã♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✸✳✷ ❖ ♠♦❞❡❧♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✸✳✷✳✶ ❆❧❣✉♥s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✸✳✷✳✷ ■♥❢❡rê♥❝✐❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✸✳✷✳✸ ▼ét♦❞♦ ❇♦♦tstr❛♣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✸✳✸ ❊st✉❞♦ ❞❡ s✐♠✉❧❛çã♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✸✳✹ ❆♣❧✐❝❛çã♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✸✳✺ ❈♦♥❝❧✉sõ❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✹ ▼♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✐♥❞✉③✐❞❛ ♣♦r ❋r❛❣✐❧✐❞❛❞❡ ❉✐s❝r❡t❛✿ ❊♥❢♦q✉❡ ❜❛②❡✲

s✐❛♥♦

✻✼

✹✳✶ ■♥❢❡rê♥❝✐❛ ❇❛②❡s✐❛♥❛ ♥♦ ♠♦❞❡❧♦ ❩■❙P✲❋❈

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✹✳✶✳✶ ❆♥á❧✐s❡ ❞❡ ❞✐❛❣♥óst✐❝♦ ❝❛s♦ ❛ ❝❛s♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵

✹✳✷ ❊st✉❞♦ ❞❡ s✐♠✉❧❛çã♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷

✹✳✸ ❆♣❧✐❝❛çã♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹

✹✳✹ ❈♦♥❝❧✉sõ❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

✺ ❈♦♥❝❧✉sõ❡s ❡ P❡sq✉✐s❛s ❋✉t✉r❛s

✼✾

✽✶

❆✳✶ ❖❜t❡♥çã♦ ❞❛ ❡st❛tíst✐❝❛ ●r❛❞✐❡♥t❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

❆✳✶✳✶ ◆♦r♠❛❧✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ❞❛s ❡st❛tíst✐❝❛s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

(9)

❇✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠♦❞❡❧♦ ❙P

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸

❇✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠♦❞❡❧♦ ❩■❙P

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹

❇✳✷✳✶ ❈á❧❝✉❧♦ ❞❛ ❢✉♥çã♦ s❝♦r❡ ❡ ❞❛ ♠❛tr✐③ ❞❡ ■❋

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

✾✵

❈✳✶ ▼ét♦❞♦ ❞♦ ❣rá✜❝♦ ❚❚❚

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵

❈✳✷ ▼❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ♦❜s❡r✈❛❞❛ ❞♦ ♠♦❞❡❧♦ ❩■❙P✲❋❈

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵

❈✳✸ ❱❡t♦r ❡s❝♦r❡ ❡ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ♦❜s❡r✈❛❞❛ ❞♦ ♠♦❞❡❧♦ ❩■P✲ ❋❈

✳ ✳ ✳ ✳ ✳ ✳ ✾✷

(10)

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✶✳✶ ❉❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦

A(θ)

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✶ ❚❛①❛s ❞❡ r❡❥❡✐çã♦ ❞❛ ❤✐♣ót❡s❡

H

o

:

φ

= 0

✱ ♥♦ ♠♦❞❡❧♦ ❩■✲P♦✐ss♦♥✱ ♣❛r❛ ❞✐❢❡✲

r❡♥t❡s ✈❛❧♦r❡s ❞❡

θ

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✷✳✷ P❡r❝❡♥t✐s ❡st✐♠❛❞♦s ❝♦♠ ❛♠♦str❛ ❞❡ t❛♠❛♥❤♦

n

= 50

s♦❜

H0

:

φ

= 0

✳ ✳ ✳ ✳ ✷✺

✷✳✸ P❡r❝❡♥t✐s ❡st✐♠❛❞♦s ❝♦♠ ❛♠♦str❛ ❞❡ t❛♠❛♥❤♦

n

= 200

s♦❜

H0

:

φ

= 0

✳ ✳ ✳ ✳ ✷✺

✷✳✹ P♦❞❡r ❞♦ t❡st❡ ❞❡

φ

q✉❛♥❞♦

θ

= 2

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✷✳✺ ❚❛①❛s ❞❡ r❡❥❡✐çã♦ ❞❛ ❤✐♣ót❡s❡ ♣❛r❛ t❡st❛r

H

o

:

φ

= 0

✱ ♥♦ ♠♦❞❡❧♦ ❩■●✱ ♣❛r❛

❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡

θ

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✷✳✻ P❡r❝❡♥t✐s ❡st✐♠❛❞♦s ❝♦♠ ❛♠♦str❛ ❞❡ t❛♠❛♥❤♦

n

= 100

s♦❜

H0

:

φ

= 0

✳ ✳ ✳ ✳ ✸✷

✷✳✼ P❡r❝❡♥t✐s ❡st✐♠❛❞♦s ❝♦♠ ❛♠♦str❛ ❞❡ t❛♠❛♥❤♦

n

= 400

s♦❜

H0

:

φ

= 0

✳ ✳ ✳ ✳ ✸✷

✷✳✽ ❚❛①❛s ❞❡ r❡❥❡✐çã♦ ❞❛ ❤✐♣ót❡s❡ ♣❛r❛ t❡st❛r

H0

:

φ

= 0

♥♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦

❩■✲P♦✐ss♦♥✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✷✳✾ P❡r❝❡♥t✐s ❡st✐♠❛❞♦s s♦❜

H0

:

φ

= 0

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✷✳✶✵ ❚❛①❛s ❞❡ r❡❥❡✐çã♦ ❞❛ ❤✐♣ót❡s❡

H0

:

β2

=

β3

= 0

♣❛r❛

S

G

, S

LR

, S

R

S

W

♣❛r❛

❞✐❢❡r❡♥t❡s t❛♠❛♥❤♦s ❞❡ ❛♠♦str❛ ♥♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❩■✲P♦✐ss♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✷✳✶✶ ❚❛①❛s ❞❡ r❡❥❡✐çã♦ ❞❛ ❤✐♣ót❡s❡

H0

:

φ

= 0

♥♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❩■✲●❡♦♠étr✐❝❛✳

✹✵

✷✳✶✷ P❡r❝❡♥t✐s ❡st✐♠❛❞♦s s♦❜

H0

:

φ

= 0

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✷✳✶✸ ❚❛①❛s ❞❡ r❡❥❡✐çã♦ ♥✉❧❛ ❞❡

H

0

:

β

2

=

β

3

= 0

♣❛r❛

S

G

, S

LR

, S

R

S

W

♣❛r❛

❞✐❢❡r❡♥t❡s t❛♠❛♥❤♦s ❞❡ ❛♠♦str❛ ♥♦ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ❩■✲●❡♦♠étr✐❝❛

✳ ✳ ✳ ✹✷

✷✳✶✹ ❊▼❱ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❝♦♠ ❛❥✉st❡ ❞♦s ♠♦❞❡❧♦s ❩■✲P♦✐ss♦♥ ❡ ❩■✲●❡♦♠étr✐❝❛

♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❧❡sõ❡s ❡♠ ❡♠♣r❡❣❛❞♦s ❞❛ ❧✐♠♣❡③❛✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✷✳✶✺ ❊st✐♠❛t✐✈❛s ❞❛s ❡st❛tíst✐❝❛s ❡ ♥í✈❡❧ ❞❡s❝r✐t✐✈♦ ❞♦ t❡st❡ ❞❡ ❤✐♣ót❡s❡s

H0

:

ω

= 0.

✹✸

(11)

✷✳✶✻ ❱❛❧♦r❡s ❞❡ ♠á①✐♠♦ ❞❛ ❧♦❣✲✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❡ ♦s ❝r✐tér✐♦s ❆■❈ ❡ ❙❇❈ ❝♦♠ ❛❥✉st❡

❞♦s ♠♦❞❡❧♦s ❩■✲P♦✐ss♦♥ ❡ ❩■✲●❡♦♠étr✐❝❛ ♣❛r❛ ❞❛❞♦s ❞❡ ❝✉❧t✐✈♦s ❞❡ ♠❛çã✳

✳ ✳ ✹✺

✷✳✶✼ ❊▼❱ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❩■✲P♦✐ss♦♥ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❝✉❧t✐✈♦s ❞❡

♠❛çã✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✷✳✶✽ ❊st✐♠❛t✐✈❛s ❞❛s ❡st❛tíst✐❝❛s ❞❡ t❡st❡ ❡ ♥í✈❡❧ ❞❡s❝r✐t✐✈♦ ♣❛r❛ ♦ t❡st❡

H0

:

ω

= 0

❞♦ ♠♦❞❡❧♦ ❩■✲P♦✐ss♦♥✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✷✳✶✾ ❊st✐♠❛t✐✈❛s ❞❛s ❡st❛tíst✐❝❛s ❡ ♥í✈❡❧ ❞❡s❝r✐t✐✈♦ ♣❛r❛ ♦ t❡st❡

H0

:

β1

=

β2

= 0

❞♦ ♠♦❞❡❧♦ ❩■✲P♦✐ss♦♥✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✷✳✷✵ ❚❡st❛♥❞♦ ❛ s✐❣♥✐✜❝â♥❝✐❛ ❞❛s ❤✐♣ót❡s❡s✱

H

0

:

β

1

= 0

H

0

:

β

2

= 0

✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✸✳✶ ▼é❞✐❛ ❞❛s ❊▼❱✱ ✈✐és ❡ r❛✐③ q✉❛❞r❛❞❛ ❞♦ ❡rr♦ q✉❛❞rát✐❝♦ ♠é❞✐♦ ✭❘❊◗▼✮ ❞♦s

♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❩■P✲❋❈✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

✸✳✷ ❊▼❱s ❝♦♠ ❛❥✉st❡ ❞♦s ♠♦❞❡❧♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❩■P✲❋❈ ❲❡✐❜✉❧❧ ❡ ❩■●✲❋❈

❲❡✐❜✉❧❧ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ♠❡❧❛♥♦♠❛✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸

✸✳✸ ❊▼❱s ❛ss✐♥tót✐❝❛s ❡ ❜♦♦tstr❛♣ ❝♦♠ ❛❥✉st❡ ❞♦ ♠♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❩■●✲❋❈

❲❡✐❜✉❧❧ s✐♠♣❧✐✜❝❛❞♦✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

✸✳✹ ❊st✐♠❛t✐✈❛s ❞❛s ❡st❛tíst✐❝❛s ❡ ♥í✈❡❧ ❞❡s❝r✐t✐✈♦ ♣❛r❛ ♦ t❡st❡ ❞❡ ❤✐♣ót❡s❡

H

0

:

ω

= 0

♥♦s ❞❛❞♦s ❞❡ ♠❡❧❛♥♦♠❛✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

✸✳✺ ❊st✐♠❛t✐✈❛ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ❢r❛çã♦ ❞❡ ❝✉r❛❞♦s ❡str❛t✐✜❝❛❞❛

♣♦r ❝❛t❡❣♦r✐❛ ❞❡ ♥ó❞✉❧♦ ❡ ✐❞❛❞❡ ❞♦s ♣❛❝✐❡♥t❡s✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✹✳✶ ❆s ♠é❞✐❛s ❞❡ ▼♦♥t❡ ❈❛r❧♦s ❞❛ ♠é❞✐❛ ❛ ♣♦st❡r✐♦r✐✱ ❞❡s✈✐♦ ♣❛❞rã♦ ❞❛s ♠é❞✐❛s

❛ ♣♦st❡r✐♦r✐ ✭❉P✮✱ r❛✐③ q✉❛❞r❛❞❛ ❞♦ ❡rr♦ q✉❛❞rát✐❝♦ ♠é❞✐♦ ✭❘❊◗▼✮ ❞❛s ♠é✲

❞✐❛s✱ ✈✐és ❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦❜❡rt✉r❛ ✭P❈✮ ❞♦ ✐♥t❡r✈❛❧♦ ❍P❉ ❞❡ ✾✺✪ ❞❡

❝♦♥✜❛♥ç❛ ♣❛r❛ ❝❛❞❛ ♣❛râ♠❡tr♦ ❞♦ ♠♦❞❡❧♦ ❩■❙P✲❋❈✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

✹✳✷ P♦r❝❡♥t❛❣❡♥s ❞❡ ✈❡③❡s q✉❡

H0

:

φ

= 0

❢♦✐ r❡❥❡✐t❛❞❛ ❡♠ ✶✵✵✵ ❛♠♦str❛s✳

✳ ✳ ✳ ✳ ✼✹

✹✳✸ ▼é❞✐❛✱ ♠❡❞✐❛♥❛ ❡ ❞❡s✈✐♦ ♣❛❞rã♦ ✭❉P✮ ❛ ♣♦st❡r✐♦r✐ ❡ ♦ ✐♥t❡r✈❛❧♦ ❍P❉ ❞❡ ✾✺✪

❞❡ ❝♦♥✜❛♥ç❛ ❞♦s ♣❛râ♠❡tr♦s ❞♦s ▼♦❞❡❧♦s ❩■P✲❋❈ ❡ ❩■●✲❋❈✱ ♣❛r❛ ♦s ❞❛❞♦s

❞❡ ♠❡❧❛♥♦♠❛✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹

✹✳✹ ❊st✐♠❛t✐✈❛s ❞❡ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❞❛s ♠❡❞✐❞❛s

ψ

✲❞✐✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦s ♠♦❞❡❧♦s

❩■P✲❋❈ ❡ ❩■●✲❋❈✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

✹✳✺ ❊st✐♠❛t✐✈❛s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❞♦ ❉■❈✱ ❊❆■❈ ❡ ❊❇■❈ ♣❛r❛ ♦ ❛❥✉st❡ ❞♦s ❞❛❞♦s

❞❡ ♠❡❧❛♥♦♠❛✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻

✹✳✻ ❙✉♠ár✐♦ ❛ ♣♦st❡r✐♦r✐✱ ♠é❞✐❛✱ ♠❡❞✐❛♥❛✱ ❞❡s✈✐♦ ♣❛❞rã♦ ❡ ✐♥t❡r✈❛❧♦ ❍P❉ ❞❡

90%

(12)

✹✳✼ ❊st✐♠❛t✐✈❛s ❇❛②❡s✐❛♥❛s ❉■❈✱ ❊❆■❈ ❡ ❊❇■❈ ♣❛r❛ ♦ ❛❥✉st❡ ❞♦s ❞❛❞♦s ❞❡ ♠❡❧❛✲

♥♦♠❛✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

✹✳✽ ❊st✐♠❛t✐✈❛ ❇❛②❡s✐❛♥❛s ♣❛r❛ ❢r❛çã♦ ❞❡ ❝✉r❛❞♦s ❡str❛t✐✜❝❛❞❛ ♣♦r ❝❛t❡❣♦r✐❛ ❞❡

(13)

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❉❛❞♦s ❞❡ ❝♦♥t❛❣❡♠ ❝♦♠ ❡①❝❡ss♦ ❞❡ ③❡r♦s ✭♦✉ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦s✮ sã♦ ❝♦♠✉♠❡♥t❡

❡♥❝♦♥tr❛❞♦s ❡♠ ♠✉✐t❛s ár❡❛s✱ ✐♥❝❧✉✐♥❞♦ ❛ ♠❡❞✐❝✐♥❛ ✭❇ö❤♥✐♥❣ ❡t ❛❧✳✱

✶✾✾✾✮✱ s❛ú❞❡ ♣ú❜❧✐❝❛

✭❩❤♦✉ ✫ ❚✉✱

✷✵✵✵✮✱ ❝✐ê♥❝✐❛ ❛♠❜✐❡♥t❛❧ ✭❆❣❛r✇❛❧ ❡t ❛❧✳✱

✷✵✵✷✮✱ ❛❣r✐❝✉❧t✉r❛ ✭❍❛❧❧✱

✷✵✵✵✮ ❡ ❛♣❧✐✲

❝❛çõ❡s ✐♥❞✉str✐❛✐s ✭▲❛♠❜❡rt✱

✶✾✾✷✮✳ ❊①❝❡ss♦ ❞❡ ③❡r♦s é ✉♠❛ ✐♥❞✐❝❛çã♦ ❞❡ s♦❜r❡❞✐s♣❡rsã♦ ♥♦s

❞❛❞♦s✱ s✐❣♥✐✜❝❛♥❞♦ q✉❡ ❛ ✐♥❝✐❞ê♥❝✐❛ ❞❡ ③❡r♦ ♥❛ ❝♦♥t❛❣❡♠ é ❣❡r❛❧♠❡♥t❡ ♠❛✐♦r ❞♦ q✉❡ ♦ ❡s♣❡✲

r❛❞♦✳ ❆ ✐♥❝✐❞ê♥❝✐❛ ❞❡ ③❡r♦s ♥❛ ❝♦♥t❛❣❡♠ t❡♠ st❛t✉s ❡s♣❡❝✐❛❧ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣rát✐❝♦✳ P♦r

❡①❡♠♣❧♦✱

❘✐❞♦✉t ❡t ❛❧✳

✭✷✵✵✶✮ ❛♣♦♥t❛r❛♠ q✉❡✱ ♥❛ ❝♦♥t❛❣❡♠ ❞❡ ❧❡sõ❡s ❞❛s ❞♦❡♥ç❛s ❡♠ ♣❧❛♥✲

t❛s✱ ✉♠❛ ♣❧❛♥t❛ ♣♦❞❡ ♥ã♦ t❡r ❧❡sõ❡s ♣♦rq✉❡ é r❡s✐st❡♥t❡ à ❞♦❡♥ç❛✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣♦rq✉❡

♥ã♦ ❤á ❡s♣♦r♦s q✉❡ ♣♦✉s❛r❛♠ s♦❜r❡ ❡❧❛s q✉❡ ❝❛✉s❛♠ ❛ ❞♦❡♥ç❛✳

❆ ❧✐t❡r❛t✉r❛ s♦❜r❡ ♦s ♠♦❞❡❧♦s ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦s é ❡①t❡♥s❛✱ ❞❡♥tr❡ ♦s q✉❛✐s ❝✐t❛♠♦s

♦s tr❛❜❛❧❤♦s

❈♦♥s✉❧ ✫ ❏❛✐♥

✭✶✾✼✸✮✱

●✉♣t❛ ❡t ❛❧✳

✭✶✾✾✺✮✱

❱❛♥ ❞❡♥ ❇r♦❡❦

✭✶✾✾✺✮✱

❨❛♥❣ ❡t ❛❧✳

✭✷✵✵✼✮✱

❞❡❧ ❈❛st✐❧❧♦ ✫ Pér❡③✲❈❛s❛♥②

✭✷✵✵✺✮✱

P❛t✐❧ ✫ ❙❤✐r❦❡

✭✷✵✵✼✮✱

❈❤♦♦✲❲♦s♦❜❛ ❡t ❛❧✳

✭✷✵✶✺✮✱

❇❛rr✐❣❛ ✫ ▲♦✉③❛❞❛

✭✷✵✶✹✮✱

❙❛♠❛♥✐ ❡t ❛❧✳

✭✷✵✶✷✮ ❡

❉❡♥❣ ✫ ❩❤❛♥❣

✭✷✵✶✺✮✳

●✉♣t❛ ❡t ❛❧✳

✭✶✾✾✺✮

❡st✉❞❛r❛♠ ❛ ❞✐str✐❜✉✐çã♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛✳

❱❛♥ ❞❡♥ ❇r♦❡❦

✭✶✾✾✺✮ ❞❡s❡♥✲

✈♦❧✈❡✉ ♦ t❡st❡ ❡s❝♦r❡ ♣❛r❛ t❡st❛r ♦ ❡①❝❡ss♦ ❞❡ ③❡r♦ ♥♦ ♠♦❞❡❧♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ P♦✐ss♦♥✳ ❏á

●✉♣t❛ ❡t ❛❧✳

✭✶✾✾✻✮ ❞❡s❡♥✈♦❧✈❡r❛♠ ♦ t❡st❡ ❞❛ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ t❡st❛r ❛ ✐♥✢❛çã♦

❞❡ ③❡r♦s ♥♦ ♠♦❞❡❧♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ P♦✐ss♦♥ ❣❡♥❡r❛❧✐③❛❞♦✳

❘✐❞♦✉t ❡t ❛❧✳

✭✷✵✵✶✮ ♣r♦♣✉s❡r❛♠

♦ ✉s♦ ❞♦ t❡st❡ ❡s❝♦r❡ ♣❛r❛ t❡st❛r ♦s ♠♦❞❡❧♦s ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ P♦✐ss♦♥ ❡ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦

❇✐♥♦♠✐❛❧ ◆❡❣❛t✐✈❛✱ ❡♥q✉❛♥t♦ q✉❡

●✉♣t❛ ❡t ❛❧✳

✭✷✵✵✺✮ ❞❡s❡♥✈♦❧✈❡r❛♠ ♦ t❡st❡ ❡s❝♦r❡ ♣❛r❛

t❡st❛r ♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥✢❛çã♦ ❞❡ ③❡r♦s ♥♦ ♠♦❞❡❧♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ P♦✐ss♦♥ ●❡♥❡r❛❧✐③❛❞♦✳

❳✐❛♥❣ ❡t ❛❧✳

✭✷✵✵✻✮ ❞❡s❡♥✈♦❧✈❡r❛♠ ♦ t❡st❡ ❡s❝♦r❡ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥✢❛çã♦ ❞❡ ③❡r♦s ❡♠

❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠ ❝♦rr❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❡①❝❡ss♦ ❞❡ ③❡r♦s✳

❉❡♥❣ ✫ ❩❤❛♥❣

✭✷✵✶✺✮ ♣r♦♣✉s❡✲

r❛♠ ♦ t❡st❡ ❡s❝♦r❡ ♣❛r❛ t❡st❛r ❛ ✐♥❝✐❞ê♥❝✐❛ ❞❡ ③❡r♦s ♥♦ ♠♦❞❡❧♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ ❇✐♥♦♠✐❛❧✳

❚♦❞❡♠ ❡t ❛❧✳

✭✷✵✶✷✮ ❡st✉❞❛r❛♠ ❛ ❡✜❝✐ê♥❝✐❛ ❞♦ t❡st❡ ❡s❝♦r❡ ♣❛r❛ ✈❡r✐✜❝❛r ❛ ❤♦♠♦❣❡♥❡✐❞❛❞❡

❡♠ ♠♦❞❡❧♦s ❣❡♥❡r❛❧✐③❛❞♦s ❝♦♠ ❡①❝❡ss♦ ❞❡ ③❡r♦s✳

❍s✉ ❡t ❛❧✳

✭✷✵✶✹✮ ❞❡s❡♥✈♦❧✈❡r❛♠ ♦ t❡st❡ ❞❡

❲❛❧❞ ♣❛r❛ ✐♥✢❛çã♦ ♦✉ ❞❡✢❛çã♦ ❞❡ ③❡r♦s ❡♠ ❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠✳ ◆♦ ❡♥t❛♥t♦ ♥ã♦ t❡♠♦s ❝♦♥❤❡✲

❝✐♠❡♥t♦ ♥❛ ❧✐t❡r❛t✉r❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ t❡st❡ ❣r❛❞✐❡♥t❡ ✭❚❡rr❡❧❧✱

✷✵✵✷✮ ❡♠ ♠♦❞❡❧♦s ♣❛r❛

(14)

❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠ ❝♦♠ ❡①❝❡ss♦ ❞❡ ③❡r♦s✳ ◆❡st❡ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈❡♠♦s ♦ t❡st❡ ❣r❛❞✐❡♥t❡

♣❛r❛ t❡st❛r ❛ ✐♥✢❛çã♦ ❞❡ ③❡r♦s ♥♦s ♠♦❞❡❧♦s ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦s ❙ér✐❡ ❞❡ P♦tê♥❝✐❛✳

❆ ✐❞❡✐❛ ❜ás✐❝❛ ❞❛ ❞❡r✐✈❛çã♦ ❞❡ ♠♦❞❡❧♦s ♣❛r❛ ❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠ ❝♦♠ ❡①❝❡ss♦s ❞❡

③❡r♦s é ♠✐st✉r❛r ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡❣❡♥❡r❛❞❛ ❡♠ ③❡r♦ ❝♦♠ ❞✐str✐❜✉✐çõ❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡

❞✐s❝r❡t❛s ✭♠♦❞❡❧♦s ❞❡ ❜❛s❡✮ ❝♦♠♦ P♦✐ss♦♥✱ ●❡♦♠étr✐❝❛✱ ❇✐♥♦♠✐❛❧✱ ❇✐♥♦♠✐❛❧ ◆❡❣❛t✐✈❛✱ ❡♥tr❡

♦✉tr❛s✳

●✉♣t❛ ❡t ❛❧✳

✭✶✾✾✺✮ ✐♥tr♦❞✉③✐r❛♠ ♦ ♠♦❞❡❧♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛ ✐♥✢❛❝✐♦♥❛❞♦ ❞❡ ③❡r♦s✱

❛♦ ♠✐st✉r❛r ❛ ❞✐str✐❜✉✐çã♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛ ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❡❣❡♥❡r❛❞❛ ❡♠ ③❡r♦✳ ❖

♠♦❞❡❧♦ t❡♠ ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡

P

(Z

=

z;

ω, θ) =

ω

+ (1

ω)

a

0

A(θ)

, z

= 0

(1

ω)

a

z

θ

z

A(θ)

, z

= 1,

2,

3, . . .

✭✶✳✶✮

❡♠ q✉❡

a

z

>

0

❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞❡

z

θ

(0, s)

s

♣♦❞❡ s❡r

✮ é ♦ ♣❛râ♠❡tr♦ ❞❡ ♣♦tê♥❝✐❛ ❞❛

sér✐❡✱

A(θ) =

X

z

∈K

a

z

θ

z

é ❛ ❢✉♥çã♦ sér✐❡ q✉❡ ❛t✉❛ ❝♦♠♦ ❝♦♥st❛♥t❡ ♥♦r♠❛❧✐③❛❞♦r❛✱ s❡♥❞♦ ✜♥✐t❛

❡ ❞✉❛s ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧ ✭

K

é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡♥✉♠❡rá✈❡❧ ❞❡ ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s✮

ω

é ♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥✢❛çã♦ ♦✉ ❞❡✢❛çã♦ ❞❡ ③❡r♦s✱ q✉❡ ♣♦❞❡ ❛ss✉♠✐r ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s✳

❆❧❣✉♥s ♠♦❞❡❧♦s ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦s ✭❩■✮ ✐♠♣♦rt❛♥t❡s sã♦ ❛s ❞✐str✐❜✉✐çõ❡s ❩■✲❇✐♥♦♠✐❛❧✱

❩■✲P♦✐ss♦♥✱ ❩■✲❇✐♥♦♠✐❛❧ ◆❡❣❛t✐✈❛ ❡ ❩■✲▲♦❣❛rít♠✐❝❛✳ P♦r ❡①❡♠♣❧♦✱ s❡

k

é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱

a

z

=

n

k

A(θ) = (1 +

θ)

k

✱ ❡♥tã♦ ✭

✶✳✶

✮ ❞❡✜♥❡ ❛ ❞✐str✐❜✉✐çã♦ ❩■✲❇✐♥♦♠✐❛❧✳ ❆❧❣✉♥s ♠❡♠❜r♦s

❞❡st❛ ❢❛♠í❧✐❛ ❞❡ ❞✐str✐❜✉✐çõ❡s sã♦ ✉s❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦ ❡ sã♦ ❡♥✉♠❡r❛❞♦s ❛ s❡❣✉✐r

a

z

= 1/z!

A(θ) =

e

θ

, θ >

0 :

❩■✲P♦✐ss♦♥

,

a

z

= 1

A(θ) = (1

θ)

1

,

0

< θ <

10 :

❩■✲●❡♦♠étr✐❝❛

,

a

z

=

k

+

z

1

k

1

A(θ) = (1

θ)

k

,

0

< θ <

1 :

❩■✲❇✐♥♦♠✐❛❧ ◆❡❣❛t✐✈❛

,

a

z

= 1/(z

+ 1)

A(θ) =

log(1

θ)/θ,

0

< θ <

1 :

❩■✲▲♦❣❛rít♠✐❝❛

.

◆♦t❡ q✉❡ q✉❛♥❞♦

ω

= 0

❛ ❞✐str✐❜✉✐çã♦ ❡♠ ✭

✶✳✶

✮ s❡ r❡❞✉③ à ❞✐str✐❜✉✐çã♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛✳ ❆

♠é❞✐❛ ❡ ❛ ✈❛r✐â♥❝✐❛ ♣❛r❛ ❛ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çã♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦s ❙ér✐❡ ❞❡ P♦tê♥❝✐❛ ✭❩■❙P✮

sã♦ ❞❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r

E[Z] =

(1

ω)θA

(θ)

A(θ)

V ar[Z] =

(1

ω)θ

A(θ)

2

(A(θ)A

(θ)A

′′

(θ)

(1

ω)θ(A

(θ))

2

),

❡♠ q✉❡

A

(θ) =

dA(θ)/dθ

A

′′

(θ) =

d

2

A(θ)/dθ

2

✳ ❆s ❡①♣r❡ssõ❡s ❞❛ ♣r✐♠❡✐r❛✱ s❡❣✉♥❞❛ ❡

t❡r❝❡✐r❛ ❞❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦

A(θ)

sã♦ ❡s❝r✐t❛s ♥❛ ❚❛❜❡❧❛

✶✳✶

♣❛r❛ ❢❛❝✐❧✐t❛r s❡✉ ✉s♦ ♥♦ ❞❡s❡♥✲

✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳

(15)

❆ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦ ♠♦❞❡❧♦ ❩■❙P é ❞❛❞❛ ♣♦r

G

Z

(ξ) =

ω

+ (1

ω)

A(θξ)

A(θ)

,

|

ξ

| ≤

1.

❖s ♠♦❞❡❧♦s ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦s tê♠ ❞✉❛s r❡♣r❡s❡♥t❛çõ❡s ✭❚♦❞❡♠ ❡t ❛❧✳✱

✷✵✶✷✮✿ ❛ r❡✲

♣r❡s❡♥t❛çã♦ ❤✐❡rárq✉✐❝❛ ❡ ❛ ♠❛r❣✐♥❛❧✳ ❖ ♠♦❞❡❧♦ ❡♠ ✭

✶✳✶

✮ s♦❜ ❛ r❡♣r❡s❡♥t❛çã♦ ❤✐❡rárq✉✐❝❛✱

ω

r❡♣r❡s❡♥t❛ ♦ ♣❡s♦ ❞❛ ♠✐st✉r❛ ❡ é ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❡

0

ω

1

✳ ❊♥q✉❛♥t♦ q✉❡

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

0

P

(Z

=

z;

ω, θ)

<

1

✱ s❡

ω

+ (1

ω)a

0

/A(θ)

0

♦ q✉❡ ✐♠♣❧✐❝❛✱

A

(

θ

)

a

0

a

0

ω <

1

✳ ◆❡ss❛ r❡♣r❡s❡♥t❛çã♦

ω

é ♦ ♣❛râ♠❡tr♦

❞❡ ✐♥✢❛çã♦ ❞❡ ③❡r♦s✱ q✉❛♥❞♦

ω >

0

✱ ❡ é ♦ ♣❛râ♠❡tr♦ ❞❡ ❞❡✢❛çã♦ ❞❡ ③❡r♦s✱ q✉❛♥❞♦

ω <

0

◆❡st❡ ❡st✉❞♦ ❝♦♥s✐❞❡r❛♠♦s ♦ ♠♦❞❡❧♦ ❡♠ ✭

✶✳✶

✮ ♥❛ r❡♣r❡s❡♥t❛çã♦ ♠❛r❣✐♥❛❧✳

❚❛❜❡❧❛ ✶✳✶✿ ❉❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦

A

(

θ

)

▼♦❞❡❧♦

A

(

θ

)

A

(

θ

)

A

′′

(

θ

)

A

′′′

(

θ

)

P♦✐ss♦♥

e

θ

e

θ

e

θ

e

θ

●❡♦♠étr✐❝♦

(1

θ

)

1

(1

θ

)

2

2(1

θ

)

3

6(1

θ

)

4

▲♦❣❛rít♠✐❝❛

log(1

θ

)

θ

θ

1

θ

+log(1

θ

)

θ

2

θ

(3

θ

2)

2(

θ

1)

2

log(1

θ

)

(1

θ

)

2

θ

3

(

11

θ

2

+15

θ

6)

(

θ

1)

3

θ

3

+

6 log(1

θ

)

θ

4

❖s ♠♦❞❡❧♦s ❞❡ ❢r❛❣✐❧✐❞❛❞❡ sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ♣❡❧❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠ ❡❢❡✐t♦ ❛❧❡❛tór✐♦✱

r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♥tí♥✉❛ ♥ã♦ ♦❜s❡r✈á✈❡❧ ♥♦ ♠♦❞❡❧♦✳ ❊st❡ ❡❢❡✐t♦

❛❧❡❛tór✐♦✱ ❞❡♥♦♠✐♥❛❞♦ ❢r❛❣✐❧✐❞❛❞❡ sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❤❡t❡r♦❣❡♥❡✐❞❛❞❡ ♥❛

❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✳ ◆❛ ❛♥á❧✐s❡ ❞❡ss❡s ❞❛❞♦s ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ❢r❛❣✐❧✐❞❛❞❡ ❡♠

❣❡♥❡r❛❧ é ❛ss✉♠✐❞❛ ❝♦♥tí♥✉❛ ✭❍♦✉❣❛❛r❞✱

✶✾✽✻✱

❱❛✉♣❡❧ ❡t ❛❧✳✱

✶✾✼✾✮✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s ♣♦❞❡ s❡r

❛♣r♦♣r✐❛❞♦ ❝♦♥s✐❞❡r❛r ❞✐str✐❜✉✐çõ❡s ❞✐s❝r❡t❛s ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ❢r❛❣✐❧✐❞❛❞❡ ✭❈❛r♦♥✐ ❡t ❛❧✳✱

✷✵✶✵✮✳ ❉❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❝♦♥t❡♥❞♦ ✉♥✐❞❛❞❡s ❡①♣❡r✐♠❡♥t❛✐s ❡♠ q✉❡ ♦ ❡✈❡♥t♦ ❞❡ ✐♥t❡r❡ss❡

♥ã♦ ❛❝♦♥t❡❝❡✉ ♠❡s♠♦ ❛♣ós ✉♠ ♣❡rí♦❞♦ ❧♦♥❣♦ ❞❡ ♦❜s❡r✈❛çã♦ ✭❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❝♦♠

❢r❛çã♦ ❞❡ ❝✉r❛ ✭■❜r❛❤✐♠ ❡t ❛❧✳✱

✷✵✵✶✱

▼❛❧❧❡r ✫ ❩❤♦✉✱

✶✾✾✻✮✳ ◆❡ss❛ s✐t✉❛çã♦ ❡ss❛s ✉♥✐❞❛❞❡s

❡①♣❡r✐♠❡♥t❛✐s tê♠ ❢r❛❣✐❧✐❞❛❞❡ ✐❣✉❛❧ ❛ ③❡r♦✳ ❆ ❢r❛❣✐❧✐❞❛❞❡ ♥❡st❡ ❝❛s♦ ♣♦❞❡ r❡♣r❡s❡♥t❛r✱ ♣♦r

❡①❡♠♣❧♦✱ ❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ♥ú♠❡r♦ ❞❡s❝♦♥❤❡❝✐❞♦ ❞❡ ❢❛t♦r❡s q✉❡ ❧❡✈❛♠ à ♦❝♦rrê♥❝✐❛ ❞♦

❡✈❡♥t♦ ❞❡ ✐♥t❡r❡ss❡✳ ❋r❛❣✐❧✐❞❛❞❡ ③❡r♦ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ♠♦❞❡❧♦ q✉❡ ❝♦♥té♠ ✉♠❛ ♣r♦♣♦rçã♦

❞❡ ✐♥❞✐✈í❞✉♦s ❧✐✈r❡ ❞♦ ❡✈❡♥t♦ ❞❡ ✐♥t❡r❡ss❡ ✭r✐s❝♦ ③❡r♦✮✳

◆❡st❡ tr❛❜❛❧❤♦✱ t❛♠❜é♠ ♣r♦♣♦♠♦s ✉♠❛ ❝❧❛ss❡ ❞❡ ♠♦❞❡❧♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✐♥❞✉✲

③✐❞♦s ♣♦r ❢r❛❣✐❧✐❞❛❞❡ ❞✐s❝r❡t❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❛❞❛ ❡♠ ✭

✶✳✶

✮ ♣❛r❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❞❛❞♦s

❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❝♦♠ ❢r❛çã♦ ❞❡ ❝✉r❛ ✭▼❛❧❧❡r ✫ ❩❤♦✉✱

✶✾✾✻✱

❘♦❞r✐❣✉❡s ❡t ❛❧✳✱

✷✵✵✾❛✮✳ ❯♠❛

❛♥á❧✐s❡ ❡st❛tíst✐❝❛ ❞♦ ♠♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✐♥❞✉③✐❞♦ ♣♦r ❢r❛❣✐❧✐❞❛❞❡ ❞✐s❝r❡t❛ é ♦❜t✐❞♦ ❝♦♥✲

s✐❞❡r❛♥❞♦ ❛s ❛❜♦r❞❛❣❡♥s ❝❧áss✐❝❛ ❡ ❇❛②❡s✐❛♥❛✳ ❆❧é♠ ❞✐ss♦✱ ♦ t❡st❡ ❣r❛❞✐❡♥t❡ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦

♣❛r❛ t❡st❛r ❛s ❤✐♣ót❡s❡s ❛ r❡s♣❡✐t♦ ❞❛ ♣r♦♣♦rçã♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❝♦♠ r✐s❝♦ ③❡r♦✳

(16)

✶✳✶ ❊st❛tíst✐❝❛ ❣r❛❞✐❡♥t❡

●r❛♥❞❡ ♣❛rt❡ ❞❛ ❧✐t❡r❛t✉r❛ tr❛t❛ ♦s ♣r♦❜❧❡♠❛s ❞❡ t❡st❡ ❞❡ ❤✐♣ót❡s❡s ❡♠ ♠♦❞❡❧♦s

♣❛r❛♠étr✐❝♦s✱ t❛✐s ❝♦♠♦ ♦s ♠♦❞❡❧♦s ❝♦♠ ❡①❝❡ss♦ ❞❡ ③❡r♦s✱ ♦s ♠♦❞❡❧♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❡♥tr❡

♦✉tr♦s✳ ❆s ❡st❛tíst✐❝❛s ❞❡ t❡st❡ ✉s✉❛✐s ♣❛r❛ ❣r❛♥❞❡s ❛♠♦str❛s sã♦ ❛s ❡st❛tíst✐❝❛s ❞❛ r❛③ã♦

❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭❲✐❧❦s✱

✶✾✸✽✮✱ ❡s❝♦r❡ ✭❘❛♦✱

✶✾✹✽✮ ❡ ❲❛❧❞ ✭❲❛❧❞✱

✶✾✹✸✮✱ ♥♦ ❡♥t❛♥t♦✱

r❡❝❡♥t❡♠❡♥t❡ ✉♠❛ ♥♦✈❛ ❡st❛tíst✐❝❛ ❢♦✐ ♣r♦♣♦st❛ ♣♦r

❚❡rr❡❧❧

✭✷✵✵✷✮ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s

❞❡ t❡st❡s ❞❡ ❤✐♣ót❡s❡s ❡♠ ❛❧❣✉♥s ♠♦❞❡❧♦s ♣❛r❛♠étr✐❝♦s ✭▲❡♠♦♥t❡✱

✷✵✶✻✱

▲❡♠♦♥t❡ ✫ ❋❡rr❛r✐✱

✷✵✶✶✱

✷✵✶✷✮✳ ❆ ❢♦r♠✉❧❛çã♦ ❞❛s ❡st❛tíst✐❝❛s ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠❛ ❤✐♣ót❡s❡ ♥✉❧❛ s✐♠♣❧❡s sã♦

❛♣r❡s❡♥t❛❞❛s ❛ s❡❣✉✐r✳

❈♦♥s✐❞❡r❡ q✉❡

y

1

, . . . , y

n

❞❡♥♦t❛♠

n

r❡❛❧✐③❛çõ❡s ❞❡ ❛❧❣✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ✉♠❛

♣♦♣✉❧❛çã♦ ❝♦♠ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♦✉ ♣r♦❜❛❜✐❧✐❞❛❞❡✱

f

(y,

ϑ

)

ϑ

Θ

R

p

✳ ◗✉❛♥❞♦ ♥ã♦ ❤á

♣❛râ♠❡tr♦s ❞❡ ♣❡rt✉❜❛çã♦✱ ♦ ✐♥t❡r❡ss❡ é t❡st❛r ❛ ❤✐♣ót❡s❡s ♥✉❧❛ s✐♠♣❧❡s

H

0

:

θ

=

ϑ

0

✈❡rs✉s

H

1

:

ϑ

6

=

ϑ

0

✱ ❡♠ q✉❡

ϑ

0

é ✉♠ ✈❡t♦r ❡s♣❡❝✐✜❝❛❞♦ ♣❛r❛

ϑ

Θ

✳ ❙❡❥❛♠

ℓ(

ϑ

)

U

(

ϑ

)

K

(

ϑ

)

❧♦❣✲✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ♦ ✈❡t♦r ❡s❝♦r❡ ❡ ❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ❞❡ ❋✐s❤❡r r❡❧❛t✐✈❛s ❛♦ ✈❡t♦r

ϑ

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆s ❡st❛tíst✐❝❛s ❝♦♠✉♠❡♥t❡ ✉s❛❞❛s ♣❛r❛ t❡st❛r

H

0

✈❡rs✉s

H

1

sã♦✿ r❛③ã♦ ❞❡

✈❡r♦ss✐♠✐❧❤❛♥ç❛ ✭

S

LR

✮✱ ❡s❝♦r❡ ❞❡ ❘❛♦ ✭

S

R

✮ ❡ ❲❛❧❞ ✭

S

W

✮✱ q✉❡ sã♦ ❞❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

♣♦r

S

LR

= 2[ℓ(

ϑ

b

)

ℓ(

ϑ

0

)],

✭✶✳✷✮

S

R

=

U

(

ϑ

0

)

K

(

ϑ

0

)

U

(

ϑ

0

)),

✭✶✳✸✮

S

W

= (

ϑ

b

ϑ

0

)

K

1

(

ϑ

b

)(

ϑ

b

ϑ

0

),

✭✶✳✹✮

❡♠ q✉❡

ϑ

b

é ♦ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞❡

θ

❚❡rr❡❧❧

✭✷✵✵✷✮ ♣r♦♣ôs ✉♠❛ ❡st❛tíst✐❝❛ ❞❡♥♦♠✐♥❛❞❛ ❣r❛❞✐❡♥t❡✳ ❊st❛ ❡st❛tíst✐❝❛ é ♦❜✲

t✐❞❛ ❞❛s ❡st❛tíst✐❝❛s ❞❡ ❡s❝♦r❡ ❡ ❞❡ ❲❛❧❞✱ ❡ é ❞❛❞❛ ♣♦r

S

G

=

U

(

ϑ

0

)

(

ϑ

b

ϑ

0

).

✭✶✳✺✮

◆♦t❡ q✉❡ ❛ ❡st❛tíst✐❝❛ ❣r❛❞✐❡♥t❡ é ♠✉✐t♦ ♠❛✐s s✐♠♣❧❡s ❞❡ s❡r ❝❛❧❝✉❧❛❞❛✱ ❥á q✉❡ ♥ã♦ ❡♥✈♦❧✈❡ ❛

♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦ ♦❜s❡r✈❛❞❛ ♦✉ ❡s♣❡r❛❞❛✱ t❛♠♣♦✉❝♦ ❛ ✐♥✈❡rs❛ ❞❛ ♠❛tr✐③✳

▲❡♠♦♥t❡ ✫ ❋❡rr❛r✐

✭✷✵✶✶✮ ❡

▲❡♠♦♥t❡ ❡t ❛❧✳

✭✷✵✶✻✮ ❝♦♠♣❛r❛r❛♠ ♣♦r ♠❡✐♦ ❞❡ s✐♠✉❧❛çõ❡s✱ ♦s t❡st❡s ❞❛ r❛③ã♦ ❞❡

✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ❡s❝♦r❡✱ ❲❛❧❞ ❡ ❣r❛❞✐❡♥t❡ ❡♠ ♠♦❞❡❧♦s ❞❡ ❇✐r❜❛✉♥✲❙❛✉♥❞❡rs ❝♦♠ r❡❣r❡ssã♦

❧✐♥❡❛r ❡ ♥ã♦ ❧✐♥❡❛r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

▲❡♠♦♥t❡ ✫ ❋❡rr❛r✐

✭✷✵✶✷✮ ♦❜t✐✈❡r❛♠ ♦s ♣♦❞❡r❡s ❧♦❝❛✐s

❞♦s t❡st❡s ❜❛s❡❛❞♦s ❝♦♠ ❛s ❡st❛tíst✐❝❛s ❞❡ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ❡s❝♦r❡✱ ❲❛❧❞ ❡ ❣r❛❞✐❡♥t❡

(17)

♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❛ ❢❛♠í❧✐❛ ❊①♣♦♥❡♥❝✐❛❧ ✉♥✐♣❛r❛♠étr✐❝❛ ❡ ❝♦♥❝❧✉ír❛♠ q✉❡ ♥❡♥❤✉♠❛ ❞❛s

❡st❛tíst✐❝❛s é ✉♥✐❢♦r♠❡♠❡♥t❡ s✉♣❡r✐♦r às ♦✉tr❛s✳ ❖ ❧✐✈r♦ ❞❡

▲❡♠♦♥t❡

✭✷✵✶✻✮ ❛♣r❡s❡♥t❛ r❡s✉❧✲

t❛❞♦s ❛♥❛❧ít✐❝♦s ❡ ❡✈✐❞❡♥❝✐❛s ♥✉♠ér✐❝❛s ❞♦ t❡st❡ ❣r❛❞✐❡♥t❡✳ ❆ ♦❜t❡♥çã♦ ❞♦ t❡st❡ ❣r❛❞✐❡♥t❡ é

♠♦str❛❞♦ ♥♦ ❆♣ê♥❞✐❝❡

❆✳✶

✶✳✷ ❖❜❥❡t✐✈♦s

◆♦ ❝♦♥t❡①t♦ ❛♣r❡s❡♥t❛❞♦ ❛❝✐♠❛✱ ♦ ♦❜❥❡t✐✈♦ ❞❡st❛ t❡s❡ é ♣r♦♣♦r ♣r♦❝❡❞✐♠❡♥t♦s ❛❧✲

t❡r♥❛t✐✈♦s ❛♦s t❡st❡s ❝❧áss✐❝♦s✱ ♣❛r❛ ♦s t❡st❡s ❞❡ ❤✐♣ót❡s❡s ❞♦s ♣❛râ♠❡tr♦s ❞❡ ✉♠❛ ❢❛♠í❧✐❛

❞❡ ❞✐str✐❜✉✐çã♦ ❞✐s❝r❡t❛ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ ✭❞✐str✐❜✉✐çã♦ ❩■❙P✮ q✉❡ ✉♥✐✜❝❛ ✈ár✐♦s ♠♦❞❡❧♦s✳

❆❧é♠ ❞✐ss♦✱ ♣r♦♣♦r ✉♠ ♥♦✈♦ ♠♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✐♥❞✉③✐❞♦ ♣♦r ❢r❛❣✐❧✐❞❛❞❡ ❞✐s❝r❡t❛

❝♦♠ ❞✐str✐❜✉✐çã♦ ❩■❙P✱ q✉❡ ✐♥❝❧✉✐ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ♦ ♠♦❞❡❧♦ ❞❡ t❡♠♣♦ ❞❡ ♣r♦♠♦çã♦

✭❨❛❦♦✈❧❡✈ ❡t ❛❧✳✱

✶✾✾✸✮ ❡ ♦ ♠♦❞❡❧♦ ♣r♦♣♦st♦ ♣♦r

❈❛♥❝❤♦ ❡t ❛❧✳

✭✷✵✶✸✮✱ ❡♥tr❡ ♦✉tr♦s✳ ❉❡ss❛

❢♦r♠❛✱ ♣♦❞❡♠♦s r❡❧❛❝✐♦♥❛r ♦s s❡❣✉✐♥t❡s ♦❜❥❡t✐✈♦s ❡s♣❡❝í✜❝♦s q✉❡ ❢♦r❛♠ ❛t✐♥❣✐❞♦s ♥♦ ❞❡❝♦rr❡r

❞❡st❛ t❡s❡✿

✭✐✮ ❞❡s❡♥✈♦❧✈❡r ♦ t❡st❡ ❣r❛❞✐❡♥t❡ ♣❛r❛ t❡st❛r ❛ ✐♥✢❛çã♦ ✭♦✉ ❞❡✢❛çã♦✮ ♥♦ ♠♦❞❡❧♦ ❩■❙P

❞❛❞❛ ❡♠ ✭

✶✳✶

✮ ❡ ❛✈❛❧✐❛r ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ♦ t❡st❡ ♣❛r❛ ♣❡q✉❡♥❛s✱

♠♦❞❡r❛❞❛s ❡ ❣r❛♥❞❡s t❛♠❛♥❤♦s ❛♠♦str❛✐s❀

✭✐✐✮ ❢❛③❡r ✉♠ ❡st✉❞♦ ❝♦♠♣❛r❛t✐✈♦ ❞♦ ♣♦❞❡r ❞♦ t❡st❡ ♣r♦♣♦st♦ ❝♦♠ ♦s t❡st❡s ❡s❝♦r❡✱ ❲❛❧❞ ❡

r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡s❝r✐t♦ ❡♠ ✭✐✮❀

✭✐✐✐✮ ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❡str✉t✉r❛✐s ❞♦ ♠♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✐♥❞✉③✐❞❛ ♣♦r ❢r❛❣✐❧✐❞❛❞❡

❞✐s❝r❡t❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❛❞❛ ❡♠ ✭

✶✳✶

✮ ❡ ❞❡s❡♥✈♦❧✈❡r ♦s ♣r♦❝❡❞✐♠❡♥t♦s ✐♥❢❡r❡♥❝✐❛✐s

♣♦r ♠❡✐♦ ❞❛ ♣❡rs♣❡❝t✐✈❛ ❝❧áss✐❝❛ ❡ ❇❛②❡s✐❛♥❛❀

✭✐✈✮ ❞❡s❡♥✈♦❧✈❡r ♦ t❡st❡ ❣r❛❞✐❡♥t❡ ♣❛r❛ ♦ ♥♦✈♦ ♠♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❝♦♠ ❢r❛çã♦ ❞❡ ❝✉r❛✳

✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞❛ t❡s❡

❆ t❡s❡ s❡ ❞❡s❡♥✈♦❧✈❡ ❝♦♠♦ s❡❣✉❡✳ ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ♦ ♠♦❞❡❧♦ ❩■❙P

♥❛ ♣r❡s❡♥ç❛ ❡ ❛✉sê♥❝✐❛ ❞❡ ❝♦✈❛r✐á✈❡✐s✱ ♦❜t❡♠♦s ❛ ❢✉♥çã♦ ❡s❝♦r❡ ❡ ❛ ♠❛tr✐③ ❞❡ ✐♥❢♦r♠❛çã♦

❞❡ ❋✐s❤❡r✱ ❞❡s❡♥✈♦❧✈❡♠♦s ♦s t❡st❡s✱ ❣r❛❞✐❡♥t❡✱ ❡s❝♦r❡✱ ❲❛❧❞ ❡ ❞❛ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳

❆❧é♠ ❞✐ss♦✱ ❝♦♠♣❛r❛♠♦s ♦ ♣♦❞❡r ❞♦s t❡st❡s ❞❛s ❡st❛tíst✐❝❛s ❞❡ t❡st❡ ❣r❛❞✐❡♥t❡✱ ❡s❝♦r❡✱ ❲❛❧❞

❡ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ❡ r❡❛❧✐③❛♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦

❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s r❡❛✐s✳ ◆♦ ❈❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♥♦✈♦ ♠♦❞❡❧♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛

✐♥❞✉③✐❞♦ ♣♦r ❢r❛❣✐❧✐❞❛❞❡ ❞✐s❝r❡t❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❩■❙P✳ ❯♠❛ ❛❜♦r❞❛❣❡♠ ❝❧áss✐❝❛ ✉t✐❧✐③❛♥❞♦

♦ ♠ét♦❞♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ é ❝♦♥s✐❞❡r❛❞❛ ♣❛r❛ ❢❛③❡r ❛ ✐♥❢❡rê♥❝✐❛ s♦❜r❡ ♦s ♣❛râ♠❡✲

tr♦s ❞♦ ♠♦❞❡❧♦✳ ❆❧é♠ ❞✐ss♦✱ ❞❡s❡♥✈♦❧✈❡♠♦s ♦ t❡st❡ ❣r❛❞✐❡♥t❡ ♣❛r❛ ❛✈❛❧✐❛r ❛ ❛❞❡q✉❛❜✐❧✐❞❛❞❡

(18)

❞♦ ♠♦❞❡❧♦ ❞❡ t❡♠♣♦ ❞❡ ♣r♦♠♦çã♦✳ ❚❛♠❜é♠✱ r❡❛❧✐③❛♠♦s ✉♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ❝♦♠ ♦ ♦❜✲

❥❡t✐✈♦ ❞❡ ❛✈❛❧✐❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦ ♠ét♦❞♦ ❞❡ ❡st✐♠❛çã♦ ❡ ❞♦ t❡st❡ ♣r♦♣♦st♦ ❡✱ ✉♠❛ ❛♣❧✐❝❛çã♦

❞♦ ♠♦❞❡❧♦ ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s r❡❛✐s ❞❛ ❧✐t❡r❛t✉r❛✳ ◆♦ ❈❛♣ít✉❧♦ ✹ ❞❡s❡♥✈♦❧✈❡♠♦s

♠ét♦❞♦s ❇❛②❡s✐❛♥♦s ❞❡ ❡st✐♠❛çã♦ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❩■❙P✲❋❈ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ▼♦♥t❡ ❈❛r❧♦

❈❛❞❡✐❛s ❞❡ ▼❛r❦♦✈ ✭▼❈▼❈✮✳ ❋✐♥❛❧♠❡♥t❡✱ ❛s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❧✉sõ❡s ❞❡st❛ t❡s❡ ❝♦♠ ❜❛s❡ ♥♦s

r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✱ ❜❡♠ ❝♦♠♦ ❛❧❣✉♠❛s ♣❡rs♣❡❝t✐✈❛s ❢✉t✉r❛s ❞❡ ♣❡sq✉✐s❛ sã♦ ❞✐s❝✉t✐❞❛s ♥♦

❈❛♣ít✉❧♦ ✺✳

(19)

❈❛♣ít✉❧♦ ✷

❚❡st❡ ●r❛❞✐❡♥t❡ ♥♦ ▼♦❞❡❧♦

❩❡r♦✲■♥✢❛❝✐♦♥❛❞♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛

◆❡st❡ ❈❛♣ít✉❧♦✱ ♠♦str❛♠♦s ❛ ♦❜t❡♥çã♦ ❞❛ ❡st❛tíst✐❝❛ ❣r❛❞✐❡♥t❡ ♣❛r❛ t❡st❛r ❛ ✐♥✢❛çã♦

✭♦✉ ❞❡✢❛çã♦✮ ❞❡ ③❡r♦s ♥♦ ♠♦❞❡❧♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛ ❞❛❞❛ ❡♠ ✭

✶✳✶

✮✱ ♦✉

s❡❥❛✱ t❡st❛r ❛ ❤✐♣ót❡s❡ ♥✉❧❛ ❝♦♠♣♦st❛✱

H

0

:

ω

=

ω

0

❝♦♥tr❛

H

1

:

ω

6

=

ω

0

,

❡♠ q✉❡

ω

0

é

✉♠❛ q✉❛♥t✐❞❛❞❡ ❡s♣❡❝✐✜❝❛❞❛ ♣❛r❛

ω.

❆❧é♠ ❞✐ss♦✱ ❝♦♠♣❛r❛♠♦s ❛ ♣❡r❢♦r♠❛♥❝❡ ❞❛ ❡st❛tíst✐❝❛

❣r❛❞✐❡♥t❡ ❝♦♠ ❛s ❡st❛tíst✐❝❛s ❞❛ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✱ ❡s❝♦r❡ ❡ ❲❛❧❞ ♣❛r❛ ♣❡q✉❡♥❛s✱

♠♦❞❡r❛❞❛s ❡ ❣r❛♥❞❡s ❛♠♦str❛s✳

✷✳✶ ❖s ♠♦❞❡❧♦s ❩❡r♦✲■♥✢❛❝✐♦♥❛❞♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛

❆ ❞✐str✐❜✉✐çã♦ ❞♦ ♠♦❞❡❧♦ ③❡r♦✲✐♥✢❛❝✐♦♥❛❞♦ ❙ér✐❡ ❞❡ P♦tê♥❝✐❛ ✭❩■❙P✮ ❞❛❞❛ ❡♠ ✭

✶✳✶

✮✱

r❡♣❛r❛♠❡tr✐③❛❞♦ ❡♠

ω

=

φ/(1 +

φ)

✭✈❡❥❛✱

❱❛♥ ❞❡♥ ❇r♦❡❦✱

✶✾✾✺✮✱ t❡♠ ❛ ❢✉♥çã♦ ❞❡ ♣r♦❜❛❜✐✲

❧✐❞❛❞❡ ❞❛❞❛ ♣♦r

P

(Z

;

θ, φ) =

φ

+

a

0

A(θ)

1

1 +

φ

, z

= 0

a

z

θ

z

A(θ)

1

1 +

φ

, z

= 1,

2,

3, . . .

✭✷✳✶✮

❡♠ q✉❡

θ

(0, s)

s

♣♦❞❡ s❡r

✮ ❡

a

0

/A(θ)

φ <

✳ ❈♦♠♦ ♦❜s❡r✈❛❞♦ ♣♦r

❱❛♥ ❞❡♥ ❇r♦❡❦

✭✶✾✾✺✮✱ ❝♦♠ ❡ss❛ ♣❛r❛♠❡tr✐③❛çã♦ ❛s ❡st✐♠❛t✐✈❛s ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦s ♣❛râ♠❡tr♦s

❞♦ ♠♦❞❡❧♦ sã♦ ♠✉✐t♦ ♠❛✐s ❡stá✈❡✐s ♥✉♠❡r✐❝❛♠❡♥t❡✳ ❆ ♠é❞✐❛ ❡ ❛ ✈❛r✐â♥❝✐❛ ❞♦ ❩■P❙ ♥❡st❛

♣❛r❛♠❡tr✐③❛çã♦ sã♦ ❞❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r

E[Z

] =

θA

(θ)

(1 +

φ)A(θ)

V ar[Z] =

θ

2

A

′′

(θ)

(1 +

φ)A(θ)

θA

(θ)

(1 +

φ)A(θ)

2

.

Referências

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