Um estudo comparativo para modelos de séries temporais de contagem

(1)

▲✉✐③❛ ❇❛r❜♦s❛ ❆♠♦r✐♠ ❋❡rr❡✐r❛

❯♠ ❡st✉❞♦ ❝♦♠♣❛r❛t✐✈♦ ♣❛r❛ ♠♦❞❡❧♦s ❞❡

sér✐❡s t❡♠♣♦r❛✐s ❞❡ ❝♦♥t❛❣❡♠

(2)

▲✉✐③❛ ❇❛r❜♦s❛ ❆♠♦r✐♠ ❋❡rr❡✐r❛

❯♠ ❡st✉❞♦ ❝♦♠♣❛r❛t✐✈♦ ♣❛r❛ ♠♦❞❡❧♦s ❞❡

sér✐❡s t❡♠♣♦r❛✐s ❞❡ ❝♦♥t❛❣❡♠

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦

❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡

❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ♣❛r❛ ❛ ♦❜✲

t❡♥çã♦ ❞❡ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❊st❛✲

tíst✐❝❛✱ ♥❛ ➪r❡❛ ❞❡ ❙ér✐❡s ❚❡♠♣♦r❛✐s✳

❖r✐❡♥t❛❞♦r❛✿ ●❧❛✉r❛ ❞❛ ❈♦♥❝❡✐çã♦

❋r❛♥❝♦

❈♦✲♦r✐❡♥t❛❞♦r✿ ❋r❛♥❦ ▼❛❣❛❧❤ã❡s ❞❡

P✐♥❤♦

(3)

▲✉✐③❛ ❇❛r❜♦s❛ ❆♠♦r✐♠ ❋❡rr❡✐r❛✳

❯♠ ❡st✉❞♦ ❝♦♠♣❛r❛t✐✈♦ ♣❛r❛ ♠♦❞❡❧♦s ❞❡ sér✐❡s t❡♠♣♦r❛✐s ❞❡ ❝♦♥t❛❣❡♠

✽✺ ♣á❣✐♥❛s

❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛✲ t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✳ ❉❡♣❛rt❛✲ ♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛✳

✶✳ Pr♦❝❡ss♦s ❆✉t♦r❡❣r❡ss✐✈♦s

✷✳ Pr♦❝❡ss♦s ▼é❞✐❛s ▼ó✈❡✐s

✸✳ ▼♦❞❡❧♦ ▲✐♥❡❛r ●❡♥❡r❛❧✐③❛❞♦

■✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✳ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

Pr♦❢✳ ❉r✳ Pr♦❢✳ ❉r✳

▼❛r❝♦s ❖❧✐✈❡✐r❛ Pr❛t❡s ▲✉✐s ▼❛✉r✐❝✐♦ ❈❛str♦ ❈❡♣❡r♦

Pr♦❢✳ ❉r✳ Pr♦❢✳ ❉r❛✳

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❆❣r❛❞❡❝✐♠❡♥t♦s

➚ ❉❡✉s ♣♦r ♠❛✐s ❡st❛ ✈✐tór✐❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛✳

➚ ◆♦ss❛ ❙❡♥❤♦r❛ ❆♣❛r❡❝✐❞❛ ♣❡❧❛ ♣♦❞❡r♦s❛ ✐♥t❡r❝❡ssã♦ ♥❡ss❛ ❥♦r♥❛❞❛✳

❆♦s ♠❡✉s ♣❛✐s✱ ❙ér❣✐♦ ❡ ❙✉❡❧②✱ ❡ ♠✐♥❤❛ ✐r♠ã ▼❛r✐❛♥❛ ♣♦r ❛❝r❡❞✐t❛r❡♠ ❡ ♠❡ ❛♣♦✐❛r❡♠ ♣❛r❛ t♦r♥❛r ❡ss❡ ♣r♦❥❡t♦ ♣♦ssí✈❡❧✳

❆♦ ♠❡✉ ♠❛r✐❞♦✱ ❛♠✐❣♦ ❡ ❝♦♠♣❛♥❤❡✐r♦ ❏♦sé ❈❛r❧♦s✱ q✉❡ s❡♠♣r❡ ❡st❡✈❡ ❛♦ ♠❡✉ ❧❛❞♦ ♣❛❝✐❡♥t❡♠❡♥t❡ ❞❛♥❞♦ ❢♦rç❛ ❡ ✐♥❝❡♥t✐✈♦✳

➚ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❡❧❛s ♦r❛çõ❡s ❡ t♦r❝✐❞❛✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ♣ós ❣r❛❞✉❛çã♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❘❛❝❤❡❧✱ ❘❡♥❛t❛✱ ▲❛r✐ss❛✱ ▼❛r❝❡❧❛ ❡ ▼❛✉rí❝✐♦✱ ✈♦❝ês ❢♦r❛♠ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❞✉❛s ♠❡t♦❞♦❧♦❣✐❛s ❞❡ sér✐❡s t❡♠♣♦r❛✐s ❞❡ ❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠ sã♦ ❛✈❛❧✐❛✲ ❞❛s✱ ♦ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ●❡♥❡r❛❧✐③❛❞♦ ✭●❆❘▼❆✮ ❡ ♦ ▼♦❞❡❧♦ ❆✉t♦r❡✲ ❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ▲✐♥❡❛r ●❡♥❡r❛❧✐③❛❞♦ ✭●▲❆❘▼❆✮✳ ❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ é ❛♥❛❧✐s❛r ❛ q✉❛❧✐❞❛❞❡ ❞♦ ❛❥✉st❡ ❞♦s ♠♦❞❡❧♦s ❡♠ q✉❡stã♦✱ ❛tr❛✈és ❞❡ ❛❧❣♦r✐t✐♠♦s ✐♠♣❧❡♠❡♥t❛❞♦s ❡♠ ❧✐♥❣✉❛❣❡♠ ❘✳ ❯♠ ♦✉tr♦ ♦❜❥❡t✐✈♦ é ❝♦♠♣❛r❛r ❡st❡s ♠♦❞❡❧♦s ❛♦ ▼♦❞❡❧♦ ▲✐♥❡❛r ●❡✲ ♥❡r❛❧✐③❛❞♦ ✭▼▲●✮✱ q✉❡ ♣❡r♠✐t❡ ♦ ❛❥✉st❡ ❞❡ ❞❛❞♦s ♥ã♦✲●❛✉ss✐❛♥♦s✱ ♠❛s ♥ã♦ ❧❡✈❛ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ❞❡♣❡♥❞ê♥❝✐❛ t❡♠♣♦r❛❧ ❡①✐st❡♥t❡ ♥❡st❡ t✐♣♦ ❞❡ ♦❜s❡r✈❛çõ❡s✳ ❯♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ é r❡❛❧✐③❛❞♦ ❛✜♠ ❞❡ ✈❡r✐✜❝❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ❡st✐♠❛t✐✈❛s✳ ❉✉❛s ❛♣❧✐✲ ❝❛çõ❡s ❛ sér✐❡s r❡❛✐s sã♦ r❡❛❧✐③❛❞❛s✱ ♦ ♥ú♠❡r♦ ❞❡ ❡♠♣r❡s❛s q✉❡ ❞❡❝r❡t❛r❛♠ ❢❛❧ê♥❝✐❛ ♥♦s ❊st❛❞♦s ❯♥✐❞♦s ♥♦s ❛♥♦s ❞❡ ✶✾✽✺ ❛ ✷✵✶✷✱ ❡ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ♠❡♥s❛✐s ❞❡ ♣♦❧✐♦♠❡❧✐t❡ ❡♠ ✉♠ ❤♦s♣✐t❛❧✳ ❖s ♠♦❞❡❧♦s ✉t✐❧✐③❛❞♦s ❞❡s❝r❡✈❡♠ ❜❡♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s sér✐❡s✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ t✇♦ ♠❡t❤♦❞s ❢♦r t✐♠❡ s❡r✐❡s ♦❢ ❝♦✉♥ts ❛r❡ ❡✈❛❧✉❛t❡❞✱ t❤❡ ❆✉t♦r❡❣r❡ss✐✈❡ ▼♦✈✐♥❣ ❆✈❡r❛❣❡ ●❡♥❡r❛❧✐③❡❞ ▼♦❞❡❧ ✭●❆❘▼❆✮ ❛♥❞ t❤❡ ❆✉t♦r❡❣r❡ss✐✈❡ ▼♦✈✐♥❣ ❆✈❡r❛❣❡ ▼♦❞❡❧ ●❡♥❡r❛❧✐③❡❞ ▲✐♥❡❛r ▼♦❞❡❧ ✭●▲❆❘▼❆✮✳ ❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ✐s t♦ ❛♥❛❧②③❡ t❤❡ q✉❛❧✐t② ♦❢ ✜t ♦❢ t❤❡ ❛❜♦✈❡ ♠♦❞❡❧s✱ ✉s✐♥❣ ❛❧❣♦r✐t❤♠s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ❘ ❧❛♥❣✉❛❣❡✳ ❆♥♦t❤❡r ♦❜❥❡❝t✐✈❡ ✐s t♦ ❝♦♠♣❛r❡ t❤❡s❡ ♠♦❞❡❧s t♦ t❤❡ ●❡♥❡r❛❧✐③❡❞ ▲✐♥❡❛r ▼♦❞❡❧ ✭●▲▼✮✱ ✇❤✐❝❤ ❛❧❧♦✇s t❤❡ ✜t ♦❢ ♥♦♥ ●❛✉ss✐❛♥ ♦❜s❡r✈❛t✐♦♥s✱ ❜✉t ❞♦❡s ♥♦t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ t✐♠❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ s✉❝❤ ❞❛t❛✳ ❆ s✐♠✉❧❛t✐♦♥ st✉❞② ✐s ❝♦♥❞✉❝t❡❞ ✐♥ ♦r❞❡r t♦ ✈❡r✐❢② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❡st✐♠❛t❡s✳ ❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s t♦ r❡❛❧ s❡r✐❡s ❛r❡ ♣❡r❢♦r♠❡❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠♣❛♥✐❡s t❤❛t ✇❡♥t ❜❛♥❦r✉♣t ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ✐♥ t❤❡ ②❡❛rs ✶✾✽✺✲✷✵✶✷✱ ❛♥❞ ♠♦♥t❤❧② ♥✉♠❜❡r ♦❢ ❝❛s❡s ♦❢ ♣♦❧✐♦ ✐♥ ❛ ❤♦s♣✐t❛❧✳ ❚❤❡ ♠♦❞❡❧s ❞❡s❝r✐❜❡ s❛t✐s❢❛❝t♦r✐❧② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s❡r✐❡s✳

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▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❙ér✐❡ t❡♠♣♦r❛❧ ❞♦ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✶✳✷ ❆✉t♦❝♦rr❡❧❛çã♦ ❡ ❆✉t♦❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ❞♦s r❡sí❞✉♦s ❞♦ ♠♦❞❡❧♦ ❛❥✉st❛❞♦

à sér✐❡ ❞❡ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✹✳✶ ❇♦①♣❧♦t ❞❛s ❡st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ❆❘✭✶✮ s❡♠ ❝♦✲ ✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s ❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦

λ = 0,5 ❡ 1,0✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ❣rá✜❝♦s ❞❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ❛♣r❡s❡♥✲

t❛♠ ❛s ❡st✐♠❛t✐✈❛s ♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0 ❡ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦

♣❛râ♠❡tr♦ φ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✹✳✷ ❇♦①♣❧♦t ❞❛s ❡st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ❆❘✭✶✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧ t❡♥❞ê♥❝✐❛✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s ❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆

❝♦♠ ♣❛râ♠❡tr♦ λ = 0,5 ❡ 1,0✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳❖s ❣rá✜❝♦s ❞❛ ♣r✐♠❡✐r❛

❧✐♥❤❛ ❛♣r❡s❡♥t❛♠ ❛s ❡st✐♠❛t✐✈❛s ♣❛r❛ ♦ ♣❛râ♠❡tr♦β0✱ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β1 ❡ ♦s ❞❛ t❡r❝❡✐r❛ ❧✐♥❤❛ ♣❛r❛φ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✹✳✸ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ▼❆✭✶✮ s❡♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ = 0,5❡1,0✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ❣rá✜❝♦s ❞❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ❛♣r❡s❡♥t❛♠ ❛s ❡st✐♠❛t✐✈❛s

(8)

✹✳✹ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ▼❆✭✶✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✸✵

✹✳✺ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ▼❆✭✶✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧✳◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ= 0,5❡1,0

r❡s♣❡❝t✐✈❛♠❡♥t❡✳❖s ❣rá✜❝♦s ❞❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ❛♣r❡s❡♥t❛♠ ❛s ❡st✐♠❛t✐✈❛s

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0 ❡ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ β1 ❡ ♦s ❞❛

t❡r❝❡✐r❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ θ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✹✳✻ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ❆❘✭✷✮ s❡♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ = 0,5❡1,0✱

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0✱ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ φ1 ❡ ♦s ❞❛

t❡r❝❡✐r❛ ♣❛r❛ φ2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✹✳✼ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ❆❘✭✷✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧✳◆♦s ●rá✲ ✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s ❡s✲

t✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦ λ = 0,5 ❡ 1,0✱

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0✱ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ β1✱ ♦s ❞❛

t❡r❝❡✐r❛ ♣❛r❛ ♣❛râ♠❡tr♦ φ1 ❡ ♦s ❞❛ q✉❛rt❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ φ2✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✹✳✽ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ▼❆✭✷✮ s❡♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ = 0,5❡1,0✱

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0✱ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ θ1 ❡ ♦s ❞❛

(9)

✹✳✾ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ▼❆✭✷✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧✳◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ= 0,5❡1,0

t❡r❝❡✐r❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ θ1 ❡ ♦s ❞❛ q✉❛rt❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ θ2 ✳ ✳ ✸✻

✹✳✶✵ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ❆❘✭✶✮ s❡♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ = 0,5❡1,0✱

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0 ❡ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦φ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✹✳✶✶ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ❆❘✭✶✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ = 0,5❡1,0✱

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0✱ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ β1 ❡ ♦s ❞❛

t❡r❝❡✐r❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ φ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✹✳✶✷ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ▼❆✭✶✮ s❡♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ= 0,5❡1,0

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0 ❡ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦θ✳✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✹✳✶✸ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ▼❆✭✶✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧✳◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ= 0,5❡1,0

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0✱ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ β1 ❡ ♦s ❞❛

(10)

✹✳✶✹ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ❆❘✭✷✮ s❡♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ= 0,5❡1,0

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0✱ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ φ1 ❡ ♦s ❞❛

t❡r❝❡✐r❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ φ2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✹✳✶✺ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ❆❘✭✷✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ = 0,5❡1,0✱

t❡r❝❡✐r❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ φ1 ❡ ♦s ❞❛ q✉❛rt❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ φ2✳ ✳ ✹✺

✹✳✶✻ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ▼❆✭✷✮ s❡♠ ❝♦✈❛r✐á✈❡❧✳ ◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ = 0,5❡1,0✱

♣❛r❛ ♦ ♣❛râ♠❡tr♦ β0✱ ♦s ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ θ1 ❡ ♦s ❞❛

t❡r❝❡✐r❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦ θ2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✹✳✶✼ ❇♦①♣❧♦t ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ▼❆✭✷✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧✳◆♦s ●rá✜❝♦s ❇♦①♣❧♦t ❛s s✐❣❧❛s ●▲❆✵✱✺ ❡ ●▲❆✶✱✵ r❡♣r❡s❡♥t❛♠ ❛s ♠é❞✐❛s ❞❛s

❡st✐♠❛t✐✈❛s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❝♦♠ ♣❛râ♠❡tr♦λ= 0,5❡1,0

t❡r❝❡✐r❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦θ1 ❡ ♦s ❞❛ q✉❛rt❛ ❧✐♥❤❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦

θ2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✺✳✶ ❆✉t♦❝♦rr❡❧❛çã♦ ❡ ❛✉t♦❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ❞❛ sér✐❡ ❞♦s ❝❛s♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡ ✺✶

✺✳✷ ❆♥á❧✐s❡ ❞♦s r❡sí❞✉♦s ♣❛r❛ ♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❆❘✭✶✮ ❛❥✉st❛❞♦ à sér✐❡ ❞❡

(11)

✺✳✸ ❆✉t♦❝♦rr❡❧♦❣r❛♠❛ ❡ ❛✉t♦❝♦rr❡❧♦❣r❛♠❛ ♣❛r❝✐❛❧ ❞♦s r❡sí❞✉♦s ❞♦ ❛❥✉st❡ à

sér✐❡ ❞❡ ♥ú♠❡r♦ ♠❡♥s❛❧ ❞❡ ❝❛s♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✺✳✹ ❙ér✐❡ t❡♠♣♦r❛❧ ❞♦ ♥ú♠❡r♦ ♠❡♥s❛❧ ❞❡ ❝❛s♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡✳ ❆ ❧✐♥❤❛ ❝♦♥tí✲ ♥✉❛ r❡♣r❡s❡♥t❛ ❛ sér✐❡ r❡❛❧ ❡ ❛ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❆❘✭✶✮

❛❥✉st❛❞♦ à sér✐❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✺✳✺ ❙ér✐❡ t❡♠♣♦r❛❧ ❞♦ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❞❡ ❢❛❧ê♥❝✐❛s ❞❡ ❡♠♣r❡s❛s ◆♦rt❡ ❆♠❡✲

r✐❝❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✺✳✻ ❋✉♥çã♦ ❞❡ ❛✉t♦❝♦rr❡❧❛çã♦ ❡ ❛✉t♦❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ❞♦ ♥ú♠❡r♦ ❞❡ ❡♠♣r❡s❛s

q✉❡ ❞❡❝r❡t❛r❛♠ ❢❛❧ê♥❝✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✺✳✼ ❘❡sí❞✉♦s ●▲❆❘▼❆ ❆❘✭✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✺✳✽ ❋❆❈ ❡ ❋❆❈P ●▲❆❘▼❆ ❆❘✭✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✺✳✾ ❙ér✐❡ t❡♠♣♦r❛❧ ❞♦ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❞❡ ❢❛❧ê♥❝✐❛✳ ❆ ❧✐♥❤❛ ❝♦♥tí♥✉❛ r❡♣r❡✲ s❡♥t❛ ❛ sér✐❡ r❡❛❧ ❡ ❛ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❆❘✭✸✮ ❛❥✉st❛❞♦

à sér✐❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸

(12)

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✶✳✶ ❆❥✉st❡ ▼▲● ♣❛r❛ ♦s ❞❛❞♦s ❞❡ P♦❧✐♦♠❡❧✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✺✳✶ ❆■❈ ❞♦s ♠♦❞❡❧♦s ❛❥✉st❛❞♦s ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❝❛s♦s ❞❡ ♣♦❧✐♦♠✐❡❧✐t❡ ✳ ✳ ✳ ✳ ✺✷

✺✳✷ ❆❥✉st❡ ●▲❆❘▼❆ ❆❘✭✶✮ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ P♦❧✐♦♠❡❧✐t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✺✳✸ Pr❡✈✐sã♦ ♣❛r❛ ❞❛❞♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡ ✉t✐❧✐③❛♥❞♦ ♦s ♠♦❞❡❧♦s ●▲❆❘▼❆

❆❘✭✶✮✱ ●❆❘▼❆ ❡ ▼▲● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

✺✳✹ ❆■❈ ❞♦s ♠♦❞❡❧♦s ❛❥✉st❛❞♦s à sér✐❡ ❞♦ ♥ú♠❡r♦ ❞❡ ❡♠♣r❡s❛s ◆♦rt❡ ❆♠❡r✐✲

❝❛♥❛s q✉❡ ❞❡❝r❡t❛r❛♠ ❢❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✺✳✺ ❆❥✉st❡ ●▲❆❘▼❆ ❆❘✭✸✮ ♣❛r❛ ♦ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❞❡ ❢❛❧ê♥❝✐❛s ❞❡ ❡♠♣r❡s❛s

◆♦rt❡ ❆♠❡r✐❝❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

✺✳✻ Pr❡✈✐sã♦ ♣❛r❛ ❞❛❞♦s ❞❡ ❢❛❧ê♥❝✐❛ ✉t✐❧✐③❛♥❞♦ ♦s ♠♦❞❡❧♦s ●▲❆❘▼❆ ❆❘✭✸✮

❡ ●❆❘▼❆ ❆❘✭✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

✼✳✶ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ❆❘✭✶✮ s❡♠ ❝♦✈❛r✐á✈❡❧ ✳ ✳ ✳ ✻✽

✼✳✷ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●❆❘▼❆ ❆❘✭✶✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧ t❡♥✲

❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾

✼✳✸ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●❆❘▼❆ ▼❆✭✶✮ s❡♠ ❝♦✈❛r✐á✈❡❧ ✼✵

✼✳✹ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●❆❘▼❆ ▼❆✭✶✮ ❝♦♠ ❝♦✈❛✲

r✐á✈❡❧ t❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

(13)

✼✳✻ ▼♦♥t❡ ❈❛r❧♦ ♣❛r❛ ●❆❘▼❆ ❆❘✭✷✮ ❝♦♠ ❝♦✈❛r✐á✈❡❧ t❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

✼✳✼ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●❆❘▼❆ ▼❆✭✷✮ s❡♠ ❝♦✈❛r✐á✈❡❧ ✼✹

✼✳✽ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●❆❘▼❆ ▼❆✭✷✮ ❝♦♠ ❝♦✈❛✲

r✐á✈❡❧ t❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

✼✳✾ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❆❘✭✶✮ s❡♠ ❝♦✈❛✲

r✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻

✼✳✶✵ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❆❘✭✶✮ ❝♦♠ ❝♦✲

✈❛r✐á✈❡❧ t❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

✼✳✶✶ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ▼❆✭✶✮ s❡♠ ❝♦✲

✈❛r✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

✼✳✶✷ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ▼❆✭✶✮ ❝♦✈❛r✐á✈❡❧

t❡♥❞ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾

✼✳✶✸ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ❆❘✭✷✮ s❡♠ ❝♦✈❛✲

r✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

✼✳✶✹ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ♣❛r❛ ●▲❆❘▼❆ ❆❘✭✷✮ ❝♦♠

❝♦✈❛r✐á✈❡❧ t❡♥❞ê♥❝✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

✼✳✶✺ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ▼❆✷ s❡♠ ❝♦✈❛r✐á✈❡❧ ✳ ✳ ✳ ✽✷

✼✳✶✻ ❊st✐♠❛t✐✈❛s ♣❛r❛ sér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ●▲❆❘▼❆ ▼❆✷ ❝♦♠ ❝♦✈❛r✐á✈❡❧ t❡♥✲

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❙✉♠ár✐♦

▲✐st❛ ❞❡ ❋✐❣✉r❛s ✈✐

▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✐

❙✉♠ár✐♦ ①✐✐✐

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❘❡✈✐sã♦ ❞❡ ▲✐t❡r❛t✉r❛ ✻

✷✳✶ Pr♦❝❡ss♦ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✷✳✷ ❊st❛❝✐♦♥❛r✐❡❞❛❞❡ ❡ ■♥✈❡rt✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✸ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷✳✸✳✶ Pr❡✈✐sã♦ ♥♦s ♠♦❞❡❧♦s ❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✹ ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✸ ▼♦❞❡❧♦s ♣❛r❛ sér✐❡s ❞❡ ❝♦♥t❛❣❡♠ ✶✺

✸✳✶ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ●❡♥❡r❛❧✐③❛❞♦ ✭●❆❘▼❆✮ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸✳✶✳✶ ❊st✐♠❛çã♦ ♥♦s ♠♦❞❡❧♦s ●❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦s ♠♦❞❡❧♦s ●❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸✳✶✳✸ Pr❡✈✐sã♦ ♥♦ ♠♦❞❡❧♦ ●❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

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✸✳✷✳✶ ❊st✐♠❛çã♦ ♥♦s ♠♦❞❡❧♦s ●▲❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸✳✷✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦s ♠♦❞❡❧♦s ●▲❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸✳✷✳✸ Pr❡✈✐sã♦ ♥♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✹ ❊st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ✷✸

✹✳✶ ❙ér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹✳✶✳✶ ●❆❘▼❆✭✶✱✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹✳✶✳✷ ●❆❘▼❆✭✵✱✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✹✳✶✳✸ ●❆❘▼❆✭✷✱✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✹✳✶✳✹ ●❆❘▼❆✭✵✱✷✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✹✳✷ ❙ér✐❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ●▲❆❘▼❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✹✳✷✳✶ ●▲❆❘▼❆✭✶✱✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✹✳✷✳✷ ●▲❆❘▼❆✭✵✱✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✹✳✷✳✸ ●▲❆❘▼❆✭✷✱✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹✳✷✳✹ ●▲❆❘▼❆✭✵✱✷✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✺ ❆♣❧✐❝❛çã♦ ❛ ❞❛❞♦s r❡❛✐s ✺✵

✺✳✶ P♦❧✐♦♠❡❧✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✺✳✷ ❋❛❧ê♥❝✐❛ ❞❡ ❡♠♣r❡s❛s ◆♦rt❡ ❆♠❡r✐❝❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✻ ❈♦♥❝❧✉sã♦ ❡ ❝♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✻✺

✼ ❆♥❡①♦ ✻✼

✼✳✶ ❆♥❡①♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✼✳✷ ❆♥❡①♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❙ér✐❡s t❡♠♣♦r❛✐s ♣❛r❛ ❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠ sã♦ r❡❣✐str♦s ❞❛ ❢r❡q✉ê♥❝✐❛ r❡❧❛t✐✈❛ ❞❛ ♦❝♦r✲ rê♥❝✐❛ ❞❡ ❞❡t❡r♠✐♥❛❞♦s ❡✈❡♥t♦s ❡♠ s✉❝❡ss✐✈♦s ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦✱ ❡ t❡♠ ❝♦♠♦ ❝❛r❛❝t❡✲ ríst✐❝❛ ✐♠♣♦rt❛♥t❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ❛s ♦❜s❡r✈❛çõ❡s✳ ❊❧❛s s✉r❣❡♠ ♥❛s ♠❛✐s ✈❛r✐❛❞❛s ár❡❛s ❞❡ ❛♣❧✐❝❛çã♦ t❛✐s ❝♦♠♦ ✐♥❞ústr✐❛✱ ♠❡❞✐❝✐♥❛✱ ❡❝♦♥♦♠✐❛ ❡ ❢❡♥ô♠❡♥♦s ♠❡t❡♦r♦❧ó❣✐❝♦s✳

◆❛ ❧✐t❡r❛t✉r❛ ❛♣❛r❡❝❡♠ ❡♠ ❞✐✈❡rs♦s tr❛❜❛❧❤♦s✱ t❛✐s ❝♦♠♦ ♥♦ ❡st✉❞♦ ❞♦s ❝❛s♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡ ♥♦s ❊st❛❞♦s ❯♥✐❞♦s ❝♦♠ ❩❡❣❡r ✭✶✾✽✽✮✱ t❛♠❜é♠ ❝♦♠ ♦s ❝❛s♦s ❞✐ár✐♦s ❞❡ ❛s♠❛ ❡♠ ✉♠ ❤♦s♣✐t❛❧ ❞❡ ❙②❞♥❡②✱ ♥♦ ❡st✉❞♦ ❞❡ ❉❛✈✐s ✫ ❲❛♥❣ ✭✷✵✵✵✮ ❡ ♦ ♥ú♠❡r♦ ❞❡ ❞❡❧✐t♦s r❡❣✐str❛❞♦s ❡♠ ❱✐tór✐❛ ❝♦♠ ❙✐❧✈❛ ✫ ❘❡✐s❡♥ ✭✷✵✶✶✮✳

❙❡ ♦ ✐♥t❡r❡ss❡ é ❛❥✉st❛r ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❣r❡ssã♦ ♣❛r❛ ✉♠❛ sér✐❡ t❡♠♣♦r❛❧ ❞❡ ❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠ é ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞✐s❝r❡t❛ ♣❛r❛ ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s✱ ❝♦♠♦ ❛ P♦✐ss♦♥ ♦✉ ❛ ❇✐♥♦♠✐❛❧ ◆❡❣❛t✐✈❛✱ ❥á q✉❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ é ✉♠❛ ❝♦♥t❛❣❡♠✳

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✷

t❡♠ s✐❞♦ ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ❡♠ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ❝♦♠ sér✐❡s ❞❡ ❝♦♥t❛❣❡♥s ❝♦♠♦ ♥♦ ❡st✉❞♦ ❞❡ ❚❛♦ ❡t ❛❧ ✭✷✵✶✶✮ à ❝❡r❝❛ ❞♦s ❡❢❡✐t♦s ❛❣✉❞♦s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ ♠♦♥ó①✐❞♦ ❞❡ ❝❛r❜♦♥♦ ♥❛ ❈❤✐♥❛✱ ❡ t❛♠❜é♠ ♣♦r ❱❡s❡❧ý ❡t ❛❧ ✭✷✵✵✾✮ s♦❜r❡ ❛ ♣♦❧✉✐çã♦ ❞♦ ❛r ♣♦r ♣❛rtí❝✉❧❛s ❡♠ s✉s♣❡♥sã♦ ♥❛ ❘❡♣ú❜❧✐❝❛ ❈❤❡❝❛✳

P❛r❛ ♠♦t✐✈❛r ❛ ✉t✐❧✐③❛çã♦ ❞♦s ♠♦❞❡❧♦s q✉❡ s❡rã♦ ❛❜♦r❞❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✱ ✐♥✐❝✐❛❧✲ ♠❡♥t❡ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞♦s ❞❛❞♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡✱ ✐♥tr♦❞✉③✐❞♦s ♥♦ ❡st✉❞♦ ❞❡ ❩❡❣❡r ✭✶✾✽✽✮✱ ✉t✐❧✐③❛♥❞♦✲s❡ ♦ ▼▲●✳

❖s ❞❛❞♦s sã♦ r❡❢❡r❡♥t❡s ❛♦ ♥ú♠❡r♦ ♠❡♥s❛❧ ❞❡ ❝❛s♦s ❞❡ ♣♦❧✐♦♠✐❡❧✐t❡ ♥♦s ❊❯❆ ♥♦s ❛♥♦s ❞❡ ✶✾✼✵ ❛ ✶✾✽✸ ❝♦♥❢♦r♠❡ r❡❧❛t❛❞♦ ♣❡❧♦ ❈❡♥tr♦ ❞❡ ❈♦♥tr♦❧❡ ❞❡ ❉♦❡♥ç❛s✳ ❙ã♦ ✶✻✽ ❞❛❞♦s

❝♦♠ ✉♠❛ ✈❛r✐❛çã♦ ❞❡ ✵ ❛ ✶✹ ❝❛s♦s✱ ✈❡r ❋✐❣✉r❛✶✳✶✳ ➱ ♣♦ssí✈❡❧ ♦❜s❡r✈❛r q✉❡ ♦s ❞❛❞♦s sã♦

✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s✱ ❥á q✉❡ tr❛t❛✲s❡ ❞❡ ✉♠❛ sér✐❡ ❞❡ ❝♦♥t❛❣❡♠✱ ♣♦r ✐ss♦ é ♥❡❝❡ssár✐❛ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞✐s❝r❡t❛✱ ❝♦♠♦ ❛ P♦✐ss♦♥✳

Tempo

Casos de P

oliomelite

0 50 100 150

0

2

4

6

8

10

12

14

❋✐❣✉r❛ ✶✳✶✿ ❙ér✐❡ t❡♠♣♦r❛❧ ❞♦ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡

❙❡❣✉✐♥❞♦ ❛ s✉❣❡stã♦ ❞♦ ❛rt✐❣♦ ❞❡ ❩❡❣❡r ✭✶✾✽✽✮✱ ♣❛r❛ ❛ ♠♦❞❡❧❛❣❡♠ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s q✉❛tr♦ ❝♦✈❛r✐á✈❡✐s ♣❛r❛ ♠♦❞❡❧❛r ❛ s❛③♦♥❛❧✐❞❛❞❡ ✭s❡♥♦ ❡ ❝♦ss❡♥♦✱ ❛♥✉❛❧ ❡ s❡♠❡str❛❧✮ ❡ ✉♠ ❝♦♠♣♦♥❡♥t❡ ❞❡ t❡♥❞ê♥❝✐❛ ❧✐♥❡❛r✳ ❖ ▼♦❞❡❧♦ ▲✐♥❡❛r ●❡♥❡r❛❧✐③❛❞♦ ❛❥✉st❛❞♦ é ❛♣r❡s❡♥t❛❞♦

(18)

✸

❚❛❜❡❧❛ ✶✳✶✿ ❆❥✉st❡ ▼▲● ♣❛r❛ ♦s ❞❛❞♦s ❞❡ P♦❧✐♦♠❡❧✐t❡

❊st✐♠❛t✐✈❛ ❊rr♦ ♣❛❞rã♦ ♣✲✈❛❧♦r

■♥t❡r❝❡♣t♦ ✵✱✷✵✼ ✵✱✵✼✺ ✵✱✵✵✻

❚❡♥❞ê♥❝✐❛ ✲✹✱✼✾✾ ✶✱✹✵✷ ✵✱✵✵✵

❈♦ss❡♥♦ ❛♥✉❛❧ ✲✵✱✶✹✽ ✵✱✵✾✼ ✵✱✶✷✻

❙❡♥♦ ❛♥✉❛❧ ✲✵✱✺✸✶ ✵✱✶✵✾ ✵✱✵✵✵

❈♦ss❡♥♦ s❡♠❡str❛❧ ✵✱✶✻✾ ✵✱✵✾✾ ✵✱✵✽✼

❙❡♥♦ s❡♠❡str❛❧ ✲✵✱✹✸✷ ✵✱✶✵✵ ✵✱✵✵✵

❖ ❣rá✜❝♦ ❞❡ r❡sí❞✉♦s é ❛♣r❡s❡♥t❛❞♦ ♥♦ ❆◆❊❳❖ ✼✳✶ ❡ é ♣♦ssí✈❡❧ ♦❜s❡r✈❛r ❛❧❣✉♥s

♦✉t❧✐❡rs✱ ♠❛s ❛ s✉♣♦s✐çã♦ ❞❡ ❤♦♠♦❝❡❞❛st✐❝✐❞❛❞❡ ♣❛r❡❝❡ ❡st❛r s❛t✐s❢❡✐t❛✱ ❥á q✉❡ ♦s r❡sí❞✉♦s ❝♦♠♣♦rt❛♠✲s❡ ❛❧❡❛t♦r✐❛♠❡♥t❡ ❡♠ t♦r♥♦ ❞❡ ③❡r♦✳

❆❧é♠ ❞❛ s✉♣♦s✐çã♦ ❞❡ ❤♦♠♦❝❡❞❛st✐❝✐❞❛❞❡ é ♥❡❝❡ssár✐♦ ✐♥✈❡st✐❣❛r t❛♠❜é♠ s❡ ♦s r❡sí✲

❞✉♦s sã♦ ♥ã♦ ❛✉t♦❝♦rr❡❧❛❝✐♦♥❛❞♦s✳ ◆❛ ❋✐❣✉r❛✶✳✷ s❡❣✉❡♠ ♦s ❣rá✜❝♦s ❞❡ ❛✉t♦❝♦rr❡❧❛çã♦

✭❋❆❈✮ ❡ ❛✉t♦❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ✭❋❆❈P✮ ❞♦s r❡sí❞✉♦s ❞♦ ▼▲● ❛❥✉st❛❞♦✳

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

Lag

A

CF

Series res

5 10 15 20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

Lag

P

ar

tial A

CF

Series res

❋✐❣✉r❛ ✶✳✷✿ ❆✉t♦❝♦rr❡❧❛çã♦ ❡ ❆✉t♦❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ❞♦s r❡sí❞✉♦s ❞♦ ♠♦❞❡❧♦ ❛❥✉st❛❞♦ à sér✐❡ ❞❡ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❞❡ ♣♦❧✐♦♠❡❧✐t❡

(19)

✹

❡st❡ ♠♦❞❡❧♦ ♥ã♦ ❢♦✐ ❝❛♣❛③ ❞❡ ❡❧✐♠✐♥❛r ❛ ❛✉t♦❝♦rr❡❧❛çã♦ ❡①✐st❡♥t❡ ♥♦s ❞❛❞♦s✱ ❥á q✉❡ ♥ã♦ ❧❡✈❛ ❡♠ ❝♦♥t❛ ❛ ❞❡♣❡♥❞ê♥❝✐❛ t❡♠♣♦r❛❧ ❞♦s ♠❡s♠♦s✳

❯♠❛ ♦♣çã♦ ♣❛r❛ ♠❡❧❤♦r❛r ♦ ❛❥✉st❡✱ ✈✐st♦ q✉❡ ♦ ▼▲● ♥ã♦ é ❝❛♣❛③ ❞❡ ❝❛♣t✉r❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ♥♦ t❡♠♣♦ ❞❡st❡s ❞❛❞♦s✱ s❡r✐❛ ♦ ✉s♦ ❞❡ ♠♦❞❡❧♦s ♣❛r❛ sér✐❡s t❡♠♣♦r❛✐s✳ ❯♠ ❞♦s ♠♦❞❡❧♦s ♠❛✐s ✉t✐❧✐③❛❞♦s é ♦ ♠♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ✭❆❘▼❆✮✱ ♣r♦♣♦st♦ ♣♦r ❇♦① ❡ ❏❡♥❦✐♥s ✭✶✾✼✻✮✳ ❊♥tr❡t❛♥t♦✱ ❡st❡ ♠♦❞❡❧♦ ✉t✐❧✐③❛ ❛ s✉♣♦s✐çã♦ ❞❡ q✉❡ ❛ sér✐❡ t❡♠ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❥✉♥t❛ ●❛✉ss✐❛♥❛ ❡ ♥♦ ❡st✉❞♦ ❞❡ sér✐❡s ❞❡ ❝♦♥t❛❣❡♠✱ ♣❛r❛ ✉♠ ❜♦♠ ❛❥✉st❡ ❞♦s ❞❛❞♦s✱ é ♥❡❝❡ssár✐♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞✐s❝r❡t❛✳

▼❛✐s r❡❝❡♥t❡♠❡♥t❡✱ ❢♦r❛♠ ♣r♦♣♦st♦s ♥♦✈♦s ♣r♦❝❡❞✐♠❡♥t♦s ♣❛r❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ sér✐❡s t❡♠♣♦r❛✐s✱ q✉❡ r❡❧❛❝✐♦♥❛♠ ♦s ♠♦❞❡❧♦s ▼▲● ❡ ❆❘▼❆✳ ◆❡st❛s ♣r♦♣♦st❛s é ❛❝r❡s❝✐❞♦ ❛♦ ▼▲● ✉♠❛ ❡str✉t✉r❛ ❛✉t♦r❡❣r❡ss✐✈❛ ♠é❞✐❛ ♠ó✈❡❧✱ ❛✜♠ ❞❡ ♠♦❞❡❧❛r ❞❛❞♦s q✉❡ t❡♥❤❛♠ ❞✐str✐❜✉✐çã♦ ♣❡rt❡♥❝❡♥t❡ à ❢❛♠í❧✐❛ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❛♣r❡s❡♥t❡♠ ❞❡♣❡♥❞ê♥❝✐❛ t❡♠♣♦r❛❧✳ ❉❛✲ ✈✐s ❡t ❛❧ ✭✷✵✵✸✮ ♣r♦♣✉s❡r❛♠ ♦ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ▲✐♥❡❛r ●❡♥❡r❛❧✐③❛❞♦ ✭●▲❆❘▼❆✮ ❡ ❇❡♥❥❛♠✐♥ ❡t ❛❧ ✭✷✵✵✸✮ ♣r♦♣✉s❡r❛♠ ♦ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ●❡♥❡r❛❧✐③❛❞♦ ✭●❆❘▼❆✮✱ q✉❡ sã♦ ❡①t❡♥sõ❡s ❞♦ ▼▲●✳

❆ ✈❛♥t❛❣❡♠ ❡♠ ❛❝r❡s❝❡♥t❛r t❡r♠♦s ❛✉t♦r❡❣r❡ss✐✈♦ ❡ ♠é❞✐❛ ♠ó✈❡❧ à ❡str✉t✉r❛ ❞♦ ♠♦✲ ❞❡❧♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦ é ♣♦❞❡r ♠♦❞❡❧❛r sér✐❡s t❡♠♣♦r❛✐s✱ ❥á q✉❡ ❡st❡s ❝♦♠♣♦♥❡♥t❡s ❝❛♣t✉r❛♠ ❛ ❞❡♣❡♥❞ê♥❝✐❛ t❡♠♣♦r❛❧ ❞♦s ❞❛❞♦s✳ ❆❧é♠ ❞✐st♦✱ ❤á ♦ ❣❛♥❤♦ ❡♠ s❡ ♠♦❞❡❧❛r sér✐❡s ♥ã♦ ●❛✉ss✐❛♥❛s✱ ❝♦♠ ❛ ❡str✉t✉r❛ ♠❛✐s ✢❡①í✈❡❧ ❞♦ ▼▲●✳

❊①✐st❡♠✱ ♥❛ ❧✐t❡r❛t✉r❛✱ ♦✉tr♦s ♣r♦❝❡❞✐♠❡♥t♦s ♣❛r❛ ♠♦❞❡❧❛r sér✐❡s ❝♦♠ ❞✐str✐❜✉✐çã♦ ◆ã♦✲●❛✉ss✐❛♥❛✱ ❝♦♠♦ ♦ ♠♦❞❡❧♦ ■◆❆❘✱ ♣r♦♣♦st♦ ♣♦r ❆❧✲❖s❤ ✫ ❆❧③❛✐❞ ✭✶✾✽✼✮✱ ♦♥❞❡ é ✉♠ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛❣❡♠ q✉❡ ❝♦♥s✐❞❡r❛ ✈❛r✐á✈❡✐s ✐♥t❡✐r❛s ♥ã♦ ♥❡❣❛t✐✈❛s✱ ♦✉ ♦s ♠♦❞❡✲ ❧♦s ❞✐♥â♠✐❝♦s ❣❡♥❡r❛❧✐③❛❞♦s✱ ❛❜♦r❞❛❞♦s ♣♦r ✈ár✐♦s ❛✉t♦r❡s✱ t❛✐s ❝♦♠♦ ●❛♠❡r♠❛♥ ❡t ❛❧ ✭✷✵✶✸✮✱ ❍❛r✈❡② ✫ ❋❡r♥❛♥❞❡s ✭✶✾✽✾✮ ❡ ❲❡st ❡t ❛❧ ✭✶✾✽✺✮✳

(20)

●❆❘▼❆ ❡ ●▲❆❘▼❆✱ ❡ ❝♦♠♣❛r❛r ❡st❡s r❡s✉❧t❛❞♦s à ♠♦❞❡❧❛❣❡♠ ♦❜t✐❞❛ ♣❡❧♦ ▼▲●✳

❆s ♣ró①✐♠❛s s❡çõ❡s ❡stã♦ ❡str✉t✉r❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♥❛ ❙❡çã♦ ✷ é ❢❡✐t❛ ✉♠❛

r❡✈✐sã♦ ❞♦s ♠♦❞❡❧♦s ❜ás✐❝♦s✿ ❆❘▼❆ ❡ ▼▲●✳ ◆❛ ❙❡çã♦✸é ❢❡✐t❛ ✉♠❛ r❡✈✐sã♦ ❞♦s ♠♦❞❡❧♦s

●▲❆❘▼❆ ❡ ●❆❘▼❆✳ ◆❛ ❙❡çã♦ ✹ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ❡st✉❞♦ ❞❡ s✐♠✉❧❛çã♦ ❛ ✜♠ ❞❡

✈❡r✐✜❝❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s ❡st✐♠❛❞♦r❡s ❞♦s ♣❛râ♠❡tr♦s✱ ♥❛ ❙❡çã♦ ✺ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s

(21)

❈❛♣ít✉❧♦ ✷

❘❡✈✐sã♦ ❞❡ ▲✐t❡r❛t✉r❛

◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ♣❛r❛ ❛♥❛❧✐s❛r ✉♠❛ sér✐❡ t❡♠♣♦r❛❧✱ ❛ss✐♠ ❝♦♠♦ ✉♠❛ ❞❡s❝r✐çã♦ r❡s✉♠✐❞❛ ❞♦ ♠♦❞❡❧♦ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧✭❆❘▼❆✮ ❡ ▼♦❞❡❧♦ ▲✐♥❡❛r ●❡♥❡r❛❧✐③❛❞♦ ✭▼▲●✮✳

✷✳✶ Pr♦❝❡ss♦ ▲✐♥❡❛r

❉❡ ❛❝♦r❞♦ ❝♦♠ ❇♦① ✫ ❏❡♥❦✐♥s ✭✶✾✼✻✮ ♦ ♣r♦❝❡ss♦ ❧✐♥❡❛r é ❜❛s❡❛❞♦ ♥❛ ✐❞é✐❛ ❞❡ q✉❡

✉♠❛ sér✐❡ t❡♠♣♦r❛❧yt ❝♦♠ s✉❝❡ss✐✈♦s ✈❛❧♦r❡s ❛❧t❛♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s ♣♦❞❡ s❡r ❣❡r❛❞❛ ❞❡

✉♠❛ sér✐❡ ❞❡ ❝❤♦q✉❡s ✐♥❞❡♣❡♥❞❡♥t❡sut✳ ❊st❡s ❝❤♦q✉❡s tê♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ✉s✉❛❧♠❡♥t❡

◆♦r♠❛❧ ❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ r✉í❞♦s ❜r❛♥❝♦s✳ ❙❡❣✉♥❞♦ ▼♦r❡tt✐♥ ✫ ❚♦❧♦✐ ✭✷✵✵✻✮

❊✭✉t) = 0,

❡

γk=E(ut,ut+k) =

    

σ2 s❡ k = 0.

0 s❡ k 6= 0

❖ r✉í❞♦ ❜r❛♥❝♦ ut é tr❛♥s❢♦r♠❛❞♦ ♥♦ ♣r♦❝❡ss♦ yt ♣❡❧♦ q✉❡ é ❝❤❛♠❛❞♦ ✜❧tr♦ ❧✐♥❡❛r✱

ψ(B)✳ ❆ ♦♣❡r❛çã♦ s✐♠♣❧❡s♠❡♥t❡ é ✉♠❛ s♦♠❛ ♣♦♥❞❡r❛❞❛ ❞❛s ♦❜s❡r✈❛çõ❡s ❛♥t❡r✐♦r❡s✱ ❞❡

(22)

✼ ✷✳✷✳ ❊st❛❝✐♦♥❛r✐❡❞❛❞❡ ❡ ■♥✈❡rt✐❜✐❧✐❞❛❞❡

yt =µ+ut+ψ1ut−1 +ψ2ut−2+...=µ+ψ(B)ut. ✭✷✳✶✮

s❡♥❞♦µ ❛ ♠é❞✐❛ ❞♦ ♣r♦❝❡ss♦ ❡ ❇ ♦ ♦♣❡r❛❞♦r ❞❡ r❡t❛r❞♦ q✉❡ r❡♣r❡s❡♥t❛ ✉♠❛ ❞❡❢❛s❛❣❡♠

❞❡ ❦ ♣❡rí♦❞♦s ❞❡ t❡♠♣♦ ❛trás✱ ❞❡✜♥✐❞♦ ♣♦r✿

Bkut=ut−k. ✭✷✳✷✮

❖ ♠♦❞❡❧♦ ❞❛❞♦ ❡♠ ✷✳✶ t❛♠❜é♠ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❛ s♦♠❛ ♣♦♥❞❡r❛❞❛ ❞♦s ✈❛❧♦r❡s

♣❛ss❛❞♦s ❞❡yt ❛❞✐❝✐♦♥❛♥❞♦ ut✱ ❡♥tã♦

yt=µ+π1yt−1+π2yt−2+...+ut. ✭✷✳✸✮

❆♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r

ut =yt−π1yt−1−π2yt−2−...−µ=π(B)yt−µ. ✭✷✳✹✮

✷✳✷ ❊st❛❝✐♦♥❛r✐❡❞❛❞❡ ❡ ■♥✈❡rt✐❜✐❧✐❞❛❞❡

❊st❛❝✐♦♥❛r✐❡❞❛❞❡

❯♠❛ ❞❛s s✉♣♦s✐çõ❡s ♠❛✐s ❢r❡q✉❡♥t❡s ♥❛ ♠♦❞❡❧❛❣❡♠ ❞❡ sér✐❡s t❡♠♣♦r❛✐s é q✉❡ ❛ sé✲ r✐❡ s❡❥❛ ❡st❛❝✐♦♥ár✐❛✱ ♥♦ q✉❛❧ ♦ ♣r♦❝❡ss♦ ♣❡r♠❛♥❡❝❡ ❡♠ ❡q✉✐❧í❜r✐♦ s♦❜r❡ ✉♠❛ ♠é❞✐❛ ❝♦♥st❛♥t❡✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❍❛r✈❡② ✭✶✾✾✸✮ ✉♠ ♣r♦❝❡ss♦ é ❡st❛❝✐♦♥ár✐♦ s❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s ♣❛r❛ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ t✿

E(yt) = µ, ✭✷✳✺✮

(23)

✽ ✷✳✸✳ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧

γk =cov[yt,yt+k] =E[(yt−µ)(yt+k−µ)]. ✭✷✳✼✮

❙✐♠✐❧❛r♠❡♥t❡ ❛ ❛✉t♦❝♦rr❡❧❛çã♦ ❞❡ ❧❛❣ ❦ é

ρk=

E[(yt−µ)(yt+k−µ)]

p

E[(yt−µ)2]E[(yt+k−µ)2]

= γk

γ0

, k = 0,±1,±2,... ✭✷✳✽✮

❯♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ yt=ψ(B)ut s❡rá ❡st❛❝✐♦♥ár✐♦ s❡

ψ(B) =P∞

k=0ψkBk✱

❝♦♥✈❡r❣❡ ♣❛r❛|B |<1✳

■♥✈❡rt✐❜✐❧✐❞❛❞❡

❉❡ ❛❝♦r❞♦ ❝♦♠ ❇♦① ✫ ❏❡♥❦✐♥s ✭✶✾✼✻✮ s❡ ❡①✐st❡ ♦ ✐♥t❡r❡ss❡ ❡♠ ❢❛③❡r ♣r❡✈✐sõ❡s ❝♦♠ ✉♠ ♠♦❞❡❧♦ é ♥❡❝❡ssár✐♦ q✉❡ ❡❧❡ s❡❥❛ ✐♥✈❡rtí✈❡❧✳

❯♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ yt s❡rá ✐♥✈❡rtí✈❡❧ s❡

π(B) = P∞

j=0πjBj✱

❝♦♥✈❡r❣❡ ♣❛r❛|B |<1✳

✷✳✸ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧

❖ ♠♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ✭❆❘▼❆✮ ♣r♦♣♦st♦ ♣♦r ❇♦① ✫ ❏❡♥❦✐♥s✭✶✾✼✻✮ é ❞❡✜♥✐❞♦ ♣♦r

˜

yt=φ1y˜t−1+...+φpy˜t−p+ut−θ1ut−1 −...−θqut−p, t= 1,...,n, ✭✷✳✾✮

(24)

✾ ✷✳✸✳ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧

φ(B)˜yt =θ(B)ut, ✭✷✳✶✵✮

♦♥❞❡

˜

yt=yt−µ✱ φ(B) = (1−φB−...−φpBp)❡ θ(B) = (1−θB−...−θqBq)✳

❉❡ ✷✳✶✵ t❡♠✲s❡

˜

yt =ψ(B)ut =

θ(B)

φ(B)ut. ✭✷✳✶✶✮

❖ ♠♦❞❡❧♦ ❡♠ ✷✳✾ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ❞❡ ♦r❞❡♠ ♣✱q✱ ♦✉

❆❘▼❆ ✭♣✱q✮✳

❆ ♠❡t♦❞♦❧♦❣✐❛ ❇♦① ✫ ❏❡♥❦✐♥s ❡stá ❢✉♥❞❛♠❡♥t❛❞❛ ♥❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❛❞❡q✉❛❞♦ ♣❛r❛ ❛ sér✐❡✱ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s✱ ✈❡r✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❡ ♣r❡✈✐sã♦ ❞❡ ✈❛❧♦r❡s ❢✉t✉r♦s ❞❛ sér✐❡✳

❆ ✐❞❡♥t✐✜❝❛çã♦ ❞♦ ♠♦❞❡❧♦ ❆❘▼❆ ✭♣✱q✮ s❡ ❢❛③ ❛tr❛✈és ❞❛s ❢✉♥çõ❡s ❞❡ ❛✉t♦❝♦rr❡❧❛çã♦

✭❋❆❈✮✱ ❞❛❞❛ ❡♠ ✷✳✽✱ ❡ ❛✉t♦❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ✭❋❆❈P✮✱ q✉❡ é ❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❞✉❛s

♦❜s❡r✈❛çõ❡s s❡r✐❛✐s✱ ❡❧✐♠✐♥❛♥❞♦ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦s t❡r♠♦s ✐♥t❡r♠❡❞✐ár✐♦s✳

P❛r❛ ✐❞❡♥t✐✜❝❛r ♦s ✈❛❧♦r❡s ❞❡ ♣ ❡ q é ❛♥❛❧✐s❛❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ❛✉t♦❝♦rr❡❧❛çõ❡s ❡ ❞❛s ❛✉t♦❝♦rr❡❧❛çõ❡s ♣❛r❝✐❛✐s ❛♠♦str❛✐s ❡ ❝♦♠♣❛r❛❞♦ ❝♦♠ ❛ ❢✉♥çã♦ ❞❡ ❛✉t♦❝♦rr❡❧❛çã♦ ❡ ❛✉t♦❝♦rr❡❧❛çã♦ ♣❛r❝✐❛❧ ❞♦s ♠♦❞❡❧♦s t❡ór✐❝♦s ❆❘▼❆✭♣✱q✮✳ ▼❛✐♦r❡s ❞❡t❛❧❤❡s ♣♦❞❡♠ s❡r

✈✐st♦s ❡♠ ❇♦①& ❏❡♥❦✐♥s ✭✶✾✼✻✮✳

❆♣ós ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s ♠♦❞❡❧♦s✱ ❞❡✈❡✲s❡ ❡st✐♠❛r ♦s ♣❛râ♠❡tr♦s✳ ◆❡st❡ ❝❛s♦ ♦s

♣❛râ♠❡tr♦s ❛ s❡r❡♠ ❡st✐♠❛❞♦s sã♦φ = (φ1,φ2,...,φp) ✱ θ = (θ1,θ2,...,θq) ❡ σ2 =V ar(ut)✳

❯♠ ❞♦s ♠ét♦❞♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❡st✐♠❛r ♦s ♣❛râ♠❡tr♦s é ♦ ♠ét♦❞♦ ❞❛ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳

P❛r❛ ♦ ♠♦❞❡❧♦ ❆❘▼❆✭♣✱q✮✱ ♦s r❡sí❞✉♦s ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦

(25)

✶✵ ✷✳✸✳ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧

❚❡♠✲s❡ q✉❡✱ s♦❜ ❛ s✉♣♦s✐çã♦ ❞❡ ♥♦r♠❛❧✐❞❛❞❡ ❞♦s ut✱ ❛ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❝♦♥❥✉♥t❛

❞❡u1,u2,...,un é

L(φ,θ,σ2) = f(u1,...,un) = (2π)

−n

2 (σ)−nexp (− n

X

t=1

u2_t

2σ2). ✭✷✳✶✷✮

❖s ❡st✐♠❛❞♦r❡s sã♦ ♦❜t✐❞♦s ♠❛①✐♠✐③❛♥❞♦ ❛ ❢✉♥çã♦ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❛❝✐♠❛✱ ♦ q✉❡ é ❢❡✐t♦ ❛tr❛✈és ❞❡ ♣r♦❝❡❞✐♠❡♥t♦s ♥✉♠ér✐❝♦s✳

❆ ✈❡r✐✜❝❛çã♦ ❞❛ ❛❞❡q✉❛çã♦ ❞♦s ♠♦❞❡❧♦s ❡st✐♠❛❞♦s ♣♦❞❡ s❡r ❢❡✐t❛ ♣♦r ❛♥á❧✐s❡ ❞❡ r❡sí✲

❞✉♦s✳ ❆ s✉♣♦s✐çã♦ ✐♥✐❝✐❛❧ ❞❡ ut∼ N(0,σ2) ❡ q✉❡ut✱ t❂✶✱✷✱✳✳✳✱♥ sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡✈❡

s❡r s❛t✐s❢❡✐t❛✳

❚❛♠❜é♠ ♣♦❞❡✲s❡ ✉t✐❧✐③❛r ♦ ❈r✐tér✐♦ ❞❡ ■♥❢♦r♠❛çã♦ ❞❡ ❆❦❛✐❦❡ ✭❆■❈✮ ♣❛r❛ ✈❡r✐✜❝❛r ❛ ❛❞❡q✉❛çã♦ ❞♦ ❛❥✉st❡ ❞♦ ♠♦❞❡❧♦✳ ❖ ❆■❈ é ❞❛❞♦ ♣♦r

AIC =−2lnL+ 2m

♦♥❞❡ ♠ é ♦ ♥ú♠❡r♦ ❞❡ ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❡ ▲ é ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❞♦ ♠♦❞❡❧♦✳

✷✳✸✳✶ Pr❡✈✐sã♦ ♥♦s ♠♦❞❡❧♦s ❆❘▼❆

❆♣ós ✐❞❡♥t✐✜çã♦ ❞♦ ♠♦❞❡❧♦ ❡ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s é ♣♦ssí✈❡❧ ✉t✐❧✐③❛r ♦ ♠♦❞❡❧♦ ♣❛r❛ ❢❛③❡r ♣r❡✈✐sõ❡s ♣❛r❛ ✈❛❧♦r❡s ❢✉t✉r♦s ❞❛ sér✐❡ t❡♠♣♦r❛❧✳

❙❡❥❛ yˆt(l) ♦ ✈❛❧♦r ♣r❡✈✐st♦ ♥❛ ♦r✐❣❡♠ t ♣❛r❛ ✉♠ ❤♦r✐③♦♥t❡ ❞❡ l ♣❡rí♦❞♦s ❞❡ t❡♠♣♦

❢✉t✉r♦s✳

❖ ♠♦❞❡❧♦ ❆❘▼❆ ❞❛❞♦ ❡♠ ✷✳✾ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥♦ t❡♠♣♦ t+l ❝♦♠♦

yt+l=φ1yt+l−1 +...+φpyt+l−p+ut+l−θ1ut+l−1−...−θqut+l−q, ✭✷✳✶✸✮

(26)

✶✶ ✷✳✹✳ ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s

yt+l = t+l

X

j=−∞

ψt+l−juj = ∞

X

j=0

ψjut+l−j, ✭✷✳✶✹✮

♦♥❞❡ψ0❂✶

❆ ♠❡❧❤♦r ♣r❡✈✐sã♦ ♣❛r❛ ♦ ✈❛❧♦r ❢✉t✉r♦ yt+l✱ ♥♦ s❡♥t✐❞♦ ❞❡ ❛♣r❡s❡♥t❛r ❡rr♦ q✉❛❞rá✲

t✐❝♦ ♠é❞✐♦ ✭❊◗▼✮ ♠í♥✐♠♦✱ é ❝❛❧❝✉❧❛❞❛ ❝♦♠♦ ❛ ❡s♣❡r❛♥ç❛ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡ yt+l ❞❛❞❛ ❛s

♦❜s❡r✈❛çõ❡s ♣❛ss❛❞❛s ❞❛ sér✐❡✱ ♦✉ s❡❥❛

ˆ

yt(l) =E[yt+l|yt,yt−1,...] =Et[yt+l]. ✭✷✳✶✺✮

❚❛♠❜é♠ é ♣♦ssí✈❡❧ r❡❡s❝r❡✈❡r ❛ ♣r❡✈✐sã♦ yˆt(l) ❝♦♠♦

ˆ

yt(l) =φ1Et[yt+l−1]+...+φpEt[yt+l−p]−θ1Et[ut+l−1]−...−θqEt[ut+l−q]+Et[ut+l], ✭✷✳✶✻✮

P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❄❄ ❞❡✈❡✲s❡ ❝♦♥s✐❞❡r❛r

• Et[yt−j] =yt−j

• Et[yt+j] = ˆyt(j)

• Et[ut−j] =ut−j

• Et[ut+j] = 0✱

♣❛r❛j >0

✷✳✹ ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s

(27)

✶✷ ✷✳✹✳ ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s

❖ ♠♦❞❡❧♦ ❧✐♥❡❛r ❣❡♥❡r❛❧✐③❛❞♦ é ✉♠ ♠♦❞❡❧♦ ❡♠ q✉❡ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❨ s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❛ ❢❛♠í❧✐❛ ❡①♣♦♥❡♥❝✐❛❧✱ ❞❛❞❛ ♣❡❧❛ ❢♦r♠❛

f(y;ϑ,ϕ) = exp{ϕ[yϑ−b(ϑ)] +c(y,ϕ)}, ✭✷✳✶✼✮

♦♥❞❡ϕ ❡ϑ sã♦ ♣❛râ♠❡tr♦s ❡ ❜ ❡ ❝ sã♦ ❢✉♥çõ❡s ❡s♣❡❝í✜❝❛s✳

❙❡❥❛η ♦ ♣r❡❞✐t♦r ❧✐♥❡❛r ❞❛❞♦ ♣♦r

η=

d

X

j=1

xjβj, ✭✷✳✶✽✮

♦♥❞❡✱xj✱ ❥❂✶✱✷✱✳✳✳✱❞✱ sã♦ ❝♦✈❛r✐á✈❡✐s ❡βj ♣❛râ♠❡tr♦s ❛ s❡r❡♠ ❡st✐♠❛❞♦s✳

❆ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ❣✱ q✉❡ ❞❡✈❡ s❡r ♠♦♥ót♦♥❛ ❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ r❡❧❛❝✐♦♥❛ ♦ ♣r❡❞✐t♦r

❧✐♥❡❛rη ❝♦♠ ❛ ❡s♣❡r❛♥ç❛✱ µ✱ ❞❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ ❨✱

g(µ) = η. ✭✷✳✶✾✮

◗✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ r❡s♣♦st❛ é ✉♠❛ ❝♦♥t❛❣❡♠✱ é ♣♦ssí✈❡❧ ✉t✐❧✐③❛r ❛ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥✳ ◆❡st❡ tr❛❜❛❧❤♦ s❡rá ✉t✐❧✐③❛❞❛ ❡ss❛ ❞✐str✐❜✉✐çã♦ ♣❛r❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ sér✐❡s t❡♠♣♦r❛✐s ❞❡ ❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠✱ ❥á q✉❡ é ✉♠❛ ❞✐str✐❜✉✐çã♦ q✉❡ ❝♦♠♣r❡❡♥❞❡ ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s✳

❙❡ ❨ t❡♠ ❞✐str✐❜✉✐çã♦ P♦✐ss♦♥ ❝♦♠ ♠é❞✐❛µ > 0✱ ❡♥tã♦

fY(y|µ) =

e−µ

µy

y! , ✭✷✳✷✵✮

q✉❡ ♥❛ ❢♦r♠❛ ❞❛ ❢❛♠í❧✐❛ ❡①♣♦♥❡♥❝✐❛❧ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

fY(y|µ) = exp (ylogµ−µ+ (−logy!)). ✭✷✳✷✶✮

(28)

✶✸ ✷✳✹✳ ▼♦❞❡❧♦s ▲✐♥❡❛r❡s ●❡♥❡r❛❧✐③❛❞♦s

logµ=η=

d

X

j=1

xjβd=x

e

′

β, ✭✷✳✷✷✮

♦♥❞❡x

e

′

= (1,x1,...,xd) ❡β = (β0, β1,...,βd)✳

❆ ❡st✐♠❛çã♦ ❞♦s ❞ ✰ ✶ ♣❛râ♠❡tr♦s (β0,β1,...,βd) é ❢❡✐t❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ♠á①✐♠❛

✈❡r♦ss✐♠✐❧❤❛♥ç❛✭❊▼❱✮✳

◆♦ ▼▲● P♦✐ss♦♥ ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ é ❞❛❞❛ ♣♦r

L(β) =

n

Y

i=1

exp (yi①✬_✐β)

yi!

exp (−exp①✬_✐β), ①_✐= (1,x1,...,xd), ✭✷✳✷✸✮

♣♦✐s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠✷✳✷✷✱ t❡♠✲s❡ µ= exp (①✬β)✳

❈♦♠♦ ♦ ❡st✐♠❛❞♦r ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♥ã♦ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❞❡ ❢♦r♠❛ ❛♥❛✲ ❧ít✐❝❛✱ é ♥❡❝❡ssár✐♦ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❛❧❣♦r✐t♠♦s ♥✉♠ér✐❝♦s ♣❛r❛ ❛ ❡st✐♠❛çã♦ ❞♦ ✈❡t♦r ❞❡

♣❛râ♠❡tr♦sβ = (β0,β1,...,βd)✳ ❖ ❛❧❣♦r✐t♠♦ ❞❡ ◆❡✇t♦♥ ❘❛♣❤s♦♥ é ✉♠ ❞♦s ♠❛✐s ✉t✐❧✐③❛❞♦s

❡ ❡stá ❞❡s❝r✐t♦✱ r❡s✉♠✐❞❛♠❡♥t❡✱ ❛ s❡❣✉✐r✳

❖ ❛❧❣♦r✐t♠♦ ✉t✐❧✐③❛✲s❡ ❞❛ ❡①♣❛♥sã♦ ❞❛ ❢✉♥çã♦ ❡s❝♦r❡Uβ =

∂L(β)

∂β ❡♠ t♦r♥♦ ❞❡ ✉♠ ✈❛❧♦r

✐♥✐❝✐❛❧β(0)✱ t❛❧ q✉❡

Uβ ≈Uβ(0)+U ′

β(0)(β−β(0)). ✭✷✳✷✹✮

❆ss✐♠ ♦❜té♠✲s❡ ♦ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦

β(m+1) =β(m)+ (−U_β′)−1(m)

U_β(m). ✭✷✳✷✺✮

❖ ♣r♦❝❡ss♦ é r❡♣❡t✐❞♦ ❛té q✉❡ s❡ ♦❜t❡♥❤❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛✳

❆❞❡q✉❛çã♦ ❞♦ ♠♦❞❡❧♦

(29)

♠❛①✐♠✐③❛❞❛s✱ q✉❡ s❡r✈❡ ♣❛r❛ ♠❡❞✐r ❛ ❞✐stâ♥❝✐❛ ❞♦s ✈❛❧♦r❡s ❛❥✉st❛❞♦s ♣❡❧♦ ♠♦❞❡❧♦ ❛♦s ❞❛❞♦s✳

❆ ❡st❛tíst✐❝❛ ❉❡✈✐❛♥❝❡ é ❞❛❞❛ ♣♦r

D∗

(Y,µ_b) = 2[L(Y;Y)−L(βb;Y)] ✭✷✳✷✻✮

s❡♥❞♦ L(Y;Y) ❛ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ s❛t✉r❛❞♦✱ ✐st♦ é✱ q✉❛♥❞♦ ♦

♥ú♠❡r♦ ❞❡ ♣❛râ♠❡tr♦s é ✐❣✉❛❧ ❛ ♥ ❡βbé ❛ ❡st✐♠❛t✐✈❛ ❞♦ ♣❛râ♠❡tr♦✳

(30)

❈❛♣ít✉❧♦ ✸

▼♦❞❡❧♦s ♣❛r❛ sér✐❡s ❞❡ ❝♦♥t❛❣❡♠

❊st❡ ❝❛♣ít✉❧♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ❞♦✐s ♠♦❞❡❧♦s q✉❡ sã♦ ❡①t❡♥sõ❡s ❞♦ ▼▲● ❢✉♥❞❛♠❡♥t❛❞♦s ♥♦s ❛rt✐❣♦s ❞❡ ❇❡♥❥❛♠✐♥ ❡t ❛❧ ✭✷✵✵✸✮ ❡ ❉❛✈✐s ❡t ❛❧ ✭✷✵✵✸✮✳

❉❛s ♠❡t♦❞♦❧♦❣✐❛s ❝✐t❛❞❛s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✱ ♦s ♠♦❞❡❧♦s ❆❘▼❆ ♠♦❞❡❧❛♠ ❞❛❞♦s ❛✉t♦❝♦rr❡❧❛❝✐♦♥❛❞♦s✱ ❝♦♠♦ é ♦ ❝❛s♦ ❞❛s sér✐❡s t❡♠♣♦r❛✐s✱ t♦❞❛✈✐❛ ❡❧❡s r❡str✐♥❣❡♠✲s❡ à s✉♣♦s✐çã♦ ❞❡ ◆♦r♠❛❧✐❞❛❞❡✳ ❖s ♠♦❞❡❧♦s ❧✐♥❡❛r❡s ❣❡♥❡r❛❧✐③❛❞♦s ♠♦❞❡❧❛♠ ❞✐str✐❜✉✐çõ❡s ❞❛ ❢❛♠í❧✐❛ ❡①♣♦♥❡♥❝✐❛❧✱ ♠❛s ♥ã♦ ❝♦♥s✐❞❡r❛♠ ❛s ❝♦rr❡❧❛çõ❡s q✉❡ ♦❝♦rr❡♠ ❡♥tr❡ ❛s ♦❜s❡r✈❛çõ❡s ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦✳ P♦r ✐ss♦✱ é ♥❡❝❡ssár✐♦ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♦✉tr♦s ♠♦❞❡❧♦s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡ sér✐❡s t❡♠♣♦r❛✐s ❞❡ ❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠✳

(31)

✶✻ ✸✳✶✳ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ●❡♥❡r❛❧✐③❛❞♦ ✭●❆❘▼❆✮

✸✳✶ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ●❡♥❡r❛❧✐③❛❞♦

✭●❆❘▼❆✮

❖ ●❆❘▼❆ ♣r♦♣♦st♦ ♣♦r ❇❡♥❥❛♠✐♥ ❡t ❛❧ ✭✷✵✵✸✮ é ✉♠❛ ❡①t❡♥sã♦ ❞♦s ♠♦❞❡❧♦s ▼▲● ♣❛r❛ ✈❛r✐á✈❡✐s r❡s♣♦st❛ q✉❡✱ ❝♦♥❞✐❝✐♦♥❛✐s à ✐♥❢♦r♠❛çã♦ ♣❛ss❛❞❛✱ ♣♦ss✉❡♠ ❞✐str✐❜✉✐çõ❡s ♣❡rt❡♥❝❡♥t❡s à ❢❛♠í❧✐❛ ❡①♣♦♥❡♥❝✐❛❧✳

❙❡❥❛yt✉♠❛ sér✐❡ t❡♠♣♦r❛❧ ❡ ①t✉♠ ✈❡t♦r ❞❡ ❝♦✈❛r✐á✈❡✐s✱ ♣❛r❛ t❂✶✱✷✱✳✳✳✱n ✳ ◆♦ ♠♦❞❡❧♦

●❆❘▼❆ ❛ ❞✐str✐❜✉✐çã♦ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡ ❝❛❞❛ ♦❜s❡r✈❛çã♦yt❞❛❞♦ ❛s ✐♥❢♦r♠❛çõ❡s ❛♥t❡r✐♦r❡s

Ht−1 = (x1,...,xt,y1,...,yt−1,µ1,...,µt−1) t❡♠ ❞✐str✐❜✉✐çã♦ ♥❛ ❢❛♠í❧✐❛ ❡①♣♦♥❡♥❝✐❛❧ ❞❛❞❛ ❡♠

✷✳✶✼✳

❈♦♠♦ ♥♦ ▼▲●✱ µt ❡stá r❡❧❛❝✐♦♥❛❞❛ ❛ ηt✱ ✉♠ ♣r❡❞✐t♦r ❧✐♥❡❛r✱ ❛tr❛✈és ❞❡ ✉♠❛ ❢✉♥çã♦

❞❡ ❧✐❣❛çã♦ ❣

g(µt) =ηt=① ′

tβ+τt, ✭✸✳✶✮

♦♥❞❡

τt= p

X

j=1

φj{g(yt−j)−①

′

t−jβ}+

q

X

j=1

θj{g(yt−j)−ηt−j}. ✭✸✳✷✮

❆ s❡❣✉✐r é ❞❡s❝r✐t♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ ♠♦❞❡❧♦ ●❆❘▼❆✱ ❡♠ q✉❡yts❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦

P♦✐ss♦♥ ❡ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ♠♦❞❡❧❛r ❞❛❞♦s ❞❡ ❝♦♥t❛❣❡♠✳

●❆❘▼❆ P♦✐ss♦♥

❖ ♠♦❞❡❧♦ ●❆❘▼❆ P♦✐ss♦♥ é ♦ ♠♦❞❡❧♦ ●❆❘▼❆ ♦♥❞❡ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦✲

❜❛❜✐❧✐❞❛❞❡ ❞❡ yt é P♦✐ss♦♥✳

➱ ♣♦ssí✈❡❧ ❞❡✜♥✐r ♦ ♠♦❞❡❧♦ ❝♦♠♦

f(yt|Ht−1) =

µyt

t e

−µt

yt!

(32)

✶✼ ✸✳✶✳ ▼♦❞❡❧♦ ❆✉t♦r❡❣r❡ss✐✈♦ ▼é❞✐❛ ▼ó✈❡❧ ●❡♥❡r❛❧✐③❛❞♦ ✭●❆❘▼❆✮

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

♥❛t✉r❛✐s ♣❛r❛ηt✳

❆ss✐♠

ηt= logµt=① ′

tβ+

p

X

j=1

φj{log (y

∗

t−j))−①

′

t−jβ}+

q

X

j=1

θjlog

y∗

t−j

µt−j

, ✭✸✳✹✮

♦♥❞❡ y∗

t−j = max(yt−j,❝) ❡ ✵❁❝❁✶✱ ✉♠❛ r❡str✐çã♦ ♥❡❝❡ssár✐❛ q✉❛♥❞♦ yt−j = 0 ♣❛r❛ q✉❡

s❡❥❛ ♣♦ssí✈❡❧ ✉t✐❧✐③❛r ❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛✳

✸✳✶✳✶ ❊st✐♠❛çã♦ ♥♦s ♠♦❞❡❧♦s ●❆❘▼❆

❉❡ ❛❝♦r❞♦ ❝♦♠ ❇❡♥❥❛♠✐♥ ❡t ❛❧ ✭✷✵✵✸✮ ♦ ✈❡t♦r ❞❡ ♣❛râ♠❡tr♦sγ′ = (β′,θ′,φ′)é ❡st✐♠❛❞♦

♣❡❧♦ ♠ét♦❞♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛✳

❆ ❢✉♥çã♦ ❧♦❣ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♥♦ ●❆❘▼❆ P♦✐ss♦♥ é

logL(γ) =

n

X

t=1

(ytlogµt−µt−logyt!), ✭✸✳✺✮

♦♥❞❡

µt = exp (ηt) = exp (① ′

tβ+

p

X

j=1

φjlog (y

∗

t−j))−①

′

t−jβ}+

q

X

j=1

θjlog

y∗

t−j

µt−j

.✭✸✳✻✮

P❛r❛ ♠❛①✐♠✐③❛r ❛ ❢✉♥çã♦ ❞❡ ❧♦❣✲✈❡r♦ss✐♠✐❧❤❛♥ç❛ sã♦ ✉t✐❧✐③❛❞♦s ♠ét♦❞♦s ♥✉♠ér✐❝♦s✳

✸✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦s ♠♦❞❡❧♦s ●❆❘▼❆

❇❡♥❥❛♠✐♥ ❡t ❛❧ ✭✷✵✵✸✮ ✐♥✈❡st✐❣❛r❛♠ ❛ ❡st❛❝✐♦♥❛r✐❡❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❢❛③❡♥❞♦ ❛ ❛♥á❧✐s❡ ♣❛r❛ ❛❧❣✉♠❛s ❢✉♥çõ❡s ❞❡ ❧✐❣❛çã♦ ❡s♣❡❝í✜❝❛s✳

❖s ❛✉t♦r❡s ♣r♦✈❛♠ q✉❡✱ ♣❛r❛ ♦ ❝❛s♦ ❞❛ ❢✉♥çã♦ ❞❡ ❧✐❣❛çã♦ ✐❞❡♥t✐❞❛❞❡✱ ❛ ♠é❞✐❛ ♠❛r❣✐♥❛❧