▼❛① ❙♦✉s❛ ❞❡ ▲✐♠❛
▼ét♦❞♦s ❆❞❛♣t❛t✐✈♦s ♣❛r❛ ❉❡t❡❝çã♦ ❞❡ ❈❧✉st❡rs ♥♦
❊s♣❛ç♦✲t❡♠♣♦
▼❛① ❙♦✉s❛ ❞❡ ▲✐♠❛
▼ét♦❞♦s ❆❞❛♣t❛t✐✈♦s ♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ❈❧✉st❡rs ♥♦
❊s♣❛ç♦✲t❡♠♣♦
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tís✲ t✐❝❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡r✲ s✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ à ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❊st❛tíst✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳❉r✳ ▲✉✐③ ❍❡♥r✐q✉❡ ❉✉❝③♠❛❧
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❊st❛tíst✐❝❛
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❊st❛tíst✐❝❛
✐
✐✐
❆❣r❛❞❡❝✐♠❡♥t♦s
• ➚ ❉❡✉s ♣❡❧❛ ❢♦rç❛✱ s❛ú❞❡ ❡ ❢❛♠í❧✐❛ q✉❡ ♠❡ ❝♦♥❝❡❞❡✉✳
• ❆♦s ♠❡✉s ♣❛✐s ♣❡❧❛ ❡❞✉❝❛çã♦ q✉❡ ♠❡ ❞❡r❛♠ ❡ q✉❡ ♠❡s♠♦ ♥❛ ❞✐✜❝✉❧❞❛❞❡ s❡♠♣r❡ s♦✉❜❡r❛♠
❝✉✐❞❛r ❞❡ ♠✐♠ ❝♦♠ ♠✉✐t♦ ❛♠♦r ❡ ❝❛r✐♥❤♦✳
• ❆♦s ♠❡✉s ✐r♠ã♦s q✉❡ ♠❡s♠♦ ❞✐st❛♥t❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♣r❡❝✐s❡✐✳ • ❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ▲✉✐③ ❉✉❝③♠❛❧✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ♠❛s ❛❝✐♠❛ ❞❡ t✉❞♦ ♣❡❧❛
♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❞❡s❡♥✈♦❧✈❡r ❝♦♠ ❡❧❡ ❡st❡ tr❛❜❛❧❤♦✳
• ❆♦s ♠❡✉s ✜é✐s ❡s❝✉❞❡✐r♦s ♥♦ ❞♦✉t♦r❛❞♦✱ ❘♦❞r✐❣♦ ❡ ▼❛r❦✉s✳
• ❆♦s ♠❡✉s ❛♠✐❣♦s ❋❛❜✐♦✱ ❈❛r❧✐t♦✱ ▲✉❝✐❛♥♦ ❡ ❘♦♥❛❧❞♦ ❝❡❛rá✱ ♣❡❧❛ ❝♦♥✈✐✈ê♥❝✐❛ ❡ t❛♠❜é♠ ♣♦r t❡r❡♠ ♠❡ s✉♣♦rt❛❞♦ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s✳
• ❆ t♦❞♦s ♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❡ ❝♦❧❡❣❛s ❞❡ ❞♦✉t♦r❛❞♦✳
• ❆♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊st❛tíst✐❝❛ ❞❛ ❯❋▼● ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡✳
• ❆ ♠✐♥❤❛ ♣r♦❢❡ss♦r❛✱ ♦r✐❡♥t❛❞♦r ❡ ❛♠✐❣❛ ❞❡ ❣r❛❞✉❛çã♦ ▼❛r✐❛ ■✈❛♥✐❧❞❡ ♣♦r ❛❝r❡❞✐t❛r ♥♦
♠❡✉ ♣♦t❡♥❝✐❛❧ ❡ s❡r ❛ ♠❛✐♦r ✐♥❝❡♥t✐✈❛❞♦r❛ ❞❛ ♠✐♥❤❛ ❝❛rr❡✐r❛ ❛❝❛❞ê♠✐❝❛✳
• ❆ ❋✉♥❞❛çã♦ ❞❡ ❆♠♣❛r♦ à P❡sq✉✐s❛ ❞♦ ❊st❛❞♦ ❞♦ ❆♠❛③♦♥❛s✲❋❆P❊❆▼✱ ♣❡❧♦ ❛♣♦✐♦
✐✐✐
❘❡s✉♠♦
❘❡s✉♠♦ ✐✈
✈
❆❜str❛❝t
✈✐
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ♣✳ ①
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ♣✳ ①✐✐
✶ ■♥tr♦❞✉çã♦ ♣✳ ✶
✶✳✶ ❆s♣❡❝t♦s ●❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶ ✶✳✷ ❏✉st✐✜❝❛t✐✈❛ ❡ ■♠♣♦rtâ♥❝✐❛ ❞♦ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹ ✶✳✸ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹ ✶✳✹ ❊str✉t✉r❛ ❞♦ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺
✷ ❈♦♥❝❡✐t♦s ❡ ♠ét♦❞♦s ♣❛r❛ ❛ ❞❡t❡❝çã♦ ❞❡ ♠✉❞❛♥ç❛s ♦✉ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✱
t❡♠♣♦ ❡ ❡s♣❛ç♦✲t❡♠♣♦ ♣✳ ✻
❙✉♠ár✐♦ ✈✐✐
✷✳✸✳✶ ❋♦r♠✉❧❛çã♦ ❞♦ Pr♦❜❧❡♠❛ ❞❡ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❊s♣❛ç♦✲❚❡♠♣♦ ✳ ✳ ✳ ♣✳ ✶✽ ✷✳✸✳✷ ❊st❛tíst✐❝❛ ❙❝❛♥ ❡s♣❛ç♦✲t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✾ ✷✳✸✳✸ ❊st❛tíst✐❝❛ ❙❝❛♥ ❡s♣❛ç♦✲t❡♠♣♦ ❜❛s❡❛❞❛ ♥♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✶ ✷✳✸✳✹ ❙❝❛♥ ❡s♣❛ç♦✲t❡♠♣♦ ❜❛s❡❛❞❛ ♥♦ t❡st❡ ❡s❝♦r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✶
✸ ❆❧❣✉♠❛s ❆❜♦r❞❛❣❡♥s ♣❛r❛ ❉❡t❡❝çã♦ ❞❡ ❈❧✉st❡r ♥♦ ❊s♣❛ç♦✲❚❡♠♣♦ ❯s❛♥❞♦
❆ ❘❛③ã♦ ❞❡ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❆❞❛♣t❛t✐✈❛ ♣✳ ✷✹
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✹ ✸✳✷ ❋♦r♠✉❧❛çã♦✱ ◆♦t❛çã♦✱ ▲❘ ❡ ❆▲❘ ♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ✳ ✳ ♣✳ ✷✹ ✸✳✷✳✶ ❋♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ✳ ✳ ✳ ✳ ♣✳ ✷✹ ✸✳✷✳✷ ◆♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✺ ✸✳✷✳✸ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ✉s❛♥❞♦ ▲❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✻ ✸✳✷✳✹ ▲❘ ❆❞❛♣t❛t✐✈❛ ♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵ ✸✳✷✳✺ ❘❡♣r❡s❡♥t❛çã♦ ❊s♣❛❝✐❛❧ ❞♦s ❈❧✉st❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶ ✸✳✷✳✻ ❆❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ❡♠❡r❣❡♥t❡s ♥♦ ❡s♣❛ç♦ t❡♠♣♦ ✳ ✳ ✳ ✳ ♣✳ ✸✸ ✸✳✸ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✸ ✸✳✹ ❊①t❡♥sõ❡s ♣❛r❛ ❛❜♦r❞❛❣❡♠ ❆❞❛♣t❛t✐✈❛ ♥♦ ❡s♣❛ç♦ ❞♦s ❝❧✉st❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✼ ✸✳✺ ❊st✉❞♦ ❝♦♠ ❞❛❞♦s s✐♠✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✵ ✸✳✺✳✶ P❡r❢♦r♠❛♥❝❡ ❞♦s ♠ét♦❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✵ ✸✳✺✳✷ ❘❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❛ s✐♠✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✶ ✸✳✻ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸
❙✉♠ár✐♦ ✈✐✐✐
✹✳✸✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ❢❛❧s♦ ❛❧❛r♠❡ ♣❛r❛ ♦ ▼❆❳✲❋❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✾ ✹✳✸✳✸ ❋❛t♦r ❞❡ ❇❛②❡s ❆❞❛♣t❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✵ ✹✳✹ ❊str✉t✉r❛ ●❡r❛❧ ❞♦s ▼♦❞❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✷ ✹✳✹✳✶ ▼♦❞❡❧♦ P❛❞rã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✷ ✹✳✹✳✷ ▼♦❞❡❧♦ ❆❧t❡r♥❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✹ ✹✳✺ ❘❡♣r❡s❡♥t❛çã♦ ❊s♣❛❝✐❛❧ ❞♦ ❝❧✉st❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✻ ✹✳✻ ❊①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✵ ✹✳✼ ❯♠❛ ❛♣❧✐❝❛çã♦ ❛ ❞❛❞♦s r❡❛✐s✿ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ❞❡ ❍❛♥s❡♥í❛s❡ ♥♦ ❆♠❛③♦♥❛s✲
❇r❛s✐❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✹ ✹✳✼✳✶ ❆♥á❧✐s❡ ❡①♣❧♦r❛rór✐❛ ❞♦s ❞❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✹ ✹✳✼✳✷ ❆♣❧✐❝❛çã♦ ❞♦ ❋❛t♦r ❞❡ ❇❛②❡s ❆❞❛♣t❛t✐✈♦ ♣❛r❛ ♦s ❞❛❞♦s ❞❡ ❍❛♥s❡♥í❛s❡ ✳ ♣✳ ✻✺ ✹✳✽ ❘❡s✉❧t❛❞♦s ❞❛ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼ ✹✳✾ ❊st✉❞♦ ❝♦♠ ❞❛❞♦s s✐♠✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✽ ✹✳✾✳✶ ❉❛❞♦s ❙✐♠✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✽ ✹✳✾✳✷ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❝♦♠ ❞❛❞♦s s✐♠✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✶ ✹✳✶✵ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✸
❙✉♠ár✐♦ ✐①
✺✳✽ ❊st✉❞♦ ❝♦♠ ❞❛❞♦s s✐♠✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✺ ✺✳✽✳✶ ❉❛❞♦s s✐♠✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✺ ✺✳✽✳✷ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❝♦♠ ❞❛❞♦s s✐♠✉❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✻ ✺✳✾ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✼
✻ ❈♦♥❝❧✉sã♦ ♣✳ ✾✾
①
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❊①❡♠♣❧♦ ✈✐s✉❛❧ ❞♦ ✉s♦ ❞♦ ♠ét♦❞♦ ●❆▼ ♠♦str❛♥❞♦ ❝❧✉st❡rs ❞❡ ár❡❛s ♣♦r ❡♠❛r❛♥✲
❤❛❞♦s ❞❡ ❝ír❝✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽ ✷✳✷ ❊①❛♠♣❧♦ ✈✐s✉❛❧ ❞♦ ✉s♦ ❞♦ ♠ét♦❞♦ ❞❡ ❱❛rr❡❞✉r❛ ❊s♣❛❝✐❛❧ ✲ ❙❝❛♥ ❝✐r❝✉❧❛r ✳ ✳ ✳ ✳ ♣✳ ✶✵ ✷✳✸ ❊①❛♠♣❧♦ ✈✐s✉❛❧ ❞♦ ✉s♦ ❞♦ ♠ét♦❞♦ ❞❡ ❱❛rr❡❞✉r❛ ❡s♣❛❝✐❛❧✲t❡♠♣♦r❛❧ ✲ ❙❝❛♥ ❡s♣❛ç♦✲
t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✶ ✸✳✶ ▼❛♣❛ ❞♦ ◆♦✈♦ ▼é①✐❝♦✳ ❋♦♥t❡✿ ❯❙ ❈❡♥s✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✹ ✸✳✷ ▼♦♥✐t♦r❛♠❡♥t♦ ♦♥✲❧✐♥❡ ✉s❛♥❞♦ ▼❆❳✱ ▼■❳ ❛♥❞ ❲❊■●❍❚✲❆▲❘ ❝♦♠ ❇✭t❤r❡s❤♦❧❞✮❂✷✵
♣✳ ✸✼
✸✳✸ ❈♦♠♣❛r❛çã♦ ❞♦ ♠♦♥✐t♦r❛♠❡♥t♦ ❖♥✲❧✐♥❡ ✉s❛♥❞♦ ♦ ▼❆❳✲❆▲❘ ❝♦♠ ●▼❆❳✲❆▲❘
❡♠ ✭❛✮ ❡ ▼■❳✲❆▲❘ ❝♦♠ ●▼■❳✲❆▲❘ ❡♠ ✭❜✮ ❝♦♠ ❇✭t❤r❡s❤♦❧❞✮❂✷✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾ ✸✳✹ ❆♠♦str❛ ❞❡ ✉♠❛ s✐♠✉❧❛çã♦ ❞♦ ♣r♦❝❡ss♦ ♣❛r❛ ●▼■❳ ❛♥❞ ●❲❊■●❍❚✲❆▲❘ ♥❛
❡s❝❛❧❛ ❧♦❣❛rtí♠✐❝❛ ❝♦♠ ❧♦❣✭❇✮❂✭t❤r❡s❤♦❧❞✮❂❧♦❣✭✷✵✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸ ✸✳✺ ❍✐st♦❣r❛♠❛ ❇✐✈❛r✐❛❞♦ ♣❛r❛ ♦ ●▼■❳✲❆▲❘✿ ❙❡♥s✐t✐✈✐❞❛❞❡ ✭❙❡♥s✐t✐✈✐t②✮ ✈❡rs✉s
❛tr❛s♦ ✭❉❡❧❛②✮ ❡♠ ✭❆✮ ❡ ❱❛❧♦r Pr❡❞✐t♦ P♦s✐t✐✈♦ ✭PP❱✲P♦s✐t✐✈❡ Pr❡❞✐❝t❡❞ ❱❛❧✉❡✮
✈❡rs✉s ❉❡❧❛②✭❛tr❛s♦✮ ❡♠ ✭❇✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸ ✸✳✻ ❍✐st♦❣r❛♠❛ ❇✐✈❛r✐❛❞♦ ♣❛r❛ ♦ ❲❊■●❍❚✲❆▲❘✿ ❙❡♥s✐t✐✈✐❞❛❞❡ ✭❙❡♥s✐t✐✈✐t②✮ ✈❡rs✉s
❛tr❛s♦ ✭❉❡❧❛②✮ ❡♠ ✭❆✮ ❡ ❱❛❧♦r Pr❡❞✐t♦ P♦s✐t✐✈♦ ✭PP❱✲P♦s✐t✐✈❡ Pr❡❞✐❝t❡❞ ❱❛❧✉❡✮
✈❡rs✉s ❉❡❧❛②✭❛tr❛s♦✮ ❡♠ ✭❇✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸ ✹✳✶ ❯♠❛ ♣♦ssí✈❡❧ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛ ❛s ❞❡♥s✐❞❛❞❡s ♣r❡❞✐t✐✈❛s ❡♠ r❡❧❛çã♦ ❛♦s ♠♦❞✲
❡❧♦s ♣❛❞rã♦ ❡ ❛❧t❡r♥❛t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✼ ✹✳✷ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✽ ✹✳✸ ❋♦r♠❛t♦ ❞❡ ❈❧✉st❡rs q✉❡ ♣♦❞❡♠ s❡r ❞❡t❡❝t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✽ ✹✳✹ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❉❡t❡❝çã♦ ❞❡ ❍❛♥s❡♥í❛s❡ ✭♣♦r ✶✵✵✵✵ ♠❧ ❤❛❜✮ ♥♦ ❊st❛❞♦ ❞♦ ❆♠❛✲
③♦♥❛s✭❡♠ ❛③✉❧✮✱ s❡❣✉♥❞♦ ♦ ♠ês✱ ♥❛s ❝✐❞❛❞❡s✿ ▼❛♥❛✉s✭❡♠ ✈❡r❞❡✮✱ ❍✉♠❛✐tá ✭❡♠
✈❡r♠❡❧❤♦✮ ❡ ■t❛❝♦❛t✐❛r❛ ✭❡♠ ♣r❡t♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✺ ✹✳✺ ▼❛♣❛ ❞♦ ❊st❛❞♦ ❞♦ ❆♠❛③♦♥❛s ❝♦♠ ♦s ♠✉♥✐❝í♣✐♦s ❡s♣❛❝✐❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛❞♦s
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐
✹✳✻ ▼♦♥✐t♦r❛♠❡♥t♦ ♦♥✲❧✐♥❡ ✉s❛♥❞♦ ❋❛t♦r ❞❡ ❇❛②❡s ❙❡q✉❡♥❝✐❛❧ ❆❞❛♣t❛t✐✈♦ ❝♦♠ ❥❛♥❡❧❛
w= 3✱α = 0,05 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✽ ✹✳✼ ❝❧✉st❡r ❞❡t❡❝t❛❞♦ ✉s❛♥❞♦ ♦ ❋❛t♦r ❞❡ ❇❛②❡s ❙❡q✉ê♥❝✐❛❧ ❊❧✐♣t✐❝♦t∈[15,17] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✽
✹✳✽ ❝❧✉st❡r ❞❡t❡❝t❛❞♦ ✉s❛♥❞♦ ♦ ❋❛t♦r ❞❡ ❇❛②❡s ❙❡q✉ê♥❝✐❛❧ ❊❧✐♣t✐❝♦t∈[16,18] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✾
✹✳✾ ▼❛♣❛ ❞♦ ◆♦✈♦ ▼é①✐❝♦✳ ❋♦♥t❡✿ ❯❙ ❈❡♥s✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✵ ✹✳✶✵ ▼á①✐♠♦ ❞♦ ❋❛t♦r ❞❡ ❇❛②❡s ❆❞❛♣t❛t✐✈♦ ❡♠ ✭❆✮✱ ❈❛r❞✐♥❛❧✐❞❛❞❡Jn❞❡Ξn❡♠ ✭❇✮
♣❛r❛ ✶✵✵✵ s✐♠✉❧❛çõ❡s ❞♦ ♣r♦❝❡ss♦ s♦❜r❡ ♦ ♠♦❞❡❧♦ ♣❛❞rã♦ ❝♦♠ ρ= 0.2❀ ▼á①✐♠♦ ❞♦ ❋❛t♦r ❞❡ ❇❛②❡s ❆❞❛♣t❛t✐✈♦ ❡♠ ✭❈✮✱ ❈❛r❞✐♥❛❧✐❞❛❞❡ Jn ❞❡ Ξn ❡♠ ✭❉✮ ♣❛r❛
✶✵✵✵ s✐♠✉❧❛çõ❡s ❞♦ ♣r♦❝❡ss♦ s♦❜r❡ ♦ ♠♦❞❡❧♦ ♣❛❞rã♦ ❝♦♠ ρ= 0.4✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✶ ✹✳✶✶ ❍✐st♦❣r❛♠❛ ❇✐✈❛r✐❛❞♦ ♣❛r❛ ♦ ❋❛t♦r ❞❡ ❇❛②❡s ❆❞❛♣t❛t✐✈♦ ❝♦♠ ρ = 0.2✿ ❙❡♥s✐✲
t✐✈✐❞❛❞❡ ✈❡rs✉s ❛tr❛s♦ ❡ ❱❛❧♦r Pr❡❞✐t♦ P♦s✐t✐✈♦ ✈❡rs✉s ❛tr❛s♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✸ ✹✳✶✷ ❍✐st♦❣r❛♠❛ ❇✐✈❛r✐❛❞♦ ♣❛r❛ ♦ ❋❛t♦r ❞❡ ❇❛②❡s ❆❞❛♣t❛t✐✈♦ ❝♦♠ ρ = 0.4✿ ❙❡♥s✐✲
t✐✈✐❞❛❞❡ ✈❡rs✉s ❛tr❛s♦ ❡ ❱❛❧♦r Pr❡❞✐t♦ P♦s✐t✐✈♦ ✈❡rs✉s ❛tr❛s♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✸ ✹✳✶✸ ❉✐str✐❜✉✐çõ❡s Pr❡❞✐t✐✈❛s ❡ P❛❞rã♦ ❛♣ós ♦ ♣♦♥t♦ ❞❡ ♠✉❞❛♥ç❛ ♥❛s ár❡❛s q✉❡ ❝♦♠✲
♣õ❡ ♦ ❝❧✉st❡r✿ ❈❤❛✈❡s ❡♠ ✭❆✮✱ ❊❞❞② ❡♠ ✭❇✮✱ ▲❡❛ ❡♠ ✭❈✮ ❡ ❖t❡r♦ ❡♠ ✭❉✮✳ ✳ ✳ ✳ ♣✳ ✼✹ ✺✳✶ ❯♠❛ ❈❧❛ss❡ ❞❡ ❝❧✉st❡rs ❝♦♠ ❢♦r♠❛t♦ ❝✐r❝✉❧❛r ❡ t❛♠❛♥❤♦ ✈❛r✐á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✻ ✺✳✷ ▼❛♣❛ ❞♦ ◆♦✈♦ ▼é①✐❝♦✳ ❋♦♥t❡✿ ❯❙ ❈❡♥s✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✽ ✺✳✸ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ✇❡✐❜✉❧❧ ✭❢✳❞✮✱ ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞♦ ♠á①✐♠♦ ❞❡ ♠✲✈❛r✐á✈❡✐s
✇❡✐❜✉❧❧ ✭❢✳❞✳ ♠❛①✮✱ ❋✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ✇❡✐❜✉❧❧ ✭❋✳❞✮✱ ❋✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦ ♠á①✐♠♦ ❞❡ ♠✲✈❛r✐á✈❡✐s ✇❡✐❜✉❧❧ ✭❋✳❞✳ ♠❛①✮✱ ✶✲❋✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ✇❡✐❜✉❧❧
✭✶✲❋✳❞✮✱ ✶✲❋✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦ ♠á①✐♠♦ ❞❡ ♠✲✈❛r✐á✈❡✐s ✇❡✐❜✉❧❧ ✭✶✲❋✳❞✳ ♠❛①✮ ♣✳ ✾✵ ✺✳✹ ❆ ❡sq✉❡r❞❛✿ ▼á①✐♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ ❛❞❛♣t❛t✐✈❛ ♣❛r❛ ♠❂✷✱ ♣♦♥t♦
❦❂✶✺ ❡ r❡❣✐ã♦ ❞❛ ♠✉❞❛♥ç❛❀ ❆ ❞✐r❡✐t❛✿ ▼á①✐♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ ❛❞❛♣✲
t❛t✐✈❛ ♣❛r❛ ♠❂✽✱ ♣♦♥t♦ ❦❂✶✾ ❡ r❡❣✐ã♦ ❞❛ ♠✉❞❛♥ç❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✹ ✺✳✺ ❊♠ ✭❆✮✿ ▼á①✐♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ ❛❞❛♣t❛t✐✈❛ ♣❛r❛ ♠❂✷✱ ✹ ❡ ✽❀ ❊♠
✭❇✮ ❘❛③ã♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ♣♦r ✈❛❧♦r ❡s♣❡r❛❞♦ xt(sl)/et(sl), t ≥ k = 1,2, ..., n
♣❛r❛ ❛s r❡❣✐õ❡s ❞♦ ❝❧✉st❡r ❡♠❡r❣❡♥t❡ ❞❡t❡❝t❛❞♦ ❡ ❝♦♥s✐❞❡r❛❞♦ s✐❣♥✐✜❝❛t✐✈♦ ✳ ✳ ✳ ♣✳ ✾✹ ✺✳✻ ❆♠♦str❛ ❞❡ ✉♠❛ s✐♠✉❧❛çã♦ ❞♦ ♣r♦❝❡ss♦ ❛ ♣♦st❡r✐♦r✐ ❛❞❛♣t❛t✐✈♦ ♣❛r❛ ❞✐str✐❜✉✐çõ❡s
①✐✐
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✸✳✶ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ❡♠❡r❣❡♥t❡s ❞❡ ❝❛s♦s ❞❡ ❝â♥❝❡r ❞❛ t✐r❡ó✐❞❡ ❡♠ ❤♦♠❡♥s q✉❡ ♦❝♦rr❡r❛♠ ♥♦ ◆♦✈♦
▼é①✐❝♦✱ ✉s❛♥❞♦ ♦ ▼■❳✲❆▲❘ ❡ ❲❊■●❍❚✲❆▲❘ ♥♦s ú❧t✐♠♦s ❝✐♥❝♦ ❛♥♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✼
✸✳✷ ❱❛❧♦r❡s ❡st✐♠❛❞♦s ♣❛r❛ ❙❡♥s✐t✐✈✐❞❛❞❡ ✭❙❙❈✮✱ ❱❛❧♦r Pr❡❞✐t♦ P♦s✐t✐✈♦ ✭❱PP✮ ❡ ❆tr❛s♦ ✭ADD✮ ❝♦♠ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ♣❛r❛ θt,j✱ ❞✐❢❡r❡♥t❡s t❛①❛s ❛❝❡✐tá✈❡✐s ❞❡
❛❧❛r♠❡s ❢❛❧s♦s ✭α✮ ✉s❛♥❞♦ ♦ ●▼■❳ ❡ ●❲❊■●❍❚✲❆▲❘✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✷ ✹✳✶ ■♥t❡r♣r❡t❛çã♦ ❞♦ ❋❛t♦r ❞❡ ❇❛②❡s ✭❑❛ss ❡ ❘❛❢t❡r②✱ ✶✾✾✺✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✷ ✹✳✷ P❡r❝❡♥t✉❛❧ ❞❡ ár❡❛ ❞❡ ✉♠❛ ❡❧✐♣s❡ q✉❡ é t❛♠❜é♠ ♣❛rt❡ ❞❡ q✉❛❧q✉❡r ♦✉tr❛ ❡❧✐♣s❡ ❝♦♠ ♦
♠❡s♠♦ ❝❡♥tr♦✱ ❢♦r♠❛ ❡ t❛♠❛♥❤♦✱♣♦ré♠ ❝♦♠ q✉❛♥t✐❞❛❞❡s ✭#✮ ❞✐❢❡r❡♥t❡s ❞❡ â♥❣✉❧♦s ✳ ✳ ♣✳ ✺✾
✹✳✸ ❱❛❧♦r❡s ❡st✐♠❛❞♦s ♣❛r❛ ❙❡♥s✐t✐✈✐❞❛❞❡ ✭❙❙❈✮✱ ❱❛❧♦r Pr❡❞✐t♦ P♦s✐t✐✈♦ ✭❱PP✮ ❡ ❆tr❛s♦ ✭ADD✮ ❝♦♠ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ♣❛r❛ δ✱ ❞✐❢❡r❡♥t❡s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡
❛❧❛r♠❡s ❢❛❧s♦s ✭α✮ ✉s❛♥❞♦ ♦ ❋❛t♦r ❞❡ ❇❛②❡s ❆❞❛♣t❛t✐✈♦ ❊❧✐♣t✐❝♦ ❝♦♠ ❥❛♥❡❧❛ ✇❂✸✳ ♣✳ ✼✷ ✺✳✶ ❱❛❧♦r❡s ♣❛r❛ p ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ✈❛❧♦r ❞❡ w ❡ p0 = 0.5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✾✶
✺✳✷ ❱❛❧♦r❡s ❡st✐♠❛❞♦s ♣❛r❛ ❙❡♥s✐t✐✈✐❞❛❞❡ ✭❙❙❈✮✱ ❱❛❧♦r Pr❡❞✐t♦ P♦s✐t✐✈♦ ✭❱PP✮ ❡ ❆tr❛s♦ ✭ADD✮ ✉s❛♥❞♦ ♦ ♣r♦❝❡ss♦ ❛ ♣♦st❡r✐♦r✐ ❛❞❛♣t❛t✐✈♦ ❝♦♠ ❥❛♥❡❧❛ ✇❂✻✱
✶
✶ ■♥tr♦❞✉çã♦
✶✳✶ ❆s♣❡❝t♦s ●❡r❛✐s
▼ét♦❞♦s ❡st❛tíst✐❝♦s ❛♣❧✐❝❛❞♦s á ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s✱ ♦❜t✐❞♦s ♣❡r✐♦❞✐❝❛♠❡♥t❡ ♣❡❧♦s s✐st❡♠❛s ❞❡ ✈✐❣✐❧â♥❝✐❛✭❡♠ s❛ú❞❡ ♣ú❜❧✐❝❛✱ ❝r✐♠❡ ♦✉ ❛♠❜✐❡♥t❛❧✮ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❞❡t❡❝t❛r ❝❧✉st❡r ❞❡ ❡✈❡♥✲ t♦s✱ ♦s q✉❛✐s ♣♦❞❡♠ ✐♥❞✐❝❛r ✉♠❛ rá♣✐❞❛ ♠✉❞❛♥ç❛ ♥♦ ♣❛❞rã♦ ❞♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s✳ ●❡r❛❧♠❡♥t❡✱ ✉♠ ❝❧✉st❡r é ✉♠❛ ✐♥❡s♣❡r❛❞❛ ❛❣❧♦♠❡r❛çã♦ ❞❡ ❡✈❡♥t♦s ♥♦ ❡s♣❛ç♦✱ t❡♠♣♦ ♦✉ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ◗✉❛♥❞♦ ♦ ♣❡rí♦❞♦ ❞❡ ✈✐❣✐❧â♥❝✐❛ é ♣ré✲❡s♣❡❝✐✜❝❛❞♦ ❡ ♦s ❡✈❡♥t♦s sã♦ ❛✈❛❧✐❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ s✉❛ ❞✐str✐❜✉✐çã♦ ❡s♣❛❝✐❛❧✱ ✉♠ ❝❧✉st❡r ✭❝❧✉st❡r ❡s♣❛❝✐❛❧✮ é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ s✉❜✲❝♦♥❥✉♥t♦ ❞❡ ❧♦✲ ❝❛❧✐③❛çõ❡s ❡s♣❛❝✐❛✐s ❞❛ r❡❣✐ã♦ ❡♠ ❡st✉❞♦ ♦♥❞❡ ❛ t❛①❛ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ t❛✐s ❡✈❡♥t♦s é ❡❧❡✈❛❞❛ ❞❡ ❢♦r♠❛ s✐❣✐♥✐✜❝❛t✐✈❛✳ ❙❡ ♦ ❡s♣❛ç♦ é ✐❣♥♦r❛❞♦✱ r❡♣r❡s❡♥t❛♠♦s ♦ ❝❧✉st❡r ♣❡❧♦ ❣r✉♣♦ ✭♦✉ s❡q✉ê♥❝✐❛✮ ❞❡ ♦❜s❡r✈❛çõ❡s q✉❡ ❝❛✉s❛♠ ✉♠❛ ♠✉❞❛♥ç❛ ♥♦ ♣❛❞rã♦ t❡♠♣♦r❛❧ ❞♦s ❞❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱ s❡ ♦ ❡s✲ ♣❛ç♦ ❡ ♦ t❡♠♣♦ sã♦ ♠♦♥✐t♦r❛❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ✉♠ ❝❧✉st❡r ❡s♣❛ç♦✲t❡♠♣♦r❛❧ é ✐❞❡♥t✐✜❝❛❞♦ ♣♦r ✉♠ ♣❛r ✭❧♦❝❛❧✐③❛çõ❡s ❡s♣❛❝✐❛✐s✱ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✮ ♣❛r❛ ♦ q✉❛❧ ♦❝♦rr❡✉ ❛ ♠✉❞❛♥ç❛✳ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡r t❡♠ s✐❞♦ ♦ ❢♦❝♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ s✐st❡♠❛s ❞❡ ✈✐❣✐❧â♥❝✐❛✿ ❙✐st❡♠❛ ❞❡ ✈✐❣✐❧â♥❝✐❛ ❡♠ ❝r✐♠❡s ✭●♦rr ❛♥❞ ❍❛rr✐❡s✱ ✷✵✵✸✮ ♣❛r❛ ❞❡t❡❝t❛r ❝❧✉st❡r ❡♠❡r❣❡♥t❡s ❞❡ ❡✈❡♥t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❛❧❣✉♠ t✐♣♦ ❞❡ ❝r✐♠❡❀ ❙✐st❡♠❛ ❞❡ ✈✐❣✐❧â♥❝✐❛ ❡♠ ❞♦❡♥ç❛s ♣❛r❛ ♠♦♥✐t♦r❛r ❞❛❞♦s ❞❡ s❛ú❞❡ ♣ú❜❧✐❝❛ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ✈✐s✐t❛s ❡♠ ❤♦s♣✐t❛✐s ✭❙❛❜❤♥❛♥✐ ❡t ❛❧✳✱ ✷✵✵✺✮ ♦✉ ❝❧✉st❡r ❞❡ ❝❛s♦s ❞❡ ❞♦❡♥ç❛s ✭❑✉❧❧❞♦r✛✱ ✷✵✵✶✮❀ ❙✐st❡♠❛s ❞❡ ♠♦♥✐t♦r❛♠❡♥t♦ ❛♠❜✐❡♥t❛❧ ♣❛r❛ ❞❡t❡❝t❛r ❛❧t♦s ♥í✈❡✐s ❞❡ ♣♦❧✉✐çã♦ ♥♦ ❛r ❡ á❣✉❛ ✭❆✐❧❛♠❛❦✐ ❡t ❛❧✳✱ ✷✵✵✸✮ ♦✉ ❞❡t❡❝t❛r ❝❧✉st❡r ❡♠❡r❣❡♥t❡s ❞❡ q✉❡✐♠❛❞❛s ❡♠ ✢♦r❡st❛s ✭❚✉✐❛ ❡t ❛❧✳✱ ✷✵✵✽✮✳
✶✳✶ ❆s♣❡❝t♦s ●❡r❛✐s ✷
❆ss✉♥çã♦✱ ✷✵✵✹❀ ❚❛♥❣♦✱ ✷✵✵✺❀ ❑✉❧❧❞♦r✛ ❡t ❛❧❧✳✱ ✷✵✵✻❀ ●❛♥❣♦♥ ❛♥❞ ❑❧❛②t♦♥✱ ✷✵✵✼❀ ❉✉❝③♠❛❧ ❡t ❛❧❧✳✱ ✷✵✵✼✮✱tê♠ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♣❛r❛ ❞❡t❡❝t❛r ❡ ❡st✐♠❛r ❛ ❧♦❝❛❧✐③❛çã♦ ❞♦ ❝❧✉st❡r✳ ❊st❡s ♠ét♦❞♦s ♥ã♦ ❧❡✈❛♠ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ✐♥❝❡rt❡③❛ s♦❜r❡ ♦ t❛♠❛♥❤♦ ❡ ❧♦❝❛❧✐③❛çã♦ ❞♦ ❝❧✉st❡r✳
◆❛ ❡s❝❛❧❛ t❡♠♣♦r❛❧✱ ❛ ❞❡t❡❝çã♦ ❞❡ ✉♠❛ ♠✉❞❛♥ç❛ ♥❛ t❛①❛ ❞❡ ♦❝♦rrê♥❝✐❛ ❞♦s ❡✈❡♥t♦s ❡♠ ✉♠ ❝❧✉st❡r ✭❣r✉♣♦✮ ❞❡ ♦❜s❡r✈❛çõ❡s ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ✉s❛♥❞♦ ♠ét♦❞♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣❛r❛ ♠♦♥✐t♦r❛✲ ♠❡♥t♦ ❞❡ ♣r♦❝❡ss♦s ✐♥❞✉str✐❛✐s ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦ ♠ét♦❞♦ ❈❯❙❯▼ ✭P❛❣❡✱ ✶✾✺✹✮✱ ❙❤✐r②❛②❡✈✲ ❘♦❜❡rts ✭❘♦❜❡rts✱ ✶✾✻✻✮ ♦✉ ♠✐st✉r❛ ❞❡ r❛③õ❡s ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛s ✭P♦❧❧❛❦✱ ✶✾✽✼✮✳ ❯♠❛ ❜♦❛ r❡✈✐sã♦ s♦❜r❡ ❡st❡s ♠ét♦❞♦s ❡ s✉❛s ✈❛r✐❛çõ❡s ♣♦❞❡ s❡r ✈✐st❛ ❡♠ ▲❛✐ ✭✶✾✾✺✮✳ ❯s✉❛❧♠❡♥t❡ ❡ss❡s ♠ét♦❞♦s ❛ss✉♠❡♠ q✉❡ ♥♦ ❡st❛❞♦ ❞❡ ❝♦♥tr♦❧❡ ❞♦ ♣r♦❝❡ss♦✱ ❡①✐st❡ ✉♠❛ t❛①❛ ♣❛❞rã♦ ❝♦♥❤❡❝✐❞❛θ0
❞❡ ❡✈❡♥t♦s✱ ❡ s♦❜ ❛ s✉♣♦s✐çã♦ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ♠✉❞❛♥ç❛ ♥♦ ♣r♦❝❡ss♦✱ ❛ t❛①❛ ♠✉❞❛ ♣❛r❛θ1t❛♠❜é♠
❛ss✉♠✐❞❛ ❝♦♥❤❡❝✐❞❛✳ ◆❛ ♣rát✐❝❛ ❡ss❛s s✉♣♦s✐çõ❡s ♥✉♥❝❛ sã♦ s❛t✐s❢❡✐t❛s ❡ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ é ✉s❛r ♠ét♦❞♦s ❛❞❛♣t❛t✐✈♦s ✭▲♦r❞❡♥ ❡ P♦❧❧❛❦✱ ✷✵✵✺❀ ❚❛rt❛❦♦✈s❦②✱ ✷✵✵✻✮ ♦✉ ❞✐♥â♠✐❝♦s ✭❲❡st✱ ✶✾✽✻✮✳
◆❛ ❡s❝❛❧❛ ❡s♣❛ç♦✲t❡♠♣♦r❛❧✱ ❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞♦ ❝♦♠♦ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡❝çã♦ ❞❡ r❡❣✐ã♦ ❡ ♣♦♥t♦ ❞❡ ♠✉❞❛♥ç❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ❊st❛tíst✐❝❛♠❡♥t❡ ❡st❡ ♣r♦❜❧❡♠❛ é ❞❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❉❡♥♦t❡ ♣♦rS ✉♠❛ r❡❣✐ã♦ ❞❡ ❡st✉❞♦ ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ L✲ár❡❛s Al
❝♦♥tí❣✉❛s ✐❞❡♥t✐✜❝❛❞❛s ♣♦r ✉♠ ♣♦♥t♦ sl✳ ❖♥❞❡✱ ♣♦r ❡①❡♠♣❧♦✱ S é ✉♠ ♠❛♣❛ ❡ sl r❡♣r❡s❡♥t❛ ♦
❝❡♥tró✐❞❡ ❞♦ ♣♦❧í❣♦♥♦ q✉❡ ❞❡❧✐♥❡✐❛ ❛ ár❡❛Al✳ ❈♦♥s✐❞❡r s♦❜r❡S♦ ♠♦♥✐t♦r❛♠❡♥t♦ ✭♦✉ ❛ ✈✐❣✐❧â♥❝✐❛✮
❡st❛tíst✐❝♦✭❛✮ ❞❡ ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ X = {Xt(sl), t = 1,2, ... ❡ l = 1,2, ...L}✳ ❆ ❝❛❞❛
t❡♠♣♦ ❞✐s❝r❡t♦ t ≥ 1 ♦❜s❡r✈❛♠♦s ✉♠ ✈❡t♦r L✲✈❛r✐❛❞♦ Xt = (Xt(s1),Xt(s2), ...,Xt(sL))′✱ ❡♠
q✉❡ Xt(sl) r❡♣r❡s❡♥t❛ ❛❧❣✉♠❛ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ♥❛ l✲és✐♠❛ ár❡❛✳ ❙❡❥❛♠ ξ = {sl ∈ S : sl ∈ ξ}
✉♠ ❝♦♥❥✉♥t♦ ❝♦♥❡①♦ ❞❡ ❧♦❝❛❧✐③❛çõ❡s ❡s♣❛❝✐❛✐s ❡♠ S✱ Hn = {X1, ...,Xn} ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s
❛❝✉♠✉❧❛❞♦s ❛té ♦ ✐♥st❛♥t❡ n✳ ❖ ♦❜❥❡t✐✈♦ ❞❛ ♠♦♥✐t♦r❛♠❡♥t♦ é ✈❡r✐✜❝❛r s❡ ❡①✐st❡ ❛❧❣✉♠ ❝❧✉st❡r ❡♠❡r❣❡♥t❡ ❡♠ S✳ ❖ ✐♥t❡r❡ss❡ é ❞❡t❡❝t❛r ✉♠ ❝❧✉st❡r q✉❡ ❝♦♠❡ç♦✉ ❡♠ ✉♠ t❡♠♣♦ ❞❡s❝♦♥❤❡❝✐❞♦
k≤n❡ q✉❡ ❡stá ♣r❡s❡♥t❡ ❛té ♦ ❡stá❣✐♦ ❛t✉❛❧n✳ ❖ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ é ❡①♣r❡ss♦ ❡♠ ❢✉♥çã♦ ❞❡ ✉♠❛ ♠✉❞❛♥ç❛ ♥♦ ♣❛❞rã♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞♦ s✉❜✲♣r♦❝❡ss♦{Xt(sl)∈X:sl∈ξ, t≥k} t❛❧ q✉❡
ξn,k ={s
l∈ S :sl∈ξ}×[k, n]r❡♣r❡s❡♥t❛ ♦ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ❝❧✉st❡r
❡♠❡r❣❡♥t❡✳
✶✳✶ ❆s♣❡❝t♦s ●❡r❛✐s ✸
♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ♠✉❞❛♥ç❛s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡♠ ♣r♦❝❡ss♦s ❞❡ P♦✐ss♦♥ ❍♦♠♦❣ê♥❡♦✳ ❆ss✉♥çã♦ ❡ ❈♦rrê❛ ✭✷✵✵✾✮ ♠♦♥✐t♦r❛♠ ❛ ❢✉♥çã♦ ❞❡ ✐♥t❡♥s✐❞❛❞❡ ❞❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ P♦✐ss♦♥ ❤❡t❡r♦❣ê♥❡♦ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❛tr❛✈és ❞❛ ❡st❛tíst✐❝❛ ❞❡ ❙❤✐r②❛②❡✈✲❘♦❜❡rts✳ ▲✐♠❛ ❡ ❉✉❝③♠❛❧✭✷✵✵✾✱✷✵✶✶✮ ♣r♦♣õ❡♠ ♦ ❋❛t♦r ❞❡ ❇❛②❡s ❈✉♠✉❧❛t✐✈♦ ♣❛r❛ ♠♦♥✐t♦r❛♠❡♥t♦ ❡ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ❡♠ ❞❛❞♦s ❞❡ ár❡❛ ❝♦♠ t❛①❛ ♣❛❞rã♦ ❝♦♥❤❡❝✐❞❛✳ ❚❛♥❣♦ ✭✷✵✶✵✮ ❯s❛ ✉♠❛ ❡st❛tíst✐❝❛ s❝❛♥ ❝♦♠❜✐♥❛❞❛ ❝♦♠ ✉♠ t❡st❡ s❝♦r❡ ♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡♠ ✉♠❛ ❥❛♥❡❧❛ t❡♠♣♦r❛❧ ✈❛r✐á✈❡❧✳
◆♦ ❣❡r❛❧✱ ♣♦✉❝♦s ♠ét♦❞♦s ♣❛r❛ ❞❡t❡❝çã♦ ❡ ♠♦♥✐t♦r❛♠❡♥t♦ ❞❡ ❝❧✉st❡rs tê♠ s✐❞♦ ♣r♦♣♦st♦s ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❞❡s❡♥✈♦❧✈❡r ✉♠ s✐st❡♠❛ ❞❡ ✈✐❣✐❧â♥❝✐❛ q✉❡ s♦❡ ✉♠ ❛❧❛r♠❡ ❛ss✐♠ q✉❡ ✉♠ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ s❡❥❛ ❞❡t❡❝t❛❞♦✳ ❯♠❛ r❡✈✐sã♦ ❞♦s ♠ét♦❞♦s s✉❣❡r✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ♣❛r❛ ❞❡t❡❝çã♦ s❡qü❡♥❝✐❛❧ ❞❡ ♠✉❞❛♥ç❛s ♥♦ ♠♦♥✐t♦r❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❞❡ s❛ú❞❡ ♣ú❜❧✐❝❛ ♥❛ ❡s❝❛❧❛ t❡♠♣♦r❛❧ é ❛♣r❡s❡♥t❛❞❛ ♣♦r ❙♦♥❡ss♦♥ ❡ ❇♦❝❦ ✭✷✵✵✸✮✳ ❋r✐s❡♥ ✭✷✵✵✸✮ ❞✐s❝✉t❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ót✐♠❛s ❞❡ ✉♠ ❜♦♠ s✐st❡♠❛ ❞❡ ✈✐❣✐❧â♥❝✐❛ ❡ r❡❢♦rç❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♠ét♦❞♦s ❡st❛tíst✐❝♦s ❡ s✐st❡♠❛s ❛♣r♦♣r✐❛❞♦s ♣❛r❛ ❛ ❞❡t❡❝çã♦ ♣r♦s♣❡❝t✐✈❛ ❞❡ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ❢♦r♠❛ rá♣✐❞❛ ❡ ❡✜❝✐❡♥t❡✳
❆ ❡s❝❛ss❡③ ❞❡ ♠ét♦❞♦s ♣❛r❛ ♠♦♥✐t♦r❛♠❡♥t♦ ❡ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦ t❡♠♣♦✱ ♣♦❞❡ s❡r ❥✉st✐✜❝❛❞❛ ♣❡❧♦ ❢❛t♦ q✉❡ ♣❛r❛ ❞❡t❡❝t❛r q✉❛❧q✉❡r ♠✉❞❛♥ç❛ ✐♥❡s♣❡r❛❞❛ ♥❛ t❛①❛ ❞❡ ❝❡rt♦s t✐♣♦s ❞❡ ❡✈❡♥t♦s✱ ❢❛③✲s❡ ♥❡❝❡ssár✐❛ ✉♠❛ ❛♥á❧✐s❡ r❡♣❡t✐❞❛ ❞❡ ❞❛❞♦s ❛❝✉♠✉❧❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ❡✱ q✉❛❧q✉❡r ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ t❡♠ q✉❡ s❡r ❛❥✉st❛❞♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡t×(2l−1)❝♦♠♣❛r❛çõ❡s
✶✳✷ ❏✉st✐✜❝❛t✐✈❛ ❡ ■♠♣♦rtâ♥❝✐❛ ❞♦ ❚r❛❜❛❧❤♦ ✹
✶✳✷ ❏✉st✐✜❝❛t✐✈❛ ❡ ■♠♣♦rtâ♥❝✐❛ ❞♦ ❚r❛❜❛❧❤♦
❉❡✈✐❞♦ ❛♦ ❛✈❛♥ç♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ tê♠ s✉r❣✐❞♦ ✉♠ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♥❛ ❝♦♥str✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ✐♥❢♦r♠❛çõ❡s ❣♦✈❡r♥❛♠❡♥t❛✐s q✉❡ ❢✉♥❝✐♦♥❡♠ ❝♦♠♦ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❢❡rr❛♠❡♥t❛ ❞❡ ❛✉①í❧✐♦ ♥♦ ❝♦♥tr♦❧❡ ❞❛ ❛❞♠✐♥✐str❛çã♦ ♣ú❜❧✐❝❛✳ ❯♠ ❡①❡♠♣❧♦ é ♦ ❡✲s✐❣❛ ✭❙✐st❡♠❛ ❞❡ ■♥❢♦r♠❛çõ❡s ●♦✈❡r♥❛♠❡♥✲ t❛❧ ❞♦ ❆♠❛③♦♥❛s✮✳ ❖ ❡✲s✐❣❛ é ✉♠ s✐st❡♠❛ ♦♥✲❧✐♥❡ ♦♣❡r❛❝✐♦♥❛❧ q✉❡ ♣❡r♠✐t❡✱ ❡♥tr❡ ♦✉tr♦s r❡❝✉rs♦s✱ ♦ ❛❝♦♠♣❛♥❤❛♠❡♥t♦ ❞♦s ♣r♦❜❧❡♠❛s ❡♣✐❞❡♠✐♦❧ó❣✐❝♦s ❡ ♠♦♥✐t♦r❛♠❡♥t♦ ❡♠ t❡♠♣♦ r❡❛❧ ❞♦s ❢❛t♦r❡s ❛❞✈❡rs♦s ❛ s❛ú❞❡ ❞♦ ❛♠❛③♦♥❡♥s❡✳ ➱ ♥❡ss❡ s❡♥t✐❞♦✱ ❡ t❡♥❞♦ ❡♠ ✈✐st❛ ♦ ❝r❡s❝❡♥t❡ ✐♥t❡r❡ss❡ ♣❡❧♦ ❡s✲ t✉❞♦ ❞❡ ❛❣r❡❣❛❞♦s ✭❝❧✉st❡r✮ ❡s♣❛ç♦✲t❡♠♣♦r❛✐s ❞❡ ❞♦❡♥ç❛s✱ q✉❡ ♥ós ♣r♦♣♦♠♦s ♥♦✈❛s ♠❡t♦❞♦❧♦❣✐❛s ♣❛r❛ ❞❡t❡❝çã♦ ❡ ✈✐❣✐❧â♥❝✐❛ ❞❡ ❝❧✉st❡rs ❡♠ ♠❛♣❛s ❞❡ ❞♦❡♥ç❛s✱ ❝r✐♠✐♥❛❧✐❞❛❞❡✱ ❡t❝✳✳✳✳ ❯♠❛ ❧❛❝✉♥❛ ♥❛ ❧✐t❡r❛t✉r❛✱ q✉❡ ❥✉st✐✜❝❛ ❛♣r❡s❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ é ❛ ❛✉sê♥❝✐❛ ❞❡ ✉♠❛ s✐st❡♠❛t✐③❛çã♦ ❝rít✐❝❛ q✉❡ ❛♠♣❧✐❡ ♦ ❤♦r✐③♦♥t❡ ❞❡ ❛♣❧✐❝❛çõ❡s ❞♦s ♠ét♦❞♦s ❞❡ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♣❛r❛ ❛❧é♠ ❞❛ ✐♥✈❡st✐✲ ❣❛çã♦ ❡t✐♦❧ó❣✐❝❛ ❞❡ ❞♦❡♥ç❛s r❛r❛s✱ ♦✉ ❡♥❞ê♠✐❝❛s✱ ❡ ♦r✐❡♥t❡ ♦ ✐♥✈❡st✐❣❛❞♦r ♥❛ ❡s❝♦❧❤❛ ❞❛ ❞❡❝✐sã♦ ♠❛✐s ❛❞❡q✉❛❞♦ ❛♦s s❡✉s ♦❜❥❡t✐✈♦s✳
✶✳✸ ❖❜❥❡t✐✈♦s
❊st❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ t❡✈❡ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❣❡r❛❧ ♦ ❡st✉❞♦ ❞♦s ♠ét♦❞♦s ❡st❛tíst✐❝♦s ♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ❛ss✐♠ ❝♦♠♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈♦s ♠ét♦❞♦s✳ ❈♦♠♦ ♦❜❥❡t✐✈♦s ❡s♣❡❝í✜❝♦s t✐✈❡♠♦s✿
✶✳ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ ♥♦✈♦ ♠ét♦❞♦ ❡st❛tíst✐❝♦ ❜❛s❡❛❞♦ ❡♠ ❡st❛tíst✐❝❛ ❜❛②❡s✐❛♥❛ ♦✉ ❝❧ás✲ s✐❝❛ ♣❛r❛ ❞❡t❡❝çã♦ ❡ ✈✐❣✐❧â♥❝✐❛ ❞❡ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡♠ ❞❛❞♦s ❞❡ ár❡❛ ❡ ♣r♦❝❡ss♦s ♣♦♥t✉❛✐s✳
✷✳ ❯t✐❧✐③❛r ♦ ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ❝♦♠♦ ♣❛rt❡ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ✈✐❣✐❧â♥❝✐❛ ❡♠ ❛❧❣✉♥s t✐♣♦s ❞❡ ❡✈❡♥t♦s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦s ❝❛s♦s ❞❡ ❍❛♥s❡♥í❛s❡ ♥♦ ❡st❛❞♦ ❞♦ ❆♠❛③♦♥❛s✳
❈♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛s ♥♦✈❛s ♠❡t♦❞♦❧♦❣✐❛s ♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ❡s♣❡r❛♠♦s ❞❛r ❛s s❡❣✉✐♥t❡s ❝♦♥tr✐❜✉✐çõ❡s✿
• ❊st❛❜❡❧❡❝❡r ♥♦✈❛s ♠❡t♦❞♦❧♦❣✐❛s té❝♥✐❝♦✲❝✐❡♥tí✜❝❛s ♣❛r❛ ✈✐❣✐❧â♥❝✐❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ❡✈❡♥✲ t♦s✳
• ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠ét♦❞♦s ❡st❛tíst✐❝♦s q✉❡ ❛❥✉❞❡♠ ❡ ✐❞❡♥t✐✜q✉❡♠ r❛♣✐❞❛♠❡♥t❡ ❛s ár❡❛s
✶✳✹ ❊str✉t✉r❛ ❞♦ ❚r❛❜❛❧❤♦ ✺
✶✳✹ ❊str✉t✉r❛ ❞♦ ❚r❛❜❛❧❤♦
✻
✷ ❈♦♥❝❡✐t♦s ❡ ♠ét♦❞♦s ♣❛r❛ ❛ ❞❡t❡❝çã♦ ❞❡
♠✉❞❛♥ç❛s ♦✉ ❝❧✉st❡r ♥♦ ❡s♣❛ç♦✱ t❡♠♣♦ ❡
❡s♣❛ç♦✲t❡♠♣♦
✷✳✶ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦
❆ ❛♥á❧✐s❡ ❞❡ ❝❧✉st❡rs ❡s♣❛❝✐❛✐s t❡♠ r❡❝❡❜✐❞♦ ✉♠❛ ❛t❡♥çã♦ ❝♦♥s✐❞❡rá✈❡❧ ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❖ ♦❜❥❡t✐✈♦ ❜ás✐❝♦ ❡♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r é ❞❡t❡r♠✐♥❛r ❛✉t♦♠❛t✐✲ ❝❛♠❡♥t❡ r❡❣✐õ❡s ❞♦ ❡s♣❛ç♦ ♦♥❞❡ t❡♥❤❛ ♦❝♦rr✐❞♦ ✉♠❛ ♠✉❞❛♥ç❛ ♥ã♦ ❡s♣❡r❛❞❛ ♥♦ ♣❛❞rã♦ ❡s♣❛❝✐❛❧ ❞♦ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ ♦❜s❡r✈❛❞♦✳ ❊ss❛s ♠✉❞❛♥ç❛s ♥♦ ♣❛❞rã♦ ❡s♣❛❝✐❛❧ ♣♦❞❡♠ ❝♦rr❡s♣♦♥❞❡r ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❢❡♥ô♠❡♥♦s✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ❝❛♠♣♦ ❞❡ ❛♣❧✐❝❛çã♦ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ❞❡ ❞♦❡♥ç❛s✱ ❞❡ ❡str❡❧❛s ♦✉ ❣❛❧á①✐❛s✱ ❞❡ ❢♦❝♦s ❞❡ q✉❡✐♠❛❞❛s ❡♠ ✢♦r❡st❛s✱ ❞❡ ✐♥❝✐❞ê♥❝✐❛s ❞❡ ❛❧❣✉♠ t✐♣♦ ❞❡ ❝r✐♠❡✱ ❞❡ ❛❧❣✉♠❛ ❡s♣é❝✐❡ ❞❡ ❛♥✐♠❛✐s✳
❖s ❡st✉❞♦s ❞❡ ❝❧✉st❡rs sã♦ ❜❛s❡❛❞♦s ❡♠ ❧♦❝❛❧✐③❛çõ❡s ❞❡ ❡✈❡♥t♦s ✭♣r♦❝❡ss♦ ♣♦♥t✉❛❧✮ ♦✉ ❛❣r❡✲ ❣❛❞♦ ❞❡ ❡✈❡♥t♦s✭❞❛❞♦s ❞❡ ár❡❛✮✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s ❛ ❤❡t❡r♦❣❡♥❡✐❞❛❞❡ ❞❛ ♣♦♣✉❧❛çã♦ ❡st✉❞❛❞❛ é ❛ss✉♠✐❞❛ ❝♦♥❤❡❝✐❞❛ ❡ ❡♠ ❝❡rt❛s s✐t✉❛çõ❡s ❛❧❣✉♠❛s ♠❡❞✐❞❛s ❞❡ ❝♦✈❛r✐á✈❡✐s t❛♠❜é♠ ♣♦❞❡♠ s❡r ✐♥✲ ❝♦r♣♦r❛❞❛s✳ ❯♠ ♣r♦❝❡ss♦ ♣♦♥t✉❛❧ ♣♦❞❡ s❡r tr❛♥s❢♦r♠❛❞♦ ❡♠ ❞❛❞♦s ❞❡ ár❡❛✳ P♦r ✐ss♦✱ ♥♦ss♦ ❢♦❝♦ sã♦ ♣r♦❝❡ss♦s ❡s♣❛❝✐❛✐s ♠♦❞❡❧❛❞♦s ❝♦♠♦ ♣r♦❝❡ss♦s ♠❡❞✐❞♦s ❡♠ ár❡❛s ✭♦✉ ❞❛❞♦s ❞❡ ár❡❛❛✮✳ ◆❡st❡ ❝❛s♦✱ s✉♣♦♠♦s q✉❡ ❡①✐st❡ ✉♠ ♣r♦❝❡ss♦ ❡st♦❝át✐❝♦X(s) ={X(sl) :l= 1,2, ..., L}✳ ❖♥❞❡X(si)é ❛
r❡❛❧✐③❛çã♦ ❞♦ ♣r♦❝❡ss♦ ♥❛ ár❡❛Al❝♦♠♣❧❡t❛♠❡♥t❡ ✐❞❡♥t✐✜❝❛❞❛ ♣♦r ✉♠ ♣♦♥t♦sl ∈ S ={s1, ..., sL}✳
❚✐♣✐❝❛♠❡♥t❡✱ S é ✉♠ ♠❛♣❛ ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ L ár❡❛s ❡ sl é ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡✮ ♦ ❝❡♥tr♦ ❞♦
♣♦❧í❣♦♥♦ q✉❡ ❞❡❧✐♥❡❛ ❛ ár❡❛Al✳
✷✳✶ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦ ✼
ár✈♦r❡s✮ ❤❛❜✐t❛♥❞♦ ♥❡st❡ ❧♦❝❛❧✳ ●❡r❛❧♠❡♥t❡✱ s♦❜ ❛ ❤✐♣ót❡s❡ ♥✉❧❛
H0 :X(sl)∼P oisson(λN(sl)).
❖♥❞❡ λ é ✉♠❛ t❛①❛ ❞❡ ♦❝♦rrê♥❝✐❛ ❣❧♦❜❛❧ ❞♦s ❡✈❡♥t♦s ❡ N(sl) r❡♣r❡s❡♥t❛ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❛
♣♦♣✉❧❛çã♦ ❡♠ r✐s❝♦ ❤❛❜✐t❛♥❞♦ ❡♠Al✳
▼✉✐t♦s ♠ét♦❞♦s ♣❛r❛ ❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ❡s♣❛❝✐❛✐s tê♠ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞♦s ❝♦♠ ❞✐❢❡r❡♥t❡s ♣r♦♣ós✐t♦s✳ ❯♥s ❛✈❛❧✐❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝❧✉st❡r ❣❧♦❜❛❧✱ s❡♠ ❡s♣❡❝✐✜❝❛r s✉❛ ❧♦❝❛❧✐③❛çã♦✳ ❖✉tr♦s ❞❡t❡r♠✐♥❛♠ ❛ ❧♦❝❛❧✐③❛çã♦ ❡ ❛✈❛❧✐❛♠ ❛ s✐❣♥✐✜❝â♥❝✐❛ ❡st❛t✐st✐❝❛ ❞♦ ❝❧✉st❡r✳ ❊st❡s ♠ét♦❞♦s ✉s❛♠ té❝♥✐❝❛s ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♥t❡♥s✐✈❛s ❝♦♠♦ ♣❡r♠✉t❛çã♦ ❛❧❡❛tór✐❛✱ ▼♦♥t❡ ❈❛r❧♦✱ ❡t❝✳ ❖s ♠❛✐s ✉s✉❛✐s✱ ❡♥tr❡ ❡st❡s ♠ét♦❞♦s✱ ❛ss✉♠❡♠ q✉❡ t❡♠♦s ❛ ❞✐s♣♦s✐çã♦ ✉♠ ♠❛♣❛ S ❞❡ ár❡❛s ✱ ❝❛❞❛ ✉♠❛
❝♦♠ ♣♦♣✉❧❛çã♦ ❞❡ r✐s❝♦ ❡ ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦s ♦❜s❡r✈❛❞♦s✳ ❊❧❡s ✉t✐❧✐③❛♠ ❥❛♥❡❧❛s ❡s♣❛❝✐❛✐s q✉❡ s✉♣❡r♣õ❡ ❝♦♥❥✉♥t♦s ❝♦♥❡①♦s✱ ❝♦♠ ❛❧❣✉♠ ❢♦r♠❛t♦ ❣❡♦♠étr✐❝♦ ✭❝ír❝✉❧♦✱ q✉❛❞r❛❞♦✱ ❡❧✐♣s❡✮✱ s♦❜r❡ ❛s ár❡❛s ❞❡ ❡st✉❞♦ ❡ ❝♦♥t❛♠ ♦ ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦s q✉❡ ♦❝♦rr❡♠ ♥❛s r❡❣✐õ❡s ❝✉❥♦s ♦s ❝❡♥tró✐❞❡s ❝❛❡♠ ❞❡♥tr♦ ❞❡ ❝❛❞❛ ❥❛♥❡❧❛ ❡s♣❛❝✐❛❧✳ ❯♠ ♣r♦❜❧❡♠❛ ❝♦♠✉♠ ♥❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r é ♦ ❛❥✉st❡ ♣❛r❛ ♠ú❧t✐♣❧❛s ❝♦♠♣❛r❛çõ❡s✳ ❊st❡ ♣r♦❜❧❡♠❛ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ❧✐st❛❞♦s t♦❞❛s ♦s ❝♦♥❥✉♥t♦s ❝♦♥❡①♦s q✉❡ ❡♠ ♣r✐♥❝í♣✐♦ ♣♦❞❡♠ r❡♣r❡s❡♥t❛r ✉♠ ❝❧✉st❡r ✭♦✉ ❝❛♥❞✐❞❛❞t♦ ❛ ❝❧✉st❡r✮ ❡ ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡❧❡s é t❡st❛❞♦ s❡ s✉❛ t❛①❛ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ❡✈❡♥t♦s ❞✐❢❡r❡ ❞❡ ❢♦r♠❛ s✐❣♥✐✜❝❛t✐✈❛ ❞❛q✉❡❧❛ ❛ss♦❝✐❛❞❛ ❝♦♠ ♦ r❡st❛♥t❡ ❞♦ ♠❛♣❛✳ ❆q✉❡❧❡s ❝♦♥❥✉♥t♦s ❡♠ q✉❡ ♦ t❡st❡ ❢♦ss❡ s✐❣♥✐✜❝❛t✐✈♦ s❡r✐❛♠ ❝♦♥s✐❞❡r❛❞♦s ✉♠ ❛❣r❡❣❛❞♦ ❞❡ t❛①❛ ❛❝✐♠❛ ✭♦✉✮ ❛❜❛✐①♦ ❞♦ ❡s♣❡r❛❞♦✳ ❊st❡ ♣r♦❝❡❞✐♠❡♥t♦ é ✐♥❛❞❡q✉❛❞♦ ♣❛r❛ ❡st❛ s✐t✉❛çã♦ ♣❡❧♦ s❡❣✉✐♥t❡ ❢❛t♦✿ s✉♣♦♥❤❛ q✉❡ ✈ár✐♦s t❡st❡s sã♦ r❡❛❧✐③❛❞♦s ❝♦♠ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛ ❣❧♦❜❛❧ α✳ ❊♥tã♦✱ ✈ár✐♦s t❡st❡ s❡rã♦ s✐❣♥✐✜❝❛t✐✈♦s ♠❡s♠♦ q✉❡ ❛ ❤✐♣ót❡s❡ ♥✉❧❛ s❡❥❛ ✈❡r❞❛❞❡✐r❛ ❡♠ t♦❞♦s ❡❧❡s✳ ■st♦ ♦❝♦rr❡ ♣♦rq✉❡ ❛❝❤❛♠♦ q✉❡ ♦ ✈❛❧♦r α ✉s❛❞♦ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡ ❝♦♥t✐♥✉❛ ♦ ♠❡s♠♦✳ ◆❛ ✈❡r❞❛❞❡✱ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s t❡st❡s✱ ❡st❡ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛ s❡rá ♠✉✐t♦ ♠❛✐♦r q✉❡ ♦ ✈❛❧♦r ♥♦♠✐♥❛❧ ♣❛r❛ ♦s t❡st❡s ✐♥❞✐✈✐❞✉❛✐s✳ ❊st❛ s✐t✉❛çã♦ só é ✈á❧✐❞❛✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s t♦❞♦s ♦s t❡st❡s s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❖ r❡s✉❧t❛❞♦ ❞❛ ♣rát✐❝❛ ❞❡ss❡ ❢❛t♦ é q✉❡ ♠✉✐t♦s ❢❛❧s♦s ❝❧✉st❡r s❡r✐❛♠ ❞❡t❡❝t❛❞♦s ♣♦r ❡st❡ ♠ét♦❞♦✳ ❱❡❥❛♠♦s✱ ❛❣♦r❛✱ ❛❧❣✉♥s ♠ét♦❞♦s ♣❛r❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r✳
✷✳✶✳✶ ▼ét♦❞♦ ●❆▼✲●❡♦❣r❛♣❤✐❝❛❧ ❆♥❛❧②s✐s ▼❛❝❤✐♥❡
P❛r❛ ❛ ❞❡t❡❝çã♦ ❞❡ ❝❧✉st❡r ❡s♣❛❝✐❛❧✱ ❖♣❡♥s❤❛✇ ❡t ❛❧✳ ✭✶✾✽✽✮ ♣r♦♣✉s❡r❛♠ ✉♠ ♠ét♦❞♦ ❝♦♠✲ ♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♥t❡♥s✐✈♦ ❝♦♠ ❣r❛♥❞❡ ❛♣❡❧♦ ✈✐s✉❛❧ ❝❤❛♠❛❞♦ ●❡♦❣r❛♣❤✐❝❛❧ ❆♥❛❧②s✐s ▼❛❝❤✐♥❡✱ ❛❜r❡✈✐❛❞♦ ♣♦r ●❆▼✳ ◆❡st❡ ♠ét♦❞♦✱ ❛ss✉♠❛ q✉❡ X(sl) r❡♣r❡s❡♥t❛ ♦ ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦s q✉❡
♦❝♦rr❡♠ ♥❛ ár❡❛Al ❝♦♠ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦✱ s♦❜H0 ❞❛❞♦ ♣♦rλN(sl)✳ ❆ss♦❝✐❡ ♦s ✈❛❧♦r❡s ♦❜s❡r✈❛✲
❞♦s ❞❡ ❝❛❞❛ ár❡❛ ❛♦s s❡✉s ❝❡♥tró✐❞❡s ✭❝❡♥tr♦ ❞♦ ♣♦❧í❣♦♥♦ q✉❡ ❞❡❧✐♥❡✐❛ ❛ ár❡❛ Al✮ ❞❡♥♦t❛❞♦s ♣♦r
sl✳ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ●❆▼ ✉s❛ ♦ s❡❣✉✐♥t❡ ❛❧❣♦rt✐♠♦✿ • ❙❡❧❡❝✐♦♥❡ ✉♠ r❛✐♦ r ✭♣♦r ❡①❡♠♣❧♦✱ r❂✶✱✷ ♦✉ ✹ ❦♠✮✳
✷✳✶ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦ ✽
• ❈❛❧❝✉❧❡
Xlr =
L
X
l=1
X(sl)I{sl∈Clr} ❡ Nlr =
L
X
l=1
N(sl)I{sl∈Clr}
♦ ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦s ❡ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❛ ♣♦♣✉❧❛çã♦ ❡♠ r✐s❝♦ ❤❛❜✐t❛♥❞♦ ♥♦ ❝ír❝✉❧♦ Clr✱
♦♥❞❡ I{B} é ❛ ❢✉♥çã♦ ✐♥❞✐❝❛❞♦r❛ ❞♦ ❡✈❡♥t♦B✳
• ❈❛❧❝✉❧❡ ♦ ✈❛❧♦r✲♣✱ plr ❞♦ t❡st❡ ❛ss♦s✐❛❞♦ ❛ X(sl)✱ s♦❜ ❛ ❤✐♣ót❡s❡ ♥✉❧❛✱ H0 : X(sl) ∼
P oisson(λN(sl))✳
• ❉❡s❡♥❤❡ ♦ ❝ír❝✉❧♦ Clr s❡plr ≤0.002
• ❘❡♣✐t❛ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❛❝✐♠❛ ❛✉♠❡♥t❛♥❞♦ ✭♦✉ ❡s❝♦❧❤❡♥❞♦✮ ♦✉tr♦ r❛✐♦ ♣❛r❛ ♦ ❝ír❝✉❧♦✳
❖ r❡s✉❧t❛❞♦ ✜♥❛❧ é ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ❝❧✉st❡rs ❞❡ ár❡❛s ♣♦r ❡♠❛r❛♥❤❛❞♦s ❞❡ ❝ír❝✉❧♦s s♦❜r❡♣♦st♦s ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✭✷✳✶✮✳
❋✐❣✉r❛ ✷✳✶✿ ❊①❡♠♣❧♦ ✈✐s✉❛❧ ❞♦ ✉s♦ ❞♦ ♠ét♦❞♦ ●❆▼ ♠♦str❛♥❞♦ ❝❧✉st❡rs ❞❡ ár❡❛s ♣♦r ❡♠❛r❛♥✲ ❤❛❞♦s ❞❡ ❝ír❝✉❧♦s
❊♠❜♦r❛ ❝❛❞❛ ❝ír❝✉❧♦ s❡❥❛ ❥✉❧❣❛❞♦ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡ s✐❣♥✐✜❝❛t✐✈♦ ❛♦ ♥í✈❡❧ ✵✳✵✵✷✱ ✉♠ ♥í✈❡❧ ❞❡ s✐❣♥✐✜❝â♥❝✐❛ ♣❛r❛ t♦❞♦s ♦s ❝ír❝✉❧♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥ã♦ é ❝♦♥❤❡❝✐❞♦✳ ❖ ♠♦t✐✈♦ é ♦ ✉s♦ ❞❡ t❡st❡s s✐♠✉❧tâ♥❡♦s ♥ã♦ ✐♥❞❡♣❡♥❞❡♥t❡s q✉❡✱ ♥❡st❡ ❝❛s♦✱ sã♦ ❞❡✈✐❞♦s ❛ ❝♦♠♣❛r❛çã♦ ❞❡ ✉♠ ✐♠❡♥s♦ ♥ú♠❡r♦ ❞❡ ❝ír❝✉❧♦s s♦❜r❡♣♦st♦s✳
✷✳✶✳✷ ▼ét♦❞♦ ❞❡ ❇❡s❛❣ ❡ ◆❡✇❡❧
◆♦ ♠ét♦❞♦ ●❆▼ ❞❡ ❖♣❡♥s❤❛✇ ✜①❛✲s❡ ♦ r❛✐♦ r ❞♦ ❝ír❝✉❧♦ ❡ ❝❛❧❝✉❧❛✲s❡ ♦ ✈❛❧♦r✲♣ s♦❜r❡ ♦ ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦s q✉❡ ♦❝♦rr❡♠ ♥♦ ❝ír❝✉❧♦ Clr✳ ❊♠ t❡r♠♦s ❣❡r❛✐s✱ ♥♦ ♠ét♦❞♦ ❇❡s❛❣ ❡ ◆❡✇❡❧
✭✶✾✾✶✮ ✜①❛✲s❡ ♦ ♥ú♠❡r♦y ❞❡ ❡✈❡♥t♦s q✉❡ ❞❡✈❡♠ s❡r ❜✉s❝❛❞♦s ❡ ❝❛❧❝✉❧❛✲s❡ ♦ r❛✐♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❡♥❣❧♦❜á✲❧♦s✳ ◆♦ ❝ír❝✉❧♦ r❡s✉❧t❛♥t❡✱ ❝❛❧❝✉❧❛✲s❡ ♦ ♣✲✈❛❧♦r ❡ ♣r♦❝❡❞❡♥❞♦ ❝♦♠♦ ♦ ●❆▼✱ ❞❡s❡♥❤❛ ❝ír❝✉❧♦s s✐❣♥✐✜❝❛t✐✈♦s ✭✈❛❧♦r✲♣≤0.002✮✳ ❘❡♣❡t❡ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ✈❛r✐❛♥❞♦ ♦ ✈❛❧♦r ❞❡ y✳
P❛r❛ ❝♦♠♣✉t❛r ♦ ✈❛❧♦r✲♣✱ s❡❥❛ X=PL
l=1X(sl) ❡N =
PL
l=1N(sl)✳ ❈❡♥tr❛❞♦ ❡♠sl✱ ❛ss✉♠❛
✷✳✶ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦ ✾
ár❡❛s ✭♦✉ ❝❡♥tró✐❞❡s✮ ♥❡❝❡ssár✐❛s ♣❛r❛ ❛❝✉♠✉❧❛r ♦sy ♣r✐♠❡✐r♦s ❝❛s♦s ♠❛✐s ♣ró①✐♠♦s ❞❡sl✳ ❙❡❥❛˜l
♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ❞❡L˜❡N˜
l♦ t♦t❛❧ ❞❛ ♣♦♣✉❧❛çã♦ ♥❡ss❛s˜lár❡❛s✳ ❙♦❜ ❛ ❤✐♣ót❡s❡ ♥✉❧❛ ✭❛ ♠❡s♠❛ ❞♦
♠ét♦❞♦ ●❆▼✮✱X˜l ♦ ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦s ♥❡ss❛s ˜l ár❡❛s s❡❣✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥ ❝♦♠
✈❛❧♦r ❡s♣❡r❛❞♦ ❞❛❞♦ ♣♦r N˜lX/N✳ ❆❣♦r❛✱ ♥♦t❛♥❞♦ q✉❡P( ˜L≤˜l) = 1−P( ˜L >˜l+ 1) r❡♣r❡s❡♥t❛
✶ ♠❡♥♦s ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❛s ˜l ♣r✐♠❡✐r❛s ár❡❛s ♣♦ss✉❛♠ ♠❡♥♦s q✉❡ y ❡✈❡♥t♦s✳ ❊♥tã♦ ♦ ✈❛❧♦r✲♣ ♣❛r❛ ✉♠ ❝ír❝✉❧♦Cly ❝❡♥tr❛❞♦s ❡♠ sl ❝♦♥t❡♥❞♦y ❡✈❡♥t♦s é ❞❛❞♦ ♣♦r✱
ply = 1−
y−1
X
j=1
P(X˜l=j) = 1−
y−1
X
j=1
(N˜lX/N)j
j! e
−N˜lX/N.
✷✳✶✳✸ ▼ét♦❞♦ ❞❡ ❈✉③✐❝❦ ❡ ❊❞✇❛r❞s
❈✉③✐❝❦ ❡ ❊❞✇❛rs✭✶✾✾✵✮ ✜③❡r❛♠ ✉♠❛ ♣r♦♣♦st❛ q✉❡ r❡♣r❡s❡♥t❛ ✉♠❛ ♣❡q✉❡♥❛ ✈❛r✐❛çã♦ ❡♠ r❡❧❛çã♦ ❛♦s ♠ét♦❞♦s ❞❡ ❇❡s❛❣ ❡ ◆❡✇❡❧ ✭✶✾✾✶✮✳ ❈♦♠♦ ❡♠ ❇❡s❛❣ ❡ ◆❡✇❡❧ ✭✶✾✾✶✮✱ ✐♥✐❝✐❛✲s❡ ✜①❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦sy✳ ❆ s❡❣✉✐r✱ ❡♠ t♦r♥♦ ❞♦ ❝❡♥tró✐❞❡ ❞❡ ❝❛❞❛ ár❡❛Alq✉❡ ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠
❡✈❡♥t♦✱ tr❛ç❛✲s❡ ✉♠ ❝ír❝✉❧♦ q✉❡ ✈❛✐ s❡ ❡①♣❛♥❞✐♥❞♦ ♣♦r ❛✉♠❡♥t❛r♠♦s ♦ s❡✉ r❛✐♦ ❛té q✉❡ ♦ ❝ír❝✉❧♦ ❝♦♥t❡♥❤❛ ✉♠❛ ♣♦♣✉❧❛çã♦ ♣❛r❛ q✉❛❧ ❡s♣❡r❛✲s❡ ♦❜s❡r✈❛r y ❡✈❡♥t♦s✳ ❉❡♣♦✐s✱ ✈❡r✐✜❝❛✲s❡ q✉❛♥t♦s ❡✈❡♥t♦sXl ❢♦r❛♠ ❞❡ ❢❛t♦ ♦❜s❡r✈❛❞♦s ❡ ❝❛❧❝✉❧❛✲s❡ ❛ ❡st❛tíst✐❝❛
Uy = L
X
l=1
(Xl−y)I{x(sl)>0}.
❈✉③✐❝❦ ❡ ❊❞✇❛r❞s ✭✶✾✾✵✮ ❞❡r✐✈❛r❛♠ ❛s ❢ór♠✉❧❛s ❞♦s ♠♦♠❡♥t♦s ❞❡st❛ ❡st❛tíst✐❝❛ s♦❜ ❛ ❤✐♣ót❡s❡ ♥✉❧❛ ❡ ♠♦str❛r❛♠ q✉❡ ❡❧❛ ♣♦ss✉✐ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ♥♦r♠❛❧ ♣♦ss✐❜✐❧✐t❛♥❞♦ ❛ss✐♠ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r✲♣ ♣❛r❛ ♦ t❡st❡✳
✷✳✶✳✹ ▼ét♦❞♦ ❞❡ ❱❛rr❡❞✉r❛ ❊s♣❛❝✐❛❧✲❙❝❛♥ ❊s♣❛❝✐❛❧
❑✉❧❧❞♦r✛ ❡ ◆❛❣❛r✇❛❧❧❛ ✭✶✾✾✺✮ ❡ ❑✉❧❧❞♦r✛✭✶✾✾✼✮ ❛♣r❡s❡♥t❛♠ ✉♠ ♠ét♦❞♦ q✉❡ ❣❡♥❡r❛❧✐③❛ ♦s ❛♥t❡r✐♦r❡s ❡ ♣❡r♠✐t❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ t❡st❡ ♠ú❧t✐♣❧♦s✳ ❉❡♥♦t❡ ♣♦r Ξ ✉♠❛ ❝❧❛ss❡ ❞❡ s✉❜✲ ❝♦♥❥✉♥t♦s ❝♦♥❡①♦s ❝❛♥❞✐❞❛t♦s ❛ ❢♦r♠❛r❡♠ ❝❧✉st❡r✳ ◆❡st❡ t❡①t♦✱ ❞❡✜♥✐♠♦s ξ ={sl ∈ S :sl∈ξ}✳
❙✉♣♦♥❤❛ ♣♦r ❡①❡♠♣❧♦ q✉❡ξ é ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ r ❛r❜✐trár✐♦ ❝❡♥tr❛❞♦ ❡♠ ❝❛❞❛ ✉♠ ❞♦s L ❝❡♥✲ tró✐❞❡s sl✳ ❊♠ t❡s❡✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ t❛✐s ❝ír❝✉❧♦s ♠❛s✱ ♥❛ ♣rát✐❝❛✱ ❡st❡s ❝ír❝✉❧♦s
♣♦❞❡♠ s❡r r❡str✐t♦s ❛♣❡♥❛s ❛q✉❡❧❡s ❝❡♥tr❛❞♦s ❡♠ si ❡ ❝♦♠ r❛✐♦s ✐❣✉❛✐s ❛s ❞✐stâ♥❝✐❛s ✭❞✐stâ♥❝✐❛
❊✉❝❧✐❞✐❛♥❛✮ ❡♥tr❡si ❡ ♦ ❞❡♠❛✐s ❝❡♥tró✐❞❡s✳ ❖ ♠♦t✐✈♦ é q✉❡ ❝ír❝✉❧♦s ❝♦♠ r❛✐♦s ❧✐❣❡✐r❛♠❡♥t❡ ❞✐❢❡r✲
✷✳✶ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦ ✶✵
❖ ♠ét♦❞♦ ❞❡ ✈❛rr❡❞✉r❛ ❡s♣❛❝✐❛❧ ❞❡ ❑✉❧❧❞♦r✛✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❙❝❛♥✱ é ❜❛s❡❛❞♦ ♥❛ r❛③ã♦ ❞❡ ✈❡r♦ss✐❧❤❛♥ç❛ ❣❡♥❡r❛❧✐③❛❞❛✳ ❖ ♣❛râ♠❡tr♦ ♥❡st❡ ❝❛s♦ é (ξ, p0, p1) ♦♥❞❡ ξ ❞❡♥♦t❛ ✉♠
❝♦♥❥✉♥t♦ ❞❡ ❧♦❝❛❧✐③❛çõ❡s ❡s♣❛❝✐❛✐s✱ ♣❛r❛♠❡tr✐③❛❞♦ ♣❡❧♦ s❡✉ r❛✐♦ ❡ ❝♦♦r❞❡♥❛❞❛s ❞♦ ❝❡♥tr♦✱ p0 é ❛
♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r ✉♠ ❡✈❡♥t♦ ❡♠ξ❡ p1 ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r ✉♠ ❡✈❡♥t♦ ♥♦ ❝♦♥❥✉♥t♦
❝♦♠♣❧❡♠❡♥t❛r ❞❡ξ ❡♠S ❞❡♥♦t❛❞♦ ♣♦r ξ={sl∈ S :sl∈/ ξ}✳ ❆s ❤✐♣ót❡s❡s t❡st❛❞❛s sã♦❀
H0 :p0 =p1, ♣❛r❛ t♦❞♦ si ∈ S
❡
H0 :p0 > p1, ♣❛r❛ t♦❞♦ si ∈ξ
❖❜s❡r✈❡ q✉❡ s♦❜H0 ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❡✈❡♥t♦ é ❛ ♠❡s♠❛ ❡♠ q✉❛❧q✉❡r ár❡❛✳
❙❡❥❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ N(ξ) ♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❡ X(ξ) ♦ ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦s ❡♠ ξ✱ ❡ ❛✐♥❞❛✱ N ❡ X ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✐♥❞✐✈í❞✉♦s ❡ ❡✈❡♥t♦s ♥❛ r❡❣✐ã♦✳ ❆ ❢✉♥çã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♣❛r❛ ♦ ♣❛râ♠❡tr♦(ξ, p0, p1) é ❞❛❞❛ ♣♦r✱
L(ξ, p0, p1) =p0x(ξ)(1−p0)N(ξ)−x(ξ)px−x1 (ξ)(1−p1)N−N(ξ)−x+x(ξ).
❆ r❛③ã♦ ❞❡ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ✉♠ ✜①♦ξ é
RV(ξ) = sup
p0>p1
L(ξ, p0, p1)
sup
p0=p1
L(ξ, p0, p1)
, p0, p1 ∈(0,1).
❊♥tã♦ ❛ ❡st❛tíst✐❝❛ ❙❝❛♥ é ❞❡✜♥✐❞❛ ❝♦♠♦ T = max
ξ∈ΞRV(ξ)✱ ❡ ♦ ❝❧✉st❡r ❡s♣❛❝✐❛❧ ❡st✐♠❛❞♦ é
ˆ ξ = arg[max
ξ∈ΞRV(ξ)]✳ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ T ❞❡♣❡♥❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛ ♣♦♣✉❧❛çã♦ ❡ é ❞✐❢í❝✐❧ ❞❡ s❡r
♦❜t✐❞❛ ❛♥❛❧✐t✐❝❛♠❡♥t❡✳ ❘❡❝❡t❡♠❡♥t❡✱ ❛❧❣✉♠❛s ❛♣r♦①✐♠❛çõ❡s tê♠ s✐❞♦ ♣r♦♣♦st❛s ❡♠ ❛❧❣✉♥s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✳ ❆ss✐♠✱ ❛ s♦❧✉çã♦ ♣r♦♣♦st❛ é ✉❧t✐❧✐③❛r ♠ét♦❞♦s ❞❡ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ♣❛r❛ ♦❜t❡r ❛ ❞✐str✐❜✉✐çã♦ ❡♠♣ír✐❝❛ ❞❡ T ❝♦♥❞✐❝✐♦♥❛❞❛ ♥♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❡✈❡♥t♦s ♦❜s❡r✈❛❞♦s X✳ ❉❡st❛ ❢♦r♠❛ ♦ ✈❛❧♦r✲♣ ❡♠♣ír✐❝♦ é ♦❜t✐❞♦ ❛♣ós ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ s✐♠✉❧❛çõ❡s ❞♦ ♣r♦❝❡ss♦ ✭♣♦r ❡①❡♠♣❧♦✱ ✶✵✵✵ ✈❡③❡s✮ s♦❜ ❛ ❤✐♣ót❡s❡ ♥✉❧❛✳ ❊①✐st❡♠ ✈ár✐❛s ❡①t❡♥sõ❡s ♣❛r❛ ❡st❡ ♠ét♦❞♦ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ❙❝❛♥ ❊❧✐♣t✐❝♦ ✭❑✉❧❧❞♦r❢ ❡t ❛❧❧✱ ✷✵✵✻ ✮✱ ❙❝❛♥ ✐rr❡❣✉❧❛r ✭t❛♥❣♦ ❡ ❚❛❦❛❤❛s❤✐✱ ✷✵✵✺✮✳ ❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ ✈❛rr❡❞✉r❛ ❡s♣❛❝✐❛❧ r❡❛❧✐③❛❞❛ ♣❡❧❛ ❡st❛tíst✐❝❛ ❙❝❛♥ é ♠♦str❛❞❛ ♥❛ ✜❣✉r❛ ✭✷✳✷✮
✷✳✶ ❉❡t❡❝çã♦ ❞❡ ❝❧✉st❡rs ♥♦ ❡s♣❛ç♦ ✶✶
✷✳✶✳✺ ▼ét♦❞♦ ❇❛②❡s✐❛♥♦
❖s ♠ét♦❞♦s ❞❡s❝r✐t♦s ❛♥t❡r✐♦r♠❡♥t❡ sã♦ ✉s❛❞♦s ♣❛r❛ ❞❡t❡❝t❛r ✉♠ ú♥✐❝♦ ❝❧✉st❡r✱ ❡ ♥ã♦ ❧❡✈❛♠ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ✐♥❝❡rt❡③❛ s♦❜r❡ ❛ ❡str✉t✉r❛ ✭❢♦r♠❛ ❡ t❛♠❛♥❤♦✮ ❞♦ ❝❧✉st❡r✳ ❈♦♠ ✐♥t✉✐t♦ ❞❡ s♦❧✉❝✐♦♥❛r ❡st❡ ♣r♦❜❧❡♠❛ ❞♦ ♣♦♥t♦ ❞❡ ❇❛②❡s✐❛♥♦✱ ●❛♥❣♥♦♥ ❡ ❈❧❛②t♦♥ ✭✷✵✵✵✮ ✉t✐❧✐③❛♠ ✉♠❛ ♠♦❞✲ ❡❧❛❣❡♠ ❜❛②❡s✐❛♥❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡❝ã♦ ❞❡ ❝❧✉st❡r✳ ❇❛s✐❝❛♠❡♥t❡✱ ♦s ❛✉t♦r❡s ❛ss✉♠❡♠ q✉❡ X(sl)|λ(sl)∼P oisson(λ(sl)N(sl))✱ ❡♠ q✉❡ ❛ ✐♥❝❡rt❡③❛ s♦❜r❡λ(sl)❡ ♦ ❝❧✉st❡r é ❞❡s❝r✐t❛ ❛tr❛✈és
❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❙✉♣♦♥❤❛ q✉❡ ♦ ✐♥t❡r❡ss❡ é ❞❡t❡❝t❛r ❝❧✉st❡r ❛tr❛✈és ❞❡ ♠✉❞❛♥ç❛s ♥❛ ❞✐str✐❜✉✐çã♦ ❞♦ ✈❡t♦r(λ(s1), ..., λ(sL))✳ ❊♥tã♦ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❜❛②❡s✐❛♥❛ ❝♦♠ ❞✐s✲
tr✐❜✉✐çõ❡s ❛ ♣r✐♦r✐ ❤✐❡rárq✉✐❝❛s é ❝♦♥str✉í❞❛✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦ ❡s♣❛ç♦ é ❞✐✈✐❞♦ ❡♠c+ 1❣r✉♣♦s ❝♦♠ ✈ár✐♦s ❝♦♠♣♦♥❡♥❡t❡s ♦✉ ár❡❛s✳ ❞❡♥♦t❡ ✉♠ ❞❡ss❡ ❣r✉♣♦s ♣♦r ❣r✉♣♦ ♣❛❞rã♦ξ0 ✭♥♦ s❡♥t✐❞♦ ❞❡
q✉❡ ❡st❡s ❝♦♠♣♦♥❡♥t❡s ♥ã♦ ❢♦r♠❛♠ ✉♠ ❝❧✉st❡r✮ ❡ ♦s ❞❡♠❛✐sb❣r✉♣♦s ❞❡ ❝❧✉st❡rs✳ ■❞❡♥t✐✜q✉❡ ✉♠ ♠♦❞❡❧♦ ❝♦♠b❝❧✉t❡rs ♣♦r ✉♠ ✈❡t♦r ξ= (ξ1, ..., ξL) ♦♥❞❡ ξl= 0s❡ sl ∈ξ0 ❡ ξl=j s❡sl ♣❡rt❡♥❝❡
❛♦ ❝❧✉st❡r j = 1,2, ..., c.✳ ❊st❛ ♠♦❞❡❧❛❣❡♠ é ✐♠♣♦rt❛♥t❡ ♣♦✐s ♣❡r♠✐t❡ ✉♠❛ ú♥✐❝❛ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛ q✉❛❧q✉❡rξ✳ ❆❣♦r❛✱ ❞❛❞♦ ✉♠ ξ ❛ss✉♠❛ q✉❡ ❝❛❞❛ sl ♣❡rt❡♥❝❡♥t❡ ❛♦ ❝❧✉st❡rj ♣♦ss✉✐ t❛①❛ ❞❡
❡✈❡♥t♦sλj ❡ s❡❥❛Λ = (λ0, λ1, ..., λc)✳
❆ss✉♠❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❛ ♣r✐♦r✐ ♣❡r♠✉tá✈❡❧ ♣❛r❛(λ1, ..., λc)✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❛ss✉♠❛
q✉❡ λ0|ξ ∼ gama(a0, b0) ❡ λj|ξ ∼ gama(a, b) ♦♥❞❡ ❣❛♠❛✭❛✱❜✮ r❡♣r❡s❡♥t❛ ❛ ❞✐str✐❜✉✐çã♦ ❣❛♠❛
❝♦♠ ♠é❞✐❛ a/b ❡ ✈❛r✐â♥❝✐❛ a/b2✳ ❉❛❞♦ X(s) = (X(s
1), X(s2), ..., X(sL)) ❡ ξ✱ λ0, λ1, ..., λc sã♦
✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ❞✐str✐❜✉✐çõ❡s
λ0|X(s),ξ∼gama(a0+X0, b0+N0)
❡
λj|X(s),ξ∼gama(a+Xj, b+Nj), j= 1,2, ..., c.
♦♥❞❡✱Xj =PLl=1X(sl)I{ξl=j} é ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❡✈❡♥t♦s ♥♦ ❝❧✉st❡r j✱Nj =PLl=1N(sl)I{ξl=j}
é ♦ t♦t❛❧ ❞❡ ✐♥❞✐✈í❞✉♦s ♥♦ ❝❧✉st❡rj✱ ❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ♠❛r❣✐♥❛❧ ❞❡ X(s)|ξ é ❞❛❞❛ ♣♦r✱
p(X(s)|ξ) = Z
Λ
f(X(s)|Λ)π(Λ|ξ)dΛ
♦♥❞❡f é ❢✉♥çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡X(s)|Λ ❡π é ❛ ❞✐str✐❜✉✐çã♦ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡Λ|ξ✳ ❆ ❞✐str✐❜✉✐çã♦
❛ ♣r✐♦r✐ ♣❛r❛ξ é ❞❛ ❢♦r♠❛
π(ξ)∝exp
−
c
X
j=1
Sj
,
♦♥❞❡ Sj é ❡s❝♦r❡ ♣❛r❛ ♦ ❝❧✉st❡r j ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s✭❢♦r♠❛✱ t❛♠❛♥❤♦✮ ❞♦
❝❧✉st❡r✳ ❉❡st❛ ❢♦r♠❛ ❛ ❞✐str✐❜✉✐çã♦ ❛ ♣♦st❡r✐♦r✐ ❞❡ξ é ♦❜t✐❞❛ ♣♦r✱
π(ξ|X(s))∝p(X(s)|ξ)π(ξ).
