■♥tr♦❞✉❝t✐♦♥ P❛rt ■✿ ■♥tr♦❞✉❝t✐♦♥ ▸ ❘✲P❛❝❦❛❣❡ ♣r❡❢♠♦❞ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✉t✐❧✐t✐❡s t♦ ✜t ❛ ✈❛r✐❡t② ♦❢ ♣❛✐r❡❞ ❝♦♠♣❛r✐s♦♥ ♠♦❞❡❧s ▸ ♣r❡❢❡r❡♥❝❡ ♠♦❞❡❧s ❜❛s❡❞ ♦♥ ♣❛✐r❡❞ ❝♦♠♣❛r✐s♦♥s ♣r❡❢♠♦❞✿ ♥❡✇s ❛♥❞ ❡①t❡♥s✐♦♥s ♦❜❥❡❝t✐✈❡ ✐s t♦ ❡st❛❜❧✐s❤ ❛ ♣r❡❢❡r❡♥❝❡ s❝❛❧❡ ❢♦r ❝❡rt❛✐♥ ♦❜❥❡❝ts ❘❡✐♥❤♦❧❞ ❍❛t③✐♥❣❡r ✫ ❘❡❣✐♥❛ ❉✐ttr✐❝❤ ✕ ❢♦♦❞✱ ❝r✐♠❡s✱ ♣❛✐♥✱ t❡❛❝❤✐♥❣ st②❧❡s✱ ♣♦rt❢♦❧✐♦s✱ ✳ ✳ ✳ ■♥st✐t✉t❡ ❢♦r ❙t❛t✐st✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s ▸ ♣❛✐r❡❞ ❝♦♠♣❛r✐s♦♥s ❲❯ ❱✐❡♥♥❛ J ♦❜❥❡❝ts ❛r❡ ❝♦♠♣❛r❡❞ ✐♥ ♣❛✐rs ❛❝❝♦r❞✐♥❣ t♦ ❛ s♣❡❝✐✜❝ ❛ttr✐❜✉t❡ ✕ t❛st❡s ❜❡tt❡r✱ ♠❛❦❡s ♠❡ ♣✉t ♦♥ ♠♦r❡ ✇❡✐❣❤t✱ ✳ ✳ ✳ ✇❡ ♦❜s❡r✈❡ ( J 2 ) ❝♦♠♣❛r✐s♦♥s ✭r❡s♣♦♥s❡s✮ Ps②❝❤♦❝♦ ✷✵✶✶ ✶ Ps②❝❤♦❝♦ ✷✵✶✶ ✷ ■♥tr♦❞✉❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ▼♦❞❡❧ ■♥❞❡♣❡♥❞❡♥❝❡✿ ▲▲❇❚ ✭❧♦❣❧✐♥❡❛r ❇r❛❞❧❡②✲❚❡rr② ♠♦❞❡❧✮ ❝♦r❡ ♠♦❞❡❧ ✐♥ ♣r❡❢♠♦❞ ✐s t❤❡ ❇r❛❞❧❡②✲❚❡rr② s♣❡❝✐✜❝❛t✐♦♥ ✇❡ ✉s❡ t❤❡ ❧♦❣❧✐♥❡❛r r❡♣r❡s❡♥t❛t✐♦♥ ✭❆♣♣❧✐❡❞ ❙t❛t✐st✐❝s✱ ✶✾✾✽✮ P { Y jk = 1 ∣ π j ,π k } = P { Y jk = − 1 ∣ π j ,π k } = π j π k π j + π k ♦r π j + π k ln m ( y jk ) = µ ( jk ) + y jk ( λ j − λ k ) Y jk = 1 ✳ ✳ ✳ ♦❜❥❡❝t j ♣r❡❢❡rr❡❞ t♦ k ✱ Y jk = − 1 ✳ ✳ ✳ ♦❜❥❡❝t k ♣r❡❢❡rr❡❞ t♦ j π j ✳ ✳ ✳ ❧♦❝❛t✐♦♥ ♦❢ ♦❜❥❡❝t j ♦♥ ♣r❡❢❡r❡♥❝❡ s❝❛❧❡ ❞❡s✐❣♥ str✉❝t✉r❡ ❢♦r ✸ ♦❜❥❡❝ts✿ ✐♥❞❡♣❡♥❞❡♥❝❡ ♠♦❞❡❧ ✭❇r❛❞❧❡②✲❚❡rr②✮✿ r❡s♣♦♥s❡ ✐s y jk µ λ 1 λ 2 λ 3 √ π j ❝♦♠♣❛r✐s♦♥ ❞❡❝✐s✐♦♥ ❝♦✉♥ts ❝♦♥st y 12 y 13 y 23 y jk n ( 1 ≻ 2 ) p ( y jk ) = c ( ) ✭✶✷✮ ✶ ✶ ✲✶ ✵ √ π k O 1 n ( 2 ≻ 1 ) ✭✶✷✮ ✶ ✲✶ ✶ ✵ O 2 n ( 1 ≻ 3 ) ✭✶✸✮ ✷ ✶ ✵ ✲✶ O 1 n ( 3 ≻ 1 ) ✭✶✸✮ ✷ ✲✶ ✵ ✶ O 3 ♣❛tt❡r♥ ♠♦❞❡❧✿ r❡s♣♦♥s❡ ✐s y = { y 12 ,y 13 ,...,y jk ,...,y J − 1 ,J } n ( 2 ≻ 3 ) ✭✷✸✮ O 2 ✸ ✵ ✶ ✲✶ √ π j n ( 3 ≻ 2 ) ✭✷✸✮ ✸ ✵ ✲✶ ✶ O 3 y jk p ( y 12 ,...,y J − 1 ,J ) = c ∏ ( ) √ π k ❢❛❝t♦r ❢♦r ♥♦r♠❛❧✐③✐♥❣ ❝♦♥st❛♥ts µ j < k Ps②❝❤♦❝♦ ✷✵✶✶ ✸ Ps②❝❤♦❝♦ ✷✵✶✶ ✹
obje t obje t ■♥tr♦❞✉❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ p rop erties p rop erties P❛tt❡r♥ ♠♦❞❡❧ ❊①t❡♥s✐♦♥s ❢♦r s✉❜❥❡❝t ❛♥❞ ♦❜❥❡❝t ❡✛❡❝ts ❧♦❣❧✐♥❡❛r ♠♦❞❡❧ ✭❈❙❉❆✱ ✷✵✵✷✮ j − 1 ⎛ ⎞ ln m ( y 12 ,...,y J − 1 ,J ) = η y = µ + J λ j x j = µ + J J y jν − ∑ ∑ ∑ ∑ O 2 ⎝ ⎠ O 4 λ j y νj j = 1 j = 1 ν = j + 1 ν = 1 Preferen e C 1 O 1 subje t C 2 e�e ts O 3 O 4 C 2 O 2 ❞❡s✐❣♥ str✉❝t✉r❡ ❢♦r ✸ ♦❜❥❡❝ts✿ C 1 O 3 O 1 µ λ 1 λ 2 λ 3 ♣❛tt❡r♥ y 12 y 13 y 23 ❝♦✉♥ts ❝♦♥st x 1 x 2 x 3 ℓ 1 1 1 1 n 1 1 ✷ ✵ ✲✷ − 1 ℓ 2 n 2 ✷ ✲✷ ✵ 1 1 1 − 1 ℓ 3 1 1 n 3 1 ✵ ✵ ✵ − 1 − 1 ✵ ✲✷ ✷ ℓ 4 1 n 4 1 − 1 s✉❜❥❡❝t ❡✛❡❝ts✿ ❞✉♣❧✐❝❛t❡ t❛❜❧❡ ❢♦r ❡❛❝❤ ❝♦✈❛r✐❛t❡ ❣r♦✉♣ s ℓ 5 1 1 n 5 1 ✵ ✷ ✲✷ ♦❜❥❡❝t ❡✛❡❝ts✿ λ j = ∑ q β C − 1 − 1 ✵ ✵ ✵ ℓ 6 1 n 6 1 q x jq − 1 − 1 ℓ 7 1 n 7 1 ✲✷ ✷ ✵ − 1 − 1 − 1 b jq ✳ ✳ ✳ ❝♦✈❛r✐❛t❡ ❢♦r ❝❤❛r❛❝t❡r✐st✐❝ C q ✲✷ ✵ ✷ ℓ 8 n 8 1 β C q ✳ ✳ ✳ ❡✛❡❝t ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ C q x j ❂ ★✭ O j ✐s ♣r❡❢❡rr❡❞ ✐♥ ℓ ✮ ✲ ★✭ O j ♥♦t ♣r❡❢❡rr❡❞ ✐♥ ℓ ✮ Ps②❝❤♦❝♦ ✷✵✶✶ ✺ Ps②❝❤♦❝♦ ✷✵✶✶ ✻ ■♥tr♦❞✉❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❊①t❡♥s✐♦♥s✿ ❖✈❡r✈✐❡✇ ❉❡r✐✈❡❞ ♣❛✐r❡❞ ❝♦♠♣❛r✐s♦♥s✿ ❊①❛♠♣❧❡✿ r❛♥❦✐♥❣ ✇✐t❤ ✸ ♦❜❥❡❝ts ❡①t❡♥s✐♦♥s ❢♦r ▲▲❇❚ ❛♥❞ ♣❛tt❡r♥ ♠♦❞❡❧ ✇❡ tr❛♥s❢♦r♠ r❛♥❦✐♥❣s t♦ ♣❛✐r❡❞ ❝♦♠♣❛r✐s♦♥s ● ✉♥❞❡❝✐❞❡❞ ✭ 3 ( J 2 ) ❞✐✛❡r❡♥t ♣❛tt❡r♥s✮✱ ♣♦s✐t✐♦♥ ❡✛❡❝ts ● s✉❜❥❡❝t ❝♦✈❛r✐❛t❡s✱ ♦❜❥❡❝t s♣❡❝✐✜❝ ❝♦✈❛r✐❛t❡s ❛❞❞✐t✐♦♥❛❧ ❡①t❡♥s✐♦♥s ❢♦r ♣❛tt❡r♥ ♠♦❞❡❧s ✇❡ ❝❛♥ ❣✐✈❡ ✉♣ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❞❡❝✐s✐♦♥s ● ❞❡♣❡♥❞❡♥❝❡ ♣❛r❛♠❡t❡rs θ ( jk )( jl ) ✭✐♥t❡r❛❝t✐♦♥s✮ ❢♦r ♣❛✐rs ♦❢ ❝♦♠♣❛r✐s♦♥s ✇✐t❤ ♦♥❡ ♦❜❥❡❝t ✐♥ ❝♦♠♠♦♥ ❛♥❞ ✇❡ ❝❛♥ ❛❧s♦ ❞❡❛❧ ✇✐t❤ ✈❛r✐♦✉s ♦t❤❡r r❡s♣♦♥s❡ ❢♦r♠❛ts ● r❛♥❦✐♥❣ ❞❛t❛ 2 ) = 8 ● ♥✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s ✐s 3! = 6 ❝♦♠♣❛r❡❞ t♦ 2 ( 3 ● r❛t✐♥❣ ✭▲✐❦❡rt✮ ❞❛t❛ ✭✏r❛♥❦✐♥❣s ✇✐t❤ t✐❡s✑✮ ● ♣❛tt❡r♥ ♠♦❞❡❧ ❜❛s❡❞ ♦♥ r❡❞✉❝❡❞ ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥t ♣❛tt❡r♥s ● ♣✐❧✐♥❣✱ ♠✉❧t✐♣❧❡ r❡s♣♦♥s❡s✱ ✳ ✳ ✳ ● ✉s✐♥❣ t❤❡ ▲▲❇❚ ❧❡❛❞s t♦ ❜✐❛s❡❞ ❡st✐♠❛t❡s ❢♦r t❤❡ λ ✬s → Ps②❝❤♦❝♦ ✷✵✶✶ ✼ Ps②❝❤♦❝♦ ✷✵✶✶ ✽
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