Inférence statistique dans le modèle de régression logistique avec fraction immune

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Submitted on 3 Jun 2013

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Inférence statistique dans le modèle de régression logistique avec fraction immune

Aba Diop

To cite this version:

Aba Diop. Inférence statistique dans le modèle de régression logistique avec fraction immune. Math- ématiques générales [math.GM]. Université de La Rochelle; Université Gaston Berger de Saint-Louis Sénégal, 2012. Français. �NNT : 2012LAROS375�. �tel-00829844�

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(✉✐ ♦♥- ❝♦♥-$✐❜✉3 0 ❧❛ $3❛❧✐&❛-✐♦♥ ❞❡ ❝❡ -$❛✈❛✐❧✳

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&❛ $3❛❧✐&❛-✐♦♥✳

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❇❊❙❙❊ ❡- ❆✈♥❡$ ❇❆❘✲❍❊◆✱ ♣♦✉$ ❛✈♦✐$ ❛❝❝❡♣-3 ❞❡ ❧✐$❡ ❝❡--❡ -❤9&❡ ❡- ♠✬❛✈♦✐$ ❢❛✐-

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❢❡&&❡✉$& ❊♠♠❛♥✉❡❧❧❡ ❆❯●❊❘❆❯❉✲❱❊❘❖◆✱ ❙♦♣❤✐❡ ❉❆❇❖✲◆■❆◆● ❡- ●❛♥❡ ❙❛♠❜

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✈♦♥- 0 ❧✬❡♥❞$♦✐- ❞✉ ❉♦❝-❡✉$ ▲❛✉$❡♥❝❡ ❇❆❘■▲ ♣♦✉$ ♥♦♥ &❡✉❧❡♠❡♥- ❛✈♦✐$ ❛❝❝❡♣-3 ❞❡

❢❛✐$❡ ♣❛$-✐❡ ❞❡ ♠♦♥ ❥✉$② ♠❛✐& ❛✉&&✐ ♣♦✉$ ♠✬❛✈♦✐$ ❢❛✐- ❜3♥3✜❝✐❡$ ❞❡ &❛ ❧❛$❣❡ ❝♦♥♥❛✐&✲

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$3❡❧❧❡& &✉$ ❧❡&(✉❡❧❧❡& ❝❡--❡ -❤9&❡ ❛ 3-3 ❛♣♣❧✐(✉3❡✳

❏❡ $❡♠❡$❝✐❡ ♣❛$-✐❝✉❧✐9$❡♠❡♥- ❧❡ ❙❡$✈✐❝❡ ❞❡ ❈♦♦♣3$❛-✐♦♥ ❡- ❞✬❆❝-✐♦♥& ❈✉❧-✉$❡❧❧❡& ❞❡

❧✬❆♠❜❛&&❛❞❡ ❞❡ ❋$❛♥❝❡ ❛✉ ❙3♥3❣❛❧✱ ❞✬❆■❘❊❙✲❙❯❉ ❡- ❙❚❆❋❆❱ ♣♦✉$ ❛✈♦✐$ ♣❛$-✐❝✐♣3

❣$❛♥❞❡♠❡♥- ❛✉ ✜♥❛♥❝❡♠❡♥- ❞❡ ❝❡--❡ -❤9&❡✳

❏❡ ✈♦✉❞$❛✐& $❡♠❡$❝✐❡$ -♦✉- ❧❡ ♣❡$&♦♥♥❡❧ ❡♥&❡✐❣♥❛♥- ❡- ❛❞♠✐♥✐&-$❛-✐❢ ❞❡ ❧✬❯❋❘ ❙❝✐❡♥❝❡&

❆♣♣❧✐(✉3❡& ❡- ❚❡❝❤♥♦❧♦❣✐❡ ❞❡ ❧✬❯♥✐✈❡$&✐-3 ●❛&-♦♥ ❇❡$❣❡$✳ ▼❡& $❡♠❡$❝✐❡♠❡♥-& ✈♦♥- 3❣❛❧❡♠❡♥- 0 ❧✬❡♥❞$♦✐- ❞❡ -♦✉& ❧❡& ♠❡♠❜$❡& ❞✉ ▲❛❜♦$❛-♦✐$❡ ▼❛-❤3♠❛-✐(✉❡& ■♠❛❣❡ ❡-

❆♣♣❧✐❝❛-✐♦♥& ❞❡ ❧✬❯♥✐✈❡$&✐-3 ❞❡ ▲❛ ❘♦❝❤❡❧❧❡ ♣♦✉$ ❧❡✉$ ❛❝❝✉❡✐❧ ❝❤❛❧❡✉$❡✉① ❛✉ ❝♦✉$&

✐✈

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.❡✉' ❋❛.♦✉♠❛.❛ ❉✐❡♥❡ ❙❛''✳

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"♦✉.✐❡♥ ♠♦'❛❧✱ ♣♦✉' ❧❡" #❝❤❛♥❣❡" ❢'✉❝.✉❡✉① ❧♦'" ❞❡" "#♠✐♥❛✐'❡" ❡. ❞✐"❝✉""✐♦♥" ❡. ❛✈❡❝

0✉✐ ❥✬❛✐ ♣❛'.❛❣# ♠❡" ❛♥❣♦✐""❡"✱ ♠❡" 'Q✈❡" ❡. ❞❡" ♠♦♠❡♥." ✐♥♦✉❜❧✐❛❜❧❡"✳

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♣'♦❝❤❡"✳ ❯♥ ❣'❛♥❞ ♠❡'❝✐ M ♠♦♥ ❝❤❡' ❜❡❛✉ ❢'J'❡ ▼❛♠❛❞♦✉ ❚❍■❆▼✱ M ♠♦♥ ❝❤❡' .♦♥" ❈❤❡✐❦❤ ▼❛❦❤.❛' ❑❡U.❛✱ M ♠❡" ❛❞♦'❛❜❧❡" ❡. ❛✐♠❛❜❧❡" "♦❡✉'" ❚♦✉.✐ ❉■❖G✱ ❉✐❛'②

❉■❖G✱ ▼❛❣❛..❡ ❉■❖G ❡. ◆❞#②❡ ▼❡U""❛ ❉■❖G ✱ M ♠❡" ❝❤❡'" ❢'J'❡" G❛♣❡ ❆♠❛❞♦✉

❉■❖G ❡. ❇❛②❧❛ ❉■❖G ❡. ❡♥✜♥ M ♠❛ ❝❤J'❡ ❜✐❡♥✲❛✐♠# ❆✇❛ ▲② ❉■❖❯▼✳

■❧ ♠✬❡". ❜✐❡♥ "Z' ✐♠♣♦""✐❜❧❡ ❞❡ .❡'♠✐♥❡' ❝❡" '❡♠❡'❝✐❡♠❡♥." "❛♥" ❛✈♦✐' ✉♥❡ ♣❡♥"#❡ M .♦✉" ♠❡" ❝❛♠❛'❛❞❡" ❞❡ ♣'♦♠♦.✐♦♥ ✷✵✵✽ ❞❡ ❉❊❆ ❙❚❆❋❆❱✳

❚❛❜❧❡ ❞❡& ♠❛(✐*+❡&

❘!"✉♠! ①✐

❆❜")*❛❝) ①✐✐✐

❆❜*!✈✐❛)✐♦♥" ❡) ◆♦)❛)✐♦♥" ①✈

✶ ■♥)*♦❞✉❝)✐♦♥ ❣!♥!*❛❧❡ ✶

■ ❘❛♣♣❡❧& &✉( )✉❡❧)✉❡& ♠♦❞-❧❡& ✶

✷ ❘❛♣♣❡❧" "✉* ❧❡ ♠♦❞9❧❡ ❞❡ *!❣*❡""✐♦♥ ❧♦❣✐")✐:✉❡ ✸

✷✳✶ ▲❡ ♠♦❞(❧❡ ❧✐♥,❛✐.❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✷✳✷ ▲❡ ♠♦❞(❧❡ ❞❡ .,❣.❡11✐♦♥ ❧♦❣✐12✐3✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✷✳✶ ■♥2.♦❞✉❝2✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✷✳✷ ■❞❡♥2✐✜❛❜✐❧✐2, ❞✉ ♠♦❞(❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷✳✷✳✸ ❊12✐♠❛2✐♦♥ ❞✉ ♠♦❞(❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✷✳✹ ❘❛♣♣❡❧ 1✉. ❧✬❛❧❣♦.✐2❤♠❡ ❞❡ ◆❡✇2♦♥✲❘❛♣❤1♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✷✳✺ F.♦♣.✐,2,1 ❛1②♠♣2♦2✐3✉❡1 ❞✉ ♠♦❞(❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✷✳✻ ■♥2❡.♣.,2❛2✐♦♥ ❞❡1 ❝♦❡✣❝✐❡♥21 ❞❡ .,❣.❡11✐♦♥ ❡2 ♦❞❞" #❛%✐♦ ✳ ✳ ✳ ✶✸

✸ ▼♦❞9❧❡" ❞❡ *!❣*❡""✐♦♥ ❛✈❡❝ ✐♥✢❛)✐♦♥ ❞❡ ③!*♦" ✶✼

✸✳✶ ▼♦❞(❧❡1 ❞❡ .,❣.❡11✐♦♥ ❩■F ❡2 ❩■◆❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✸✳✶✳✶ ▼♦❞(❧❡1 ❞❡ .,❣.❡11✐♦♥ ❞❡ F♦✐11♦♥ ❡2 ❇✐♥♦♠✐❛❧ ◆,❣❛2✐❢ ✳ ✳ ✳ ✳ ✳ ✶✾

✸✳✶✳✷ ▼♦❞(❧❡1 ❞❡ .,❣.❡11✐♦♥ ❩■F ❡2 ❩■◆❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸✳✶✳✸ F.♦♣.✐,2,1 ❛1②♠♣2♦2✐3✉❡1 ❞✉ ♠♦❞(❧❡ ❩■F ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸✳✷ ▼♦❞(❧❡ ❞❡ .,❣.❡11✐♦♥ ❩■❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

❚❛❜❧❡ ❞❡& ♠❛(✐*+❡& ✈✐✐✐

✸✳✷✳✶ ▼♦❞'❧❡ ❜✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✸✳✷✳✷ ❉1✜♥✐3✐♦♥ ❞✉ ♠♦❞'❧❡ ❩■❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸✳✷✳✸ ■❞❡♥3✐✜❛❜✐❧✐31 ❞❡ ♠♦❞'❧❡ ❞❡ 91❣9❡;;✐♦♥ ❩■❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸✳✸ ▼♦❞'❧❡ ❞❡ 91❣9❡;;✐♦♥ ❩■=❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

■■ ▼♦❞$❧❡ ❞❡ '(❣'❡**✐♦♥ ❧♦❣✐*-✐.✉❡ ❛✈❡❝ ❢'❛❝-✐♦♥ ✐♠♠✉♥❡ ✷✾

✹ ❊&(✐♠❛(✐♦♥ ♣❛+ ♠❛①✐♠✉♠ ❞❡ ✈+❛✐&❡♠❜❧❛♥❝❡ ❞❛♥& ❧❡ ♠♦❞*❧❡ ❞❡ +5✲

❣+❡&&✐♦♥ ❧♦❣✐&(✐8✉❡ ❛✈❡❝ ❢+❛❝(✐♦♥ ✐♠♠✉♥❡ ✸✶

✹✳✶ ■♥39♦❞✉❝3✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✹✳✷ ▲♦❣✐;3✐❝ 9❡❣9❡;;✐♦♥ ✇✐3❤ ❛ ❝✉9❡ ❢9❛❝3✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✹✳✷✳✶ ◆♦3❛3✐♦♥; ❛♥❞ 3❤❡ ♠♦❞❡❧ ;❡3✲✉♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✹✳✷✳✷ ❚❤❡ ♣9♦♣♦;❡❞ ❡;3✐♠❛3✐♦♥ ♣9♦❝❡❞✉9❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✹✳✷✳✸ ❙♦♠❡ ❢✉93❤❡9 ♥♦3❛3✐♦♥; ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✹✳✸ ■❞❡♥3✐✜❛❜✐❧✐3② ❛♥❞ 9❡❣✉❧❛9✐3② ❝♦♥❞✐3✐♦♥; ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✹✳✹ ❆;②♠♣3♦3✐❝ 3❤❡♦9② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹✳✺ ❆ ;✐♠✉❧❛3✐♦♥ ;3✉❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✹✳✺✳✶ ❙3✉❞② ❞❡;✐❣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✹✳✺✳✷ ❘❡;✉❧3; ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✹✳✻ ❉✐;❝✉;;✐♦♥ ❛♥❞ ♣❡9;♣❡❝3✐✈❡; ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✺ ❇❛♥❞❡& ❞❡ ❝♦♥✜❛♥❝❡ &✐♠✉❧(❛♥5❡& ❞❛♥& ❧❡ ♠♦❞*❧❡ ❞❡ +5❣+❡&&✐♦♥ ❧♦✲

❣✐&(✐8✉❡ ❛✈❡❝ ❢+❛❝(✐♦♥ ✐♠♠✉♥❡ ✻✺

✺✳✶ ■♥39♦❞✉❝3✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✺✳✷ ▼♦❞'❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✺✳✷✳✶ ◆♦3❛3✐♦♥; ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

✺✳✷✳✷ ❙✉♣9❡♠✉♠ ❞❡ ♣9♦❝❡;;✉; ●❛✉;;✐❡♥; ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

✺✳✸ ❇❛♥❞❡; ❞❡ ❝♦♥✜❛♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷

✺✳✸✳✶ ▼13❤♦❞❡ ✶ ✿ ▼13❤♦❞❡ ❞❡ ❙❝❤❡✛1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

❚❛❜❧❡ ❞❡& ♠❛(✐*+❡& ✐①

✺✳✸✳✷ ▼%&❤♦❞❡ ✷ ✿ ➱❣❛❧✐&% ❞❡ ▲❛♥❞❛✉ ❡& ❙❤❡❡♣ ✭✶✾✼✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

✺✳✸✳✸ ▼%&❤♦❞❡ ✸ ✿ ❇♦♦&=&>❛♣ ✲ ▼♦♥&❡ ❈❛>❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

✺✳✹ ➱&✉❞❡ ❞❡ =✐♠✉❧❛&✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

✺✳✹✳✶ D❧❛♥ ❞❡ =✐♠✉❧❛&✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

✺✳✹✳✷ ❘%=✉❧&❛&= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷

■■■ ❆♣♣❧✐❝❛'✐♦♥* ✽✾

✻ ➱(✉❞❡ 0♣✐❞0♠✐♦❧♦❣✐4✉❡ ❞❡ ❧❛ ❞❡♥❣✉❡ ✿ ♣+♦❥❡( ❉❊◆❋❘❆▼❊ ✾✶

✻✳✶ ■♥&>♦❞✉❝&✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸

✻✳✷ ▼❡&❤♦❞= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺

✻✳✷✳✶ ❖❜❥❡❝&✐✈❡= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺

✻✳✷✳✷ ❙&✉❞② =✐&❡= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺

✻✳✷✳✸ ❙&✉❞② ❞❡=✐❣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻

✻✳✷✳✹ ❈❧✐♥✐❝❛❧ ❞❛&❛ ❛♥❞ ❜❧♦♦❞ =❛♠♣❧❡ ❝♦❧❧❡❝&✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼

✻✳✷✳✺ ❈❧❛==✐✜❝❛&✐♦♥ ♦❢ ❞❡♥❣✉❡ ❝❛=❡= ♦♥ &❤❡ ❜❛=✐= ♦❢ ❛❝✉&❡ ❞❡♥❣✉❡

❞✐❛❣♥♦=✐= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽

✻✳✷✳✻ ❊&❤✐❝= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾

✻✳✷✳✼ ❙&❛&✐=&✐❝❛❧ ♠❡&❤♦❞= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾

✻✳✸ ❘❡=✉❧&= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵

✻✳✹ ❉✐=❝✉==✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺

✻✳✺ ❆♣♣❡♥❞✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶

✼ ❆♣♣❧✐❝❛(✐♦♥ ❛✉① ❞♦♥♥0❡& ❞❡ ❉❊◆❋❘❆▼❊ ✶✶✾

✼✳✶ ■♥&>♦❞✉❝&✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

✼✳✷ ❉❡=❝>✐♣&✐♦♥ ❞❡= ❞♦♥♥%❡= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵

✼✳✸ ▼♦❞T❧❡= ❡& >%=✉❧&❛&= ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✷

❈♦♥❝❧✉&✐♦♥ ❡( ♣❡+&♣❡❝(✐✈❡& ✶✷✺

❚❛❜❧❡ ❞❡& ♠❛(✐*+❡& ①

■❱ ❆♥♥❡①❡& ✶✷✾

❆ ❘❛♣♣❡❧& ❡( ♣+❡✉✈❡& ❝♦♠♣❧4♠❡♥(❛✐+❡& ✶✸✶

❆✳✶ ❘❛♣♣❡❧( ♠❛*❤,♠❛*✐.✉❡( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✶

❆✳✷ ❇♦♦*(*4❛♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✶

❆✳✸ 54❡✉✈❡( ❝♦♠♣❧,♠❡♥*❛✐4❡( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✷

❇ ❘4&✉❧(❛(& ❝♦♠♣❧❡(& ❞❡& &✐♠✉❧❛(✐♦♥& ✶✸✺

❇✳✶ ❘,(✉❧*❛*( ❞❡ (✐♠✉❧❛*✐♦♥( ❞✉ ❝❤❛♣✐*4❡ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✺

❇✳✶✳✶ ▼♦❞=❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✺

❇✳✶✳✷ ❘,(✉❧*❛*( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✻

❇✳✷ ❘,(✉❧*❛*( ❞❡ (✐♠✉❧❛*✐♦♥( ❞✉ ❝❤❛♣✐*4❡ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✵

❇✐❜❧✐♦❣+❛♣❤✐❡ ✶✻✸

❘!"✉♠!

▲❡" ♠♦❞&❧❡" ❧✐♥*❛✐,❡" ❣*♥*,❛❧✐"*" "♦♥. ✉♥❡ ❣*♥*,❛❧✐"❛.✐♦♥ ❞❡" ♠♦❞&❧❡" ❞❡ ,*❣,❡"✲

"✐♦♥ ❧✐♥*❛✐,❡✱ ❡. "♦♥. .,&" ✉.✐❧✐"*" ❞❛♥" ❧❡ ❞♦♠❛✐♥❡ ❞✉ ✈✐✈❛♥.✳ ▲❡ ♠♦❞&❧❡ ❞❡ ,*❣,❡""✐♦♥

❧♦❣✐".✐4✉❡✱ ❧✬✉♥ ❞❡" ♠♦❞&❧❡" ❞❡ ❝❡..❡ ❝❧❛""❡✱ .,&" "♦✉✈❡♥. ✉.✐❧✐"* ❞❛♥" ❧❡" *.✉❞❡" ❜✐♦♠*✲

❞✐❝❛❧❡" ❞❡♠❡✉,❡ ❧❡ ♠♦❞&❧❡ ❞❡ ,*❣,❡""✐♦♥ ❧❡ ♣❧✉" ❛♣♣,♦♣,✐* 4✉❛♥❞ ✐❧ "✬❛❣✐. ❞❡ ♠♦❞*❧✐"❡,

✉♥❡ ✈❛,✐❛❜❧❡ ❞✐"❝,&.❡ ❞❡ ♥❛.✉,❡ ❜✐♥❛✐,❡✳ ❉❛♥" ❝❡..❡ .❤&"❡✱ ♥♦✉" ♥♦✉" ✐♥.*,❡""♦♥" ❛✉

♣,♦❜❧&♠❡ ❞❡ ❧✬✐♥❢*,❡♥❝❡ ".❛.✐".✐4✉❡ ❞❛♥" ❧❡ ♠♦❞&❧❡ ❞❡ ,*❣,❡""✐♦♥ ❧♦❣✐".✐4✉❡✱ ❡♥ ♣,*✲

"❡♥❝❡ ❞✬✐♥❞✐✈✐❞✉" ✐♠♠✉♥❡" ❞❛♥" ❧❛ ♣♦♣✉❧❛.✐♦♥ ❞✬*.✉❞❡✳

❉❛♥" ✉♥ ♣,❡♠✐❡, .❡♠♣"✱ ♥♦✉" ❝♦♥"✐❞*,♦♥" ❧❡ ♣,♦❜❧&♠❡ ❞❡ ❧✬❡".✐♠❛.✐♦♥ ❞❛♥" ❧❡

♠♦❞&❧❡ ❞❡ ,*❣,❡""✐♦♥ ❧♦❣✐".✐4✉❡ ❡♥ ♣,*"❡♥❝❡ ❞✬✐♥❞✐✈✐❞✉" ✐♠♠✉♥❡"✱ 4✉✐ ❡♥.,❡ ❞❛♥"

❧❡ ❝❛❞,❡ ❞❡" ♠♦❞&❧❡" ❞❡ ,*❣,❡""✐♦♥ < ❡①❝&" ❞❡ ③*,♦" ✭♦✉ ③*,♦✲✐♥✢❛.*"✮✳ ❯♥ ✐♥❞✐✈✐❞✉

❡". ❞✐. ✐♠♠✉♥❡ "✬✐❧ ♥✬❡". ♣❛" ❡①♣♦"* < ❧✬*✈*♥❡♠❡♥. ❞✬✐♥.*,C.✳ ▲❡ ".❛.✉. ❞✬✐♠♠✉♥✐.*

❡". ✐♥❝♦♥♥✉ "❛✉❢ "✐ ❧✬*✈*♥❡♠❡♥. ❞✬✐♥.*,C. ❛ *.* ♦❜"❡,✈*✳ ◆♦✉" ❞*✈❡❧♦♣♣♦♥" ✉♥❡ ♠*✲

.❤♦❞❡ ❞✬❡".✐♠❛.✐♦♥ ♣❛, ♠❛①✐♠✉♠ ❞❡ ✈,❛✐"❡♠❜❧❛♥❝❡ ❡♥ ♣,♦♣♦"❛♥. ✉♥❡ ♠♦❞*❧✐"❛.✐♦♥

❝♦♥❥♦✐♥.❡ ❞❡ ❧✬✐♠♠✉♥✐.* ❡. ❞❡" ,✐"4✉❡" ❞✬✐♥❢❡❝.✐♦♥✳ ◆♦✉" *.❛❜❧✐""♦♥" ❞✬❛❜♦,❞ ❧✬✐❞❡♥✲

.✐✜❛❜✐❧✐.* ❞✉ ♠♦❞&❧❡ ♣,♦♣♦"*✳ G✉✐"✱ ♥♦✉" ♠♦♥.,♦♥" ❧✬❡①✐".❡♥❝❡ ❞❡ ❧✬❡".✐♠❛.❡✉, ❞✉

♠❛①✐♠✉♠ ❞❡ ✈,❛✐"❡♠❜❧❛♥❝❡ ❞❡" ♣❛,❛♠&.,❡" ❞❡ ❝❡ ♠♦❞&❧❡✳ ◆♦✉" ♠♦♥.,♦♥" ❡♥"✉✐.❡✱

❧❛ ❝♦♥"✐".❛♥❝❡ ❞❡ ❝❡. ❡".✐♠❛.❡✉,✱ ❡. ♥♦✉" *.❛❜❧✐""♦♥" "❛ ♥♦,♠❛❧✐.* ❛"②♠♣.♦.✐4✉❡✳ ❊♥✲

✜♥✱ ♥♦✉" *.✉❞✐♦♥"✱ ❛✉ ♠♦②❡♥ ❞❡ "✐♠✉❧❛.✐♦♥"✱ ❧❡✉, ❝♦♠♣♦,.❡♠❡♥. "✉, ❞❡" *❝❤❛♥.✐❧❧♦♥"

❞❡ .❛✐❧❧❡ ✜♥✐❡✳

❉❛♥" ✉♥ ❞❡✉①✐&♠❡ .❡♠♣"✱ ♥♦✉" ♥♦✉" ✐♥.*,❡""♦♥" < ❧❛ ❝♦♥".,✉❝.✐♦♥ ❞❡ ❜❛♥❞❡" ❞❡

❝♦♥✜❛♥❝❡ "✐♠✉❧.❛♥*❡" ♣♦✉, ❧❛ ♣,♦❜❛❜✐❧✐.* ❞✬✐♥❢❡❝.✐♦♥✱ ❞❛♥" ❧❡ ♠♦❞&❧❡ ❞❡ ,*❣,❡""✐♦♥

❧♦❣✐".✐4✉❡ ❛✈❡❝ ❢,❛❝.✐♦♥ ✐♠♠✉♥❡✳ ◆♦✉" ♣,♦♣♦"♦♥" .,♦✐" ♠*.❤♦❞❡" ❞❡ ❝♦♥".,✉❝.✐♦♥" ❞❡

①✐✐

❜❛♥❞❡% ❞❡ ❝♦♥✜❛♥❝❡ ♣♦✉+ ❧❛ ❢♦♥❝.✐♦♥ ❞❡ +0❣+❡%%✐♦♥✳ ▲❛ ♣+❡♠✐5+❡ ♠0.❤♦❞❡ ✭♠0.❤♦❞❡

❞❡ ❙❝❤❡✛0✮ ✉.✐❧✐%❡ ❧❛ ♣+♦♣+✐0.0 ❞❡ ♥♦+♠❛❧✐.0 ❛%②♠♣.♦.✐<✉❡ ❞❡ ❧✬❡%.✐♠❛.❡✉+ ❞✉ ♠❛①✐✲

♠✉♠ ❞❡ ✈+❛✐%❡♠❜❧❛♥❝❡✱ ❡. ✉♥❡ ❛♣♣+♦①✐♠❛.✐♦♥ ♣❛+ ✉♥❡ ❧♦✐ ❞✉ ❦❤✐ ❞❡✉① ♣♦✉+ ❛♣♣+♦✲

❝❤❡+ ❧❡ <✉❛♥.✐❧❡ ♥0❝❡%%❛✐+❡ C ❧❛ ❝♦♥%.+✉❝.✐♦♥ ❞❡% ❜❛♥❞❡%✳ ▲❛ ❞❡✉①✐5♠❡ ♠0.❤♦❞❡ ✉.✐❧✐%❡

0❣❛❧❡♠❡♥. ❧❛ ♣+♦♣+✐0.0 ❞❡ ♥♦+♠❛❧✐.0 ❛%②♠♣.♦.✐<✉❡ ❞❡ ❧✬❡%.✐♠❛.❡✉+ ❞✉ ♠❛①✐♠✉♠ ❞❡

✈+❛✐%❡♠❜❧❛♥❝❡ ❡. ❡%. ❜❛%0❡ %✉+ ✉♥❡ 0❣❛❧✐.0 ❝❧❛%%✐<✉❡ ❞❡ ✭▲❛♥❞❛✉ ✫ ❙❤❡❡♣ ✶✾✼✵✮✳ ▲❛

.+♦✐%✐5♠❡ ♠0.❤♦❞❡ ✭♠0.❤♦❞❡ ❜♦♦.%.+❛♣✮ +❡♣♦%❡ %✉+ ❞❡% %✐♠✉❧❛.✐♦♥%✱ ♣♦✉+ ❡%.✐♠❡+

❧❡ <✉❛♥.✐❧❡ ❛♣♣+♦♣+✐0 ❞❡ ❧❛ ❧♦✐ ❞✉ %✉♣+❡♠✉♠ ❞✬✉♥ ♣+♦❝❡%%✉% ❣❛✉%%✐❡♥✳ ❊♥✜♥✱ ♥♦✉%

0✈❛❧✉♦♥%✱ ❛✉ ♠♦②❡♥ ❞❡ %✐♠✉❧❛.✐♦♥%✱ ❧❡✉+% ♣+♦♣+✐0.0% %✉+ ❞❡% 0❝❤❛♥.✐❧❧♦♥% ❞❡ .❛✐❧❧❡ ✜♥✐❡✳

❊♥✜♥✱ ♥♦✉% ❛♣♣❧✐<✉♦♥% ❧❡% +0%✉❧.❛.% ❞❡ ♠♦❞0❧✐%❛.✐♦♥ C ❞❡% ❞♦♥♥0❡% +0❡❧❧❡% %✉+

❧❛ ❞❡♥❣✉❡✳ ■❧ %✬❛❣✐. ❞✬✉♥❡ ♠❛❧❛❞✐❡ ✈❡❝.♦+✐❡❧❧❡ .+♦♣✐❝❛❧❡ C .+❛♥%♠✐%%✐♦♥ %.+✐❝.❡♠❡♥.

✐♥.❡+✲❤✉♠❛✐♥❡✳ ▲❡% +0%✉❧.❛.% ♠♦♥.+❡♥. <✉❡ ❧❡% ♣+♦❜❛❜✐❧✐.0% ❞✬✐♥❢❡❝.✐♦♥ ❡%.✐♠0❡% C

♣❛+.✐+ ❞❡ ♥♦.+❡ ❛♣♣+♦❝❤❡ ❞❡ ♠♦❞0❧✐%❛.✐♦♥ %♦♥. ♣❧✉% 0❧❡✈0❡% <✉❡ ❝❡❧❧❡% ♦❜.❡♥✉❡% C

♣❛+.✐+ ❞✬✉♥ ♠♦❞5❧❡ ❞❡ +0❣+❡%%✐♦♥ ❧♦❣✐%.✐<✉❡ %.❛♥❞❛+❞ <✉✐ ♥❡ .✐❡♥. ♣❛% ❝♦♠♣.❡ ❞✬✉♥❡

♣♦%%✐❜❧❡ ✐♠♠✉♥✐.0✳ ❊♥ ♣❛+.✐❝✉❧✐❡+✱ ❧❡% ❡%.✐♠❛.✐♦♥% ❢♦✉+♥✐❡% ♣❛+ ♥♦.+❡ ❛♣♣+♦❝❤❡ %✉❣✲

❣5+❡♥. <✉❡ ❧❡ %♦✉%✲♣♦✐❞% ❝♦♥%.✐.✉❡ ✉♥ ❢❛❝.❡✉+ ❞❡ +✐%<✉❡ ♠❛❥❡✉+ ❞❡ ❧✬✐♥❢❡❝.✐♦♥ ♣❛+ ❧❛

❞❡♥❣✉❡✱ ✐♥❞0♣❡♥❞❛♠♠❡♥. ❞❡ ❧✬L❣❡✳

▼♦"# ❝❧&#

❘0❣+❡%%✐♦♥ ❧♦❣✐%.✐<✉❡✱ ❋+❛❝.✐♦♥ ✐♠♠✉♥❡✱ ▼❛①✐♠✉♠ ❞❡ ✈+❛✐%❡♠❜❧❛♥❝❡✱ ■❞❡♥.✐✜❛❜✐❧✐.0✱

❈♦♥%✐%.❛♥❝❡✱ ◆♦+♠❛❧✐.0 ❛%②♠♣.♦.✐<✉❡✱ ❇❛♥❞❡% ❞❡ ❝♦♥✜❛♥❝❡✱ ❇♦♦.%.+❛♣✱ ❙✐♠✉❧❛.✐♦♥%✱

❉❡♥❣✉❡✳

❆❜"#$❛❝#

●❡♥❡#❛❧✐③❡❞ ❧✐♥❡❛# ♠♦❞❡❧+ ❛#❡ ❛ ❣❡♥❡#❛❧✐③❛-✐♦♥ ♦❢ ❧✐♥❡❛# #❡❣#❡++✐♦♥ ♠♦❞❡❧+✱ ❛♥❞

❛#❡ ✇✐❞❡❧② ✉+❡❞ ✐♥ -❤❡ ✜❡❧❞ ♦❢ ❧✐❢❡✳ ❚❤❡ ❧♦❣✐+-✐❝ #❡❣#❡++✐♦♥ ♠♦❞❡❧✱ ♦♥❡ ♦❢ -❤✐+ ❝❧❛++ ♦❢

♠♦❞❡❧+✱ ✇✐❞❡❧② ✉+❡❞ ✐♥ ❜✐♦♠❡❞✐❝❛❧ +-✉❞✐❡+ #❡♠❛✐♥+ -❤❡ ♠♦+- ❛♣♣#♦♣#✐❛-❡ #❡❣#❡++✐♦♥

♠♦❞❡❧ ✇❤❡♥ ✐- ❝♦♠❡+ -♦ ♠♦❞❡❧ ❞✐+❝#❡-❡ ✈❛#✐❛❜❧❡✱ ❜✐♥❛#② ✐♥ ♥❛-✉#❡✳ ■♥ -❤✐+ -❤❡+✐+✱ ✇❡

✐♥✈❡+-✐❣❛-❡ -❤❡ ♣#♦❜❧❡♠ ♦❢ +-❛-✐+-✐❝❛❧ ✐♥❢❡#❡♥❝❡ ✐♥ -❤❡ ❧♦❣✐+-✐❝ #❡❣#❡++✐♦♥ ♠♦❞❡❧✱ ✐♥

-❤❡ ♣#❡+❡♥❝❡ ♦❢ ✐♠♠✉♥❡ ✐♥❞✐✈✐❞✉❛❧+ ✐♥ -❤❡ +-✉❞② ♣♦♣✉❧❛-✐♦♥✳

❆- ✜#+-✱ ✇❡ ❝♦♥+✐❞❡# -❤❡ ♣#♦❜❧❡♠ ♦❢ ❡+-✐♠❛-✐♦♥ ✐♥ -❤❡ ❧♦❣✐+-✐❝ #❡❣#❡++✐♦♥ ♠♦✲

❞❡❧ ✐♥ -❤❡ ♣#❡+❡♥❝❡ ♦❢ ✐♠♠✉♥❡ ✐♥❞✐✈✐❞✉❛❧+ -❤❛- ❡♥-❡#+ ✐♥ -❤❡ ❝❛+❡ ♦❢ ③❡#♦✲✐♥✢❛-❡❞

#❡❣#❡++✐♦♥ ♠♦❞❡❧+✳ ❆ +✉❜❥❡❝- ✐+ +❛✐❞ -♦ ❜❡ ✐♠♠✉♥❡ ✐❢ ❤❡ ❝❛♥♥♦- ❡①♣❡#✐❡♥❝❡ -❤❡ ♦✉-✲

❝♦♠❡ ♦❢ ✐♥-❡#❡+-✳ ❚❤❡ ✐♠♠✉♥❡ +-❛-✉+ ✐+ ✉♥❦♥♦✇♥ ✉♥❧❡++ -❤❡ ❡✈❡♥- ♦❢ ✐♥-❡#❡+- ❤❛+

❜❡❡♥ ♦❜+❡#✈❡❞✳ ❲❡ ❞❡✈❡❧♦♣ ❛ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡+-✐♠❛-✐♦♥ ♣#♦❝❡❞✉#❡ ❢♦# -❤✐+ ♣#♦✲

❜❧❡♠✱ ❜❛+❡❞ ♦♥ -❤❡ ❥♦✐♥- ♠♦❞❡❧✐♥❣ ♦❢ -❤❡ ❜✐♥❛#② #❡+♣♦♥+❡ ♦❢ ✐♥-❡#❡+- ❛♥❞ -❤❡ ❝✉#❡

+-❛-✉+✳ ❲❡ ✐♥✈❡+-✐❣❛-❡ -❤❡ ✐❞❡♥-✐✜❛❜✐❧✐-② ♦❢ -❤❡ #❡+✉❧-✐♥❣ ♠♦❞❡❧✳ ❚❤❡♥✱ ✇❡ ❡+-❛❜❧✐+❤

-❤❡ ❡①✐+-❡♥❝❡✱ ❝♦♥+✐+-❡♥❝② ❛♥❞ ❛+②♠♣-♦-✐❝ ♥♦#♠❛❧✐-② ♦❢ -❤❡ ♣#♦♣♦+❡❞ ❡+-✐♠❛-♦#✱ ❛♥❞

✇❡ ❝♦♥❞✉❝- ❛ +✐♠✉❧❛-✐♦♥ +-✉❞② -♦ ✐♥✈❡+-✐❣❛-❡ ✐-+ ✜♥✐-❡✲+❛♠♣❧❡ ❜❡❤❛✈✐♦#✳

■♥ ❛ +❡❝♦♥❞ -✐♠❡✱ ✇❡ ❢♦❝✉+ ♦♥ -❤❡ ❝♦♥+-#✉❝-✐♦♥ ♦❢ +✐♠✉❧-❛♥❡♦✉+ ❝♦♥✜❞❡♥❝❡ ❜❛♥❞+

❢♦# -❤❡ ♣#♦❜❛❜✐❧✐-② ♦❢ ✐♥❢❡❝-✐♦♥ ✐♥ -❤❡ ❧♦❣✐+-✐❝ #❡❣#❡++✐♦♥ ♠♦❞❡❧ ✇✐-❤ ✐♠♠✉♥❡ ❢#❛❝✲

-✐♦♥✳ ❲❡ ♣#♦♣♦+❡ -❤#❡❡ ♠❡-❤♦❞+ ♦❢ ❝♦♥+-#✉❝-✐♦♥ ♦❢ ❝♦♥✜❞❡♥❝❡ ❜❛♥❞+ ❢♦# -❤❡ #❡❣#❡+✲

+✐♦♥ ❢✉♥❝-✐♦♥✳ ❚❤❡ ✜#+- ♠❡-❤♦❞ ✭❙❝❤❡✛❡✬+ ♠❡-❤♦❞✮ ✉+❡+ -❤❡ ❛+②♠♣-♦-✐❝ ♥♦#♠❛❧✐-②

♦❢ -❤❡ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡+-✐♠❛-♦#✱ ❛♥❞ ❛♥ ❛♣♣#♦①✐♠❛-✐♦♥ ❜② -❤❡ ❝❤✐✲+H✉❛#❡❞ ❞✐+✲

-#✐❜✉-✐♦♥ -♦ ❛♣♣#♦①✐♠❛-❡ -❤❡ ♥❡❝❡++❛#② H✉❛♥-✐❧❡ ❢♦# -❤❡ ❝♦♥+-#✉❝-✐♦♥ ♦❢ ❜❛♥❞+✳ ❚❤❡

+❡❝♦♥❞ ♠❡-❤♦❞ ✉+❡+ ❛❧+♦ -❤❡ ❛+②♠♣-♦-✐❝ ♥♦#♠❛❧✐-② ♦❢ -❤❡ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡+✲

①✐✈

✐♠❛ ♦% ❛♥❞ ✐( ❜❛(❡❞ ♦♥ ❛ ❝❧❛((✐❝❛❧ ❡-✉❛❧✐ ② ❜② ✭▲❛♥❞❛✉ ✫ ❙❤❡❡♣ ✶✾✼✵✮✳ ❚❤❡ ❤✐%❞

♠❡ ❤♦❞ ✭❜♦♦ ( %❛♣ ♠❡ ❤♦❞✮ ✐( ❜❛(❡❞ ♦♥ (✐♠✉❧❛ ✐♦♥(✱ ♦ ❡( ✐♠❛ ❡ ❤❡ ❛♣♣%♦♣%✐❛ ❡ -✉❛♥ ✐❧❡ ♦❢ ❤❡ ❧❛✇ ♦❢ ❛ (✉♣%❡♠✉♠ ♦❢ ❛ ●❛✉((✐❛♥ ♣%♦❝❡((✳ ❋✐♥❛❧❧②✱ ✇❡ ❝♦♥❞✉❝ ❛ (✐♠✉❧❛ ✐♦♥ ( ✉❞② ♦ ✐♥✈❡( ✐❣❛ ❡ ✐ ( ✜♥✐ ❡✲(❛♠♣❧❡ ♣%♦♣❡% ✐❡(✳

❋✐♥❛❧❧②✱ ✇❡ ❝♦♥(✐❞❡% ❛ ( ✉❞② ♦❢ ❞❡♥❣✉❡ ❢❡✈❡%✱ ✇❤✐❝❤ ✐( ❛ %♦♣✐❝❛❧ ♠♦(-✉✐ ♦✲❜♦%♥❡

✈✐%❛❧ ❤✉♠❛♥ ❞✐(❡❛(❡✱ ( %✐❝ ❧② ✐♥ ❡%✲❤✉♠❛♥✳ ❚❤❡ %❡(✉❧ ( (❤♦✇ ❤❛ ✱ ❤❡ ❡( ✐♠❛ ❡❞ ♣%♦✲

❜❛❜✐❧✐ ✐❡( ♦❢ ✐♥❢❡❝ ✐♦♥ ♦❜ ❛✐♥❡❞ ❢%♦♠ ♦✉% ❛♣♣%♦❛❝❤ ❛%❡ ❧❛%❣❡% ❤❛♥ ❤❡ ♦♥❡( ❞❡%✐✈❡❞

❢%♦♠ ❛ ( ❛♥❞❛%❞ ❛♥❛❧②(✐( ❤❛ ❞♦❡( ♥♦ ❛❦❡ ❛❝❝♦✉♥ ♦❢ ❤❡ ♣♦((✐❜❧❡ ✐♠♠✉♥✐ ②✳ ■♥

♣❛% ✐❝✉❧❛%✱ ❤❡ ❡( ✐♠❛ ❡( ♣%♦✈✐❞❡❞ ❜② ♦✉% ❛♣♣%♦❛❝❤ (✉❣❣❡( ❤❛ ✉♥❞❡%✇❡✐❣❤ ❝♦♥( ✐✲

✉ ❡( ❛ ♠❛❥♦% %✐(❦ ❢❛❝ ♦% ❢♦% ❞❡♥❣✉❡ ✐♥❢❡❝ ✐♦♥✱ ✐%%❡(♣❡❝ ✐✈❡❧② ♦❢ ❛❣❡✳

❑❡② ✇♦%❞'

▲♦❣✐( ✐❝ %❡❣%❡((✐♦♥✱ ■♠♠✉♥❡ ❢%❛❝ ✐♦♥✱ ▼❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞✱ ■❞❡♥ ✐✜❛❜✐❧✐ ②✱ ❈♦♥(✐(✲

❛♥❝②✱ ❆(②♠♣ ♦ ✐❝ ♥♦%♠❛❧✐ ②✱ ❈♦♥✜❞❛♥❝❡ ❇❛♥❞(✱ ❇♦♦ ( %❛♣✱ ❙✐♠✉❧❛ ✐♦♥(✱ ❉❡♥❣✉❡✳

❆❜"#✈✐❛'✐♦♥* ❡' ◆♦'❛'✐♦♥*

P(A) ✿ ▲❛ ♣$♦❜❛❜✐❧✐)* ❞❡ ❧✬*✈*♥❡♠❡♥)A✳

E(X) ✿ ▲✬❡2♣*$❛♥❝❡ ♠❛)❤*♠❛)✐5✉❡ ❞❡ ❧❛ ✈❛$✐❛❜❧❡ ❛❧*❛)♦✐$❡X✳ var (X) ✿ ▲❛ ✈❛$✐❛♥❝❡ ❞❡ ❧❛ ✈❛$✐❛❜❧❡ ❛❧*❛)♦✐$❡X✳

cov (X, Y) ✿ ▲❛ ❝♦✈❛$✐❛♥❝❡ ❞❡2 ✈❛$✐❛❜❧❡2 ❛❧*❛)♦✐$❡2X ❡)Y✳

X_n=⇒Y ✿ ▲❛ 2✉✐)❡ ❞❡ ✈❛$✐❛❜❧❡2 ❛❧*❛)♦✐$❡2(Xn)_n≥0 ❝♦♥✈❡$❣❡ ❢❛✐❜❧❡♠❡♥) ✈❡$2Y✳ X_n−→^p.s Y ✿ ▲❛ 2✉✐)❡ ❞❡ ✈❛$✐❛❜❧❡2 ❛❧*❛)♦✐$❡2(X_n)_n≥0 ❝♦♥✈❡$❣❡ ♣$❡25✉❡ 29$❡♠❡♥) ✈❡$2Y✳ N ✿ ❊♥2❡♠❜❧❡ ❞❡2 ❡♥)✐❡$2 ♥❛)✉$❡❧2✳

N^∗ ✿ ❊♥2❡♠❜❧❡ ❞❡2 ❡♥)✐❡$2 ♥❛)✉$❡❧2 ♥♦♥ ♥✉❧2✳

R ✿ ❊♥2❡♠❜❧❡ ❞❡2 $*❡❧2 ❡)R^d=R×. . .×R

| {z }

d❢♦✐#

✳

X^⊤ ✿ ❚$❛♥2♣♦2*❡ ❞✉ ✈❡❝)❡✉$X✳

✐✳✐✳❞✳ ✿ ✐♥❞*♣❡♥❞❛♥)❡2 ❡) ✐❞❡♥)✐5✉❡♠❡♥) ❞✐2)$✐❜✉*❡2✳

M(n×p) ✿ ▲✬❡♥2❡♠❜❧❡ ❞❡2 ♠❛)$✐❝❡2 $*❡❧❧❡2 ❞❡n ❧✐❣♥❡2 ❡)p❝♦❧♦♥♥❡2✳

I_p ✿ ♠❛)$✐❝❡ ✐❞❡♥)✐)* ❞✬♦$❞$❡ p✳

❊▼❱ ✿ ❊2)✐♠❛)❡✉$ ❞✉ ♠❛①✐♠✉♠ ❞❡ ✈$❛✐2❡♠❜❧❛♥❝❡✳

❩■❇ ✿ ❩❡$♦✲■♥✢❛)❡❞ ❇✐♥♦♠✐❛❧✳

❩■D ✿ ❩❡$♦✲■♥✢❛)❡❞ D♦✐22♦♥✳

❩■◆❇ ✿ ❩❡$♦✲■♥✢❛)❡❞ ◆❡❣❛)✐✈❡ ❇✐♥♦♠✐❛❧✳

❩■D❖ ✿ ❩❡$♦✲■♥✢❛)❡❞ D$♦♣♦$)✐♦♥❛❧ ❖❞❞2✳

❈❤❛♣✐%&❡ ✶

■♥"#♦❞✉❝"✐♦♥ ❣*♥*#❛❧❡

▲✬❛♥❛❧②&❡ &(❛(✐&(✐*✉❡ ❞❡& ❞♦♥♥.❡& ❝♦♥&(✐(✉❡ ❛✉❥♦✉1❞✬❤✉✐ ✉♥ ♦✉(✐❧ ❢♦♥❞❛♠❡♥(❛❧

❞❛♥& ❧❡ ❞♦♠❛✐♥❡ ❞✉ ✈✐✈❛♥(✱ ❡( 1❡♣♦&❡ (18& &♦✉✈❡♥( &✉1 ❧✬✉(✐❧✐&❛(✐♦♥ ❞❡ ♠♦❞8❧❡& ❞❡ 1.✲

❣1❡&&✐♦♥✳ ▲✬✉♥❡ ❞❡ ❝❡& ❝❧❛&&❡& ❞❡ ♠♦❞8❧❡&✱ ♣❛1♠✐ ❧❡& ♣❧✉& ✉(✐❧✐&.❡&✱ ❞❡♠❡✉1❡ ❧❛ ❝❧❛&&❡

❞❡& ♠♦❞8❧❡& ❧✐♥.❛✐1❡& ❣.♥.1❛❧✐&.& *✉✐ &♦♥( ✉♥❡ ❣.♥.1❛❧✐&❛(✐♦♥ ❜✐❡♥ ❝♦♥♥✉❡ ❞❡& ♠♦❞8❧❡&

❞❡ 1.❣1❡&&✐♦♥ ❧✐♥.❛✐1❡ ❡♥ (❡1♠❡& ❞❡ ❧♦✐✳ ❊❧❧❡ ♣❡1♠❡( ❧✬❛♥❛❧②&❡ ❞❡ ❞♦♥♥.❡& ❞✐&❝18(❡&

♠❛✐& ❛✉&&✐ ❞❡ ❞♦♥♥.❡& ❝♦♥(✐♥✉❡& ♣♦✉1 ❧❡&*✉❡❧❧❡& ❧❛ ❧♦✐ ♥♦1♠❛❧❡ ♥✬❡&( ♣❛& (18& ❛❞❛♣(.❡✳

✭▼❝❈✉❧❧❛❣❤ ✫ ◆❡❧❞❡1 ✶✾✽✾✮ ❡♥ ❢♦♥( ✉♥❡ ♣1.&❡♥(❛(✐♦♥ ❞.(❛✐❧❧.❡✳

▲✬✉(✐❧✐&❛(✐♦♥ ❞❡& ♠♦❞8❧❡& ❧✐♥.❛✐1❡& ❣.♥.1❛❧✐&.& ❡&( (18& ❝♦✉1❛♥(❡ ❡♥ &❛♥(. ♣✉❜❧✐*✉❡

♣♦✉1 ❧❛ 1.&♦❧✉(✐♦♥ ❞❡ ♣1♦❜❧8♠❡& 1.❡❧&✳ ❊♥ ♣❛1(✐❝✉❧✐❡1✱ ❧❡& .♣✐❞.♠✐♦❧♦❣✐&(❡&✱ ❧♦1& ❞❡

❧✬❡①♣❧♦✐(❛(✐♦♥ ❞❡ ❞♦♥♥.❡& ♠.❞✐❝❛❧❡&✱ ❞.❝1✐✈❡♥( &♦✉✈❡♥( ✉♥ .✈.♥❡♠❡♥( ♦✉ ✉♥ ♣❤.♥♦✲

♠8♥❡ ❧✉✐✲♠H♠❡ ✐♥✢✉❡♥❝. ♣❛1 ❧❛ &✉1✈❡♥✉❡ ❞✬❛✉(1❡& .✈.♥❡♠❡♥(& ♦✉ ♣❤.♥♦♠8♥❡& ❛♣♣❡✲

❧.& ❢❛❝(❡✉1& ❞✬❡①♣♦&✐(✐♦♥✳ ▲❛ &✉1✈❡♥✉❡ ❞✬✉♥❡ ✐♥❢❡❝(✐♦♥ 1❡♣1.&❡♥(.❡ ♣❛1 ✉♥❡ ✈❛1✐❛❜❧❡

❞✐❝❤♦(♦♠✐*✉❡ ✭✶ ♣♦✉$ ✐♥❢❡❝*+ ❡* ✵ ♣♦✉$ ♥♦♥ ✐♥❢❡❝*+✮ ♣❡✉( ❡♥ ❡✛❡( H(1❡ ❝❡ ♣❤.♥♦✲

♠8♥❡ .(✉❞✐. ♣❛1 ❧❡& .♣✐❞.♠✐♦❧♦❣✐&(❡& ❡♥ ♣1.&❡♥❝❡ ❞❡ ❝❡1(❛✐♥& ❢❛❝(❡✉1& ❞✬❡①♣♦&✐(✐♦♥✳

❈❡♣❡♥❞❛♥( ✐❧ ♣❡✉( ❡①✐&(❡1 ❞❛♥& ❧❛ ♣♦♣✉❧❛(✐♦♥ ❞✬.(✉❞❡ ✉♥❡ ♣1♦♣♦1(✐♦♥ ❞✬✐♥❞✐✈✐❞✉&

♥♦♥ &✉&❝❡♣(✐❜❧❡& K ❧✬✐♥❢❡❝(✐♦♥ *✉✐ ♣❡✉( H(1❡ ❞✉ K ✉♥❡ ✐♠♠✉♥✐(. ♥❛(✉1❡❧❧❡ ♦✉ K ✉♥❡

❛❝(✐♦♥ ♣1.✈❡♥(✐✈❡✳ ❊♥ ❡✛❡(✱❧✬✐♠♠✉♥✐(. ✐♥♥.❡ ❝♦♥&(✐(✉❡ ❧❛ ♣1❡♠✐81❡ ❧✐❣♥❡ ❞❡ ❞.❢❡♥&❡

❡( ❞✬✐♥(❡1❛❝(✐♦♥ ❡♥(1❡ ❧✬❤L(❡ ❤✉♠❛✐♥ ❡( ❧❡& ♣❛(❤♦❣8♥❡&✳ ❈❡ ♣❤.♥♦♠8♥❡ ❡♥(1❛M♥❡ ❛✐♥&✐

❧❛ ♣1.&❡♥❝❡ ❞❡ ❜❡❛✉❝♦✉♣ ❞❡ ③.1♦& ❞❛♥& ❧❛ ✈❛1✐❛❜❧❡ ❞✐❝❤♦(♦♠✐*✉❡ ❀ ❝❡ *✉✐ ❡♥(1❛M♥❡

❧✬❛♣♣❛1✐(✐♦♥ ❞✬✉♥ ❜✐❛✐& ♥❡( &✉1 ❧✬❡&(✐♠❛(✐♦♥ ❞❡& ♣❛1❛♠8(1❡& ❞✉ ♠♦❞8❧❡ ❡( *✉✐ ♣♦✉1✲

1❛✐( ❛✐♥&✐ H(1❡ K ❧✬♦1✐❣✐♥❡ ❞✬✉♥❡ ♠❛✉✈❛✐&❡ ✐♥(❡1♣1.(❛(✐♦♥ ❞❡& 1.&✉❧(❛(& ❛((❡♥❞✉&✳ ❈❡

♣❤.♥♦♠8♥❡ ❝♦♥♥✉ &♦✉& ❧❡ ♥♦♠ ❞❡ ✧③❡$♦✲✐♥✢❛*❡❞✧ ❛ .(. K ❧✬♦1✐❣✐♥❡ ❞❡ ❧✬❡①(❡♥&✐♦♥ ❞❡&

✷

♠♦❞#❧❡& ❞❡ '(❣'❡&&✐♦♥ ❞❡ ❝♦♠♣.❛❣❡ ❞❡ 0♦✐&♦♥ ❡. ❇✐♥♦♠✐❛❧❡ ◆(❣❛.✐✈❡✱ ❞✉ ♠♦❞#❧❡ ❞❡

'(❣'❡&&✐♦♥ ❇✐♥♦♠✐❛❧❡ ❡. ❞✉ ♠♦❞#❧❡ ❞❡ '(❣'❡&&✐♦♥ 6 ❖❞❞& 0'♦♣♦'.✐♦♥♥❡❧&✳

▲❡ ♠♦❞#❧❡ ❞❡ '(❣'❡&&✐♦♥ ❧♦❣✐&.✐:✉❡ &.❛♥❞❛'❞ ❡&. ♥❛.✉'❡❧❧❡♠❡♥. ✉.✐❧✐&( ❡♥ (♣✐❞(♠✐♦✲

❧♦❣✐❡ ❧♦'&:✉✬✐❧ &✬❛❣✐. ❞❡ ♠♦❞(❧✐&❡' ✉♥❡ ✈❛'✐❛❜❧❡ ❞(♣❡♥❞❛♥.❡ ❜✐♥❛✐'❡✳ ▲✬♦❜❥❡❝.✐❢ ❞❛♥&

❝❡..❡ .❤#&❡ ❡&. ❞✬(.❡♥❞'❡ ❝❡ ♠♦❞#❧❡ ❝❧❛&&✐:✉❡ ❡♥ ♣'❡♥❛♥. ❡♥ ❝♦♠♣.❡ ❧❛ ♣'(&❡♥❝❡ ❞❡

❝❡..❡ ❢'❛❝.✐♦♥ ✐♠♠✉♥❡ ❞❛♥& ❧❛ ♣♦♣✉❧❛.✐♦♥ ❞✬(.✉❞❡ .♦✉. ❡♥ ♣'♦♣♦&❛♥. ✉♥❡ ♠♦❞(❧✐&❛✲

.✐♦♥ ❝♦♥❥♦✐♥.❡ ❞❡ ❧❛ ♣'♦❜❛❜✐❧✐.( ❞✬✐♥❢❡❝.✐♦♥ ❡. ❞❡ ❧❛ ♣'♦❜❛❜✐❧✐.( ❞✬✐♠♠✉♥✐.(✳

▲❡ ♣'❡♠✐❡' ❝❤❛♣✐.'❡ ❞❡ ❝❡ ♠❛♥✉&❝'✐. ❡&. ❝♦♥&❛❝'( 6 ❞❡ ❜'❡❢& '❛♣♣❡❧& &✉' ❧❡ ♠♦❞#❧❡

❧✐♥(❛✐'❡✱ ❡. 6 ❞❡& '❛♣♣❡❧& ❞(.❛✐❧❧(& &✉' ❧❡ ♠♦❞#❧❡ ❞❡ '(❣'❡&&✐♦♥ ❧♦❣✐&.✐:✉❡✳ ◆♦✉& ♥♦✉&

✐♥.('❡&&♦♥& ❡♥ ♣❛'.✐❝✉❧✐❡' 6 ❧✬✐❞❡♥.✐✜❛❜✐❧✐.( ❞❡ ❝❡ ♠♦❞#❧❡✱ ❡. 6 ❧✬❡&.✐♠❛.✐♦♥ ❞❡ &❡& ♣❛✲

'❛♠#.'❡&✳ 0✉✐&✱ ♥♦✉& '❛♣♣❡❧♦♥& ❧❡& '(&✉❧.❛.& ❞✬❡①✐&.❡♥❝❡ ❞✬❡&.✐♠❛.❡✉'& ❞✉ ♠❛①✐♠✉♠

❞❡ ✈'❛✐&❡♠❜❧❛♥❝❡ ❞❛♥& ❝❡& ♠♦❞#❧❡&✳ ❉❛♥& ❧❡ ❝❤❛♣✐.'❡ ✷✱ ♥♦✉& ❞(✜♥✐&&♦♥& ❧❡& ♠♦❞#❧❡&

③('♦✲✐♥✢❛.(& ❡①✐&.❛♥. ✿ ❩■0✱ ❩■◆❇✱ ❩■❇ ❡. ❩■0❖✳ 0✉✐&✱ ♥♦✉& ♣'(&❡♥.♦♥& ❧❡& '(&✉❧.❛.&

❞✬❡①✐&.❡♥❝❡ ❞✬❡&.✐♠❛.❡✉' ❞✉ ♠❛①✐♠✉♠ ❞❡ ✈'❛✐&❡♠❜❧❛♥❝❡ ❡. ❧❡& ♣'♦♣'✐(.(& ❛&②♠♣.♦✲

.✐:✉❡& ❞❡ ❝❡& ❡&.✐♠❛.❡✉'&✳

▲❡& ♣❛'.✐❡& ✷ ❡. ✸ ❞❡ ❝❡ ♠❛♥✉&❝'✐. ❝♦♥.✐❡♥♥❡♥. ❧❡& ❝♦♥.'✐❜✉.✐♦♥& ♦'✐❣✐♥❛❧❡& ❞❡ ❝❡..❡

.❤#&❡✳ ❉❛♥& ❧❛ ❞❡✉①✐#♠❡ ♣❛'.✐❡ ❞❡ ❝❡..❡ .❤#&❡✱ ♥♦✉& ♥♦✉& ✐♥.('❡&&♦♥& ❛✉ ♣'♦❜❧#♠❡ ❞❡

❧✬✐♥❢('❡♥❝❡ &.❛.✐&.✐:✉❡ ❞❛♥& ❧❡ ♠♦❞#❧❡ ❞❡ '(❣'❡&&✐♦♥ ❧♦❣✐&.✐:✉❡ ❛✈❡❝ ❢'❛❝.✐♦♥ ✐♠♠✉♥❡✳

❉❛♥& ❧❡ ❝❤❛♣✐.'❡ ✸✱ ♥♦✉& ❝♦♠♠❡♥L♦♥& ♣❛' ❞(❝'✐'❡ ❧❡ ♠♦❞#❧❡ ❞❡ '(❣'❡&&✐♦♥ ❧♦❣✐&.✐:✉❡

❛✈❡❝ ❢'❛❝.✐♦♥ ✐♠♠✉♥❡✱ :✉✐ ❡♥.'❡ ❞❛♥& ❧❛ ❢❛♠✐❧❧❡ ❞❡& ♠♦❞#❧❡& ③('♦✲✐♥✢❛.(&✱ ♣✉✐&✱ ♥♦✉&

♣'♦♣♦&♦♥& ❞❡& ❡&.✐♠❛.❡✉'& ♣❛' ♠❛①✐♠✉♠ ❞❡ ✈'❛✐&❡♠❜❧❛♥❝❡ ♣♦✉' ❝❡ ♠♦❞#❧❡✳ ◆♦✉& ♣'(✲

&❡♥.♦♥& ❡♥✜♥✱ ❧❡& '(&✉❧.❛.& ❛&②♠♣.♦.✐:✉❡& ♦❜.❡♥✉& ♣♦✉' ❝❡& ❡&.✐♠❛.❡✉'& ❞✉ ♠❛①✐♠✉♠

❞❡ ✈'❛✐&❡♠❜❧❛♥❝❡✳ ❉❛♥& ❧❡ ❝❤❛♣✐.'❡ ✹✱ ♥♦✉& ✉.✐❧✐&❡'♦♥& ❝❡& '(&✉❧.❛.& ❛&②♠♣.♦.✐:✉❡&

♣♦✉' ❝♦♥&.'✉✐'❡ ❞❡& ❜❛♥❞❡& ❞❡ ❝♦♥✜❛♥❝❡ &✐♠✉❧.❛♥(❡& ♣♦✉' ❧❛ ♣'♦❜❛❜✐❧✐.( ❞✬✐♥❢❡❝.✐♦♥✳

❉❛♥& ❝❡ ❝❤❛♣✐.'❡✱ ♥♦✉& ♣'(&❡♥.♦♥& .'♦✐& ♠(.❤♦❞❡& ♣♦✉' ❧❛ ❝♦♥&.'✉❝.✐♦♥ ❞❡& ❜❛♥❞❡&

❞❡ ❝♦♥✜❛♥❝❡ ♣♦✉' ❧❛ ❢♦♥❝.✐♦♥ '(♣♦♥&❡✳

❉❛♥& ❧❛ .'♦✐&✐#♠❡ ♣❛'.✐❡ ❞❡ ❝❡..❡ .❤#&❡✱ ♥♦✉& ❛♣♣❧✐:✉♦♥& ❧❡ ♠♦❞#❧❡ ❝♦♥❥♦✐♥. ❞(✜♥✐

❞❛♥& ❧❡ ❝❤❛♣✐.'❡ ✸ &✉' ✉♥ ❥❡✉ ❞❡ ❞♦♥♥(❡& '(❡❧❧❡&✳ ◆♦✉& ❝♦♠♠❡♥L♦♥& ♣❛' ✉♥❡ (.✉❞❡

✸ ♣✐❞ ♠✐♦❧♦❣✐(✉❡ ❞❡ ❧❛ ❞❡♥❣✉❡ ❞❛♥- ❧❡ ❝❤❛♣✐01❡ ✺✱ ♦4 ♥♦✉- ❞ 0❛✐❧❧♦♥- ❧❡- ❝❛1❛❝0 1✐-✲

0✐(✉❡- ❞❡ ❝❡00❡ ✐♥❢❡❝0✐♦♥ ✈✐1❛❧❡ ❡0 ♣1 -❡♥0♦♥- ❞❡- 1 -✉❧0❛0- ❞✬❛♥❛❧②-❡- -0❛0✐-0✐(✉❡-

✉♥✐✈❛1✐ ❡- ❡0 ♠✉❧0✐✈❛1✐ ❡-✳ ❉❛♥- ❧❡ ❝❤❛♣✐01❡ ✻✱ ♥♦✉- ❛♣♣❧✐(✉♦♥- ❧❡ ♠♦❞=❧❡ ❝♦♥❥♦✐♥0 ?

❧✬❛♥❛❧②-❡ -0❛0✐-0✐(✉❡ ❞❡- ❞♦♥♥ ❡- ❞❡ ❞❡♥❣✉❡✳

▲❛ ❞❡1♥✐=1❡ ♣❛10✐❡✱ ❡-0 ❝♦♥-0✐0✉ ❡ ❞❡- ❞✐✛ 1❡♥0❡- ❛♥♥❡①❡- ✿ ❞❡- 1❛♣♣❡❧- ♠❛0❤ ♠❛✲

0✐(✉❡-✱ ❞❡- 1 -✉❧0❛0- ❝♦♠♣❧ ♠❡♥0❛✐1❡- ❝♦♥❝❡1♥❛♥0 ❝❡10❛✐♥- ❧❡♠♠❡- ♥♦♥❝ - ❞❛♥- ❧❛

0❤=-❡✱ ❞❡ 0❛❜❧❡❛✉① ❞❡- 1 -✉❧0❛0- ❞❡ -✐♠✉❧❛0✐♦♥✳

!❡♠✐%!❡ ♣❛!(✐❡

❘❛♣♣❡❧+ +✉! -✉❡❧-✉❡+ ♠♦❞%❧❡+

❈❤❛♣✐%&❡ ✷

❘❛♣♣❡❧% %✉' ❧❡ ♠♦❞+❧❡ ❞❡ ',❣'❡%%✐♦♥

❧♦❣✐%0✐1✉❡

❙♦♠♠❛✐%❡

✷✳✶ ▲❡ ♠♦❞(❧❡ ❧✐♥,❛✐.❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✷✳✷ ▲❡ ♠♦❞(❧❡ ❞❡ .,❣.❡11✐♦♥ ❧♦❣✐12✐3✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✷✳✶ ■♥%&♦❞✉❝%✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✷✳✷ ■❞❡♥%✐✜❛❜✐❧✐%2 ❞✉ ♠♦❞4❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷✳✷✳✸ ❊8%✐♠❛%✐♦♥ ❞✉ ♠♦❞4❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✷✳✹ ❘❛♣♣❡❧ 8✉& ❧✬❛❧❣♦&✐%❤♠❡ ❞❡ ◆❡✇%♦♥✲❘❛♣❤8♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✷✳✺ D&♦♣&✐2%28 ❛8②♠♣%♦%✐F✉❡8 ❞✉ ♠♦❞4❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✷✳✻ ■♥%❡&♣&2%❛%✐♦♥ ❞❡8 ❝♦❡✣❝✐❡♥%8 ❞❡ &2❣&❡88✐♦♥ ❡% ♦❞❞" #❛%✐♦ ✳ ✳ ✳ ✶✸

◆♦"♦♥$ y ❧❡ ✈❡❝"❡✉* ❞❡$ ♦❜$❡*✈❛"✐♦♥$ ❞❡ "❛✐❧❧❡ n✱ *0❛❧✐$❛"✐♦♥ ❞✉ ✈❡❝"❡✉* ❛❧0❛"♦✐*❡

Y✱ ✈❛*✐❛❜❧❡ 1 ❡①♣❧✐4✉❡*✳ ❯♥ ♠♦❞8❧❡ ❧✐♥0❛✐*❡ ❣0♥0*❛❧✐$0 $❡ ❝❛*❛❝"0*✐$❡ ♣❛* ❧❡$ "*♦✐$

❤②♣♦"❤8$❡$ $✉✐✈❛♥"❡$ ✿

✕ ❖♥ $✉♣♣♦$❡ 4✉❡ ❧❡$ ❝♦♠♣♦$❛♥"❡$ Y_i (i = 1, ..., n) ❞❡ Y $♦♥" ✐♥❞0♣❡♥❞❛♥"❡$

❡" ❞✐$"*✐❜✉0❡$ $❡❧♦♥ ✉♥❡ ❧♦✐ ❛♣♣❛*"❡♥❛♥" 1 ❧❛ ❢❛♠✐❧❧❡ ❡①♣♦♥❡♥"✐❡❧❧❡ ❛✉ $❡♥$

❞❡ ✭◆❡❧❞❡* ✫ ❲❡❞❞❡*❜✉*♥ ✶✾✼✷✮✱ ❝✬❡$"✲1✲❞✐*❡ 4✉❡ ❧❛ ❢♦♥❝"✐♦♥ ❞❡ ❞❡♥$✐"0 ❞❡ ❧❛

✹

✈❛"✐❛❜❧❡ ❛❧'❛(♦✐"❡ Yi *✬'❝"✐( ✿ fYi(yi, θi) = exp

yiθi−b(θi)

ai(φ) +c(yi, φ)

♦. θi ❡*( ✉♥ ♣❛"❛♠3("❡ ❝❛♥♦♥✐4✉❡ ❡( φ ✉♥ ♣❛"❛♠3("❡ ❞❡ ❞✐*♣❡"*✐♦♥✳ ▲❡* ❢♦♥❝✲

(✐♦♥* b ❡(c *♦♥( *♣'❝✐✜4✉❡* ; ❝❤❛4✉❡ ❞✐*("✐❜✉(✐♦♥ ❡( ❧❛ ❢♦♥❝(✐♦♥ a_i ❡*( ❞♦♥♥'❡

♣❛" ✿ ai(φ) = _w^φ_i✱ ♦. wi ❡*( ✉♥ ♣♦✐❞* ❝♦♥♥✉ ❛**♦❝✐' ; ❧❛ "'❛❧✐*❛(✐♦♥ yi✳

▲✬❡*♣'"❛♥❝❡ ❡( ❧❛ ✈❛"✐❛♥❝❡ ❞❡ ❧❛ ✈❛"✐❛❜❧❡ Y_i *✬❡①♣"✐♠❡ ❝♦♠♠❡ *✉✐( ; ❧✬❛✐❞❡

❞❡* ❢♦♥❝(✐♦♥* ai ❡( b ✿

E(Yi) = b^′(θi), var(Yi) = ai(φ)b^′′(θi).

✕ ▲❡ ♣"'❞✐❝(❡✉" ❧✐♥'❛✐"❡ ❡*( ❞'✜♥✐( ❝♦♠♠❡ *✉✐( ✿ η=β^⊤X,

♦. β ❡*( ✉♥ ✈❡❝(❡✉" ❞❡ ♣❛"❛♠3("❡* ✐♥❝♦♥♥✉* ❞❡ (❛✐❧❧❡ p ❡( X ❧❡ ✈❡❝(❡✉" ❞❡* p

✈❛"✐❛❜❧❡* ❡①♣❧✐❝❛(✐✈❡*✳

✕ ▲❡ ❧✐❡♥ ❡♥("❡ ❧✬❡*♣'"❛♥❝❡ ❞❡ Yi ❡( ❧❛ ✐^me ❝♦♠♣♦*❛♥(❡ ❞✉ ♣"'❞✐❝(❡✉" ❧✐♥'❛✐"❡ ❡*(

"'❛❧✐*' ♣❛" ❧❛ ❢♦♥❝(✐♦♥ g ✭♠♦♥♦(♦♥❡ ❡( ❞✐✛'"❡♥(✐❛❜❧❡✮ ❛♣♣❡❧'❡ ❢♦♥❝$✐♦♥ ❞❡ ❧✐❡♥ ✿ ηi =g(E(Yi)).

❯♥❡ ❢♦♥❝(✐♦♥ ❞❡ ❧✐❡♥ ♣♦✉" ❧❛4✉❡❧❧❡ η_i = θ_i ❡*( ❛♣♣❡❧'❡ ❢♦♥❝$✐♦♥ ❞❡ ❧✐❡♥ ❝❛♥♦✲

♥✐+✉❡✳

▲❡* ♠♦❞3❧❡* ❣❛✉**✐❡♥* ✭❧✐♥'❛✐"❡* ♦✉ ♥♦♥✮ ; ✈❛"✐❛♥❝❡ ❝♦♥*(❛♥(❡ ✭a(φ) = σ²✮✱ ❧❡* ♠♦✲

❞3❧❡* ❧♦❣✐*(✐4✉❡* ❜✐♥❛✐"❡* ♦✉ ♣♦❧②(♦♠✐4✉❡*✱ ❧❡* ♠♦❞3❧❡* ❞❡ "'❣"❡**✐♦♥ F♦✐**♦♥ ♦✉

●❛♠♠❛ *♦♥( ❞❡* ❡①❡♠♣❧❡* ❞❡ ♠♦❞3❧❡* ❧✐♥'❛✐"❡* ❣'♥'"❛❧✐*'*✳ F❛"♠✐ ❝❡* ♠♦❞3❧❡* ♥♦✉*

♥♦✉* ✐♥('"❡**❡"♦♥* ♣❛"(✐❝✉❧✐3"❡♠❡♥( ❞❛♥* ❧❛ *✉✐(❡ ❛✉ ♠♦❞3❧❡ ❞❡ "'❣"❡**✐♦♥ ❧♦❣✐*(✐4✉❡✳

✷✳✶✳ ▲❡ ♠♦❞(❧❡ ❧✐♥,❛✐.❡ ✺

✷✳✶ ▲❡ ♠♦❞(❧❡ ❧✐♥,❛✐.❡

▲❡ ♠♦❞%❧❡ ❧✐♥)❛✐+❡ ❡,- ,♦✉✈❡♥- ❧❡ ♣+❡♠✐❡+ ♦✉-✐❧ ❞❡ ,-❛-✐,-✐1✉❡ ✐♥❢)+❡♥-✐❡❧❧❡ ♠✐,

❡♥ ♦❡✉✈+❡✳ ■❧ ♣❡+♠❡- ❞✬❡①♣❧✐1✉❡+ ✉♥❡ ✈❛+✐❛❜❧❡ Y ✭❛♣♣❡❧)❡ ✈❛+✐❛❜❧❡ ❞)♣❡♥❞❛♥-❡✮ ♣❛+

p ✈❛+✐❛❜❧❡, ❡①♣❧✐❝❛-✐✈❡, X = (X1, X2, ..., Xp)✳ ;♦✉+ ❝❡ ❢❛✐+❡✱ ♥♦✉, ❞✐,♣♦,♦♥, ❞❡ n +)❛❧✐,❛-✐♦♥, ✐♥❞)♣❡♥❞❛♥-❡, (X1, Y1), . . . ,(Xn, Yn) ❞✉ ❝♦✉♣❧❡ (X, Y)✳ ▲❡ ❜✉- ❡,- ❞❡

♠♦❞)❧✐,❡+ ❧❛ ❞)♣❡♥❞❛♥❝❡ ❞❡ ❧❛ ✈❛+✐❛❜❧❡ +)♣♦♥,❡ Y ♣❛+ +❛♣♣♦+- ❛✉① ✈❛+✐❛❜❧❡, ❡①♣❧✐❝❛✲

-✐✈❡,X1, X2, . . . , Xp✳ ❉❡, )-✉❞❡, ❞✐✈❡+,❡, ❡- ❞)-❛✐❧❧)❡, ♦♥- )-) ❡✛❡❝-✉)❡, ,✉+ ❧❡ ♠♦❞%❧❡

❧✐♥)❛✐+❡✳ ;♦✉+ ♣❧✉, ❞❡ ❞)-❛✐❧, ♦♥ ♣❡✉- ,❡ +)❢)+❡+ @ ✭❆③❛C, ✫ ❇❛+❞❡- ✷✵✵✻✮✳

▲❡ ♠♦❞%❧❡ ❧✐♥)❛✐+❡ ,-❛♥❞❛+❞ -+❛❞✉✐- ❧❛ ❞)♣❡♥❞❛♥❝❡ ❧✐♥)❛✐+❡ ❞❡ ❧✬❡,♣)+❛♥❝❡ ❡♥ β = (β1, . . . , βp)^⊤✱ ✉♥ ♣❛+❛♠%-+❡ ✐♥❝♦♥♥✉ ♥♦♥✲❝♦♥-+❛✐♥- ❞❡ R^p ❡- ✐❧ ,✬)❝+✐- ✿

Yi =β^⊤❳i+εi, i= 1, ..., n ✭✷✳✶✮

▲❡ ♠♦❞%❧❡ ❧✐♥)❛✐+❡ ❡,- ❝❛+❛❝-)+✐,) ♣❛+ ❧❡, ❤②♣♦-❤%,❡, ,✉✐✈❛♥-❡, ✿

✕ E(ε_i|❳i) = 0✱ ♣♦✉+ i= 1, ..., n ✭❧❡, ❡++❡✉+, ,♦♥- ❝❡♥-+)❡,✮ ❀

✕ var(εi) = σ²✱ ♣♦✉+i= 1, ..., n✭❧❛ ✈❛+✐❛♥❝❡ ❞❡, ❡++❡✉+, ❡,- ❝♦♥,-❛♥-❡ ✿ ❤②♣♦-❤%,❡

❞✬❤♦♠♦,❝)❞❛,-✐❝✐-)✮ ❀

✕ ❧❡, ✈❛+✐❛❜❧❡,εi, i= 1, ..., n ,♦♥- ✐♥❞)♣❡♥❞❛♥-❡, ❡- ❞❡ ❧♦✐ ❣❛✉,,✐❡♥♥❡✳

;❧❛P♦♥, ♥♦✉, ♠❛✐♥-❡♥❛♥- ❞❛♥, ❧❡ ❝❛, ♦Q ❧❛ ✈❛+✐❛❜❧❡ @ ❡①♣❧✐1✉❡+ ❨ ❡,- 1✉❛❧✐-❛-✐✈❡ ♦✉

❞❡ -②♣❡ ❢❛❝-❡✉+ ✭,❡①❡✱ ❝♦✉❧❡✉+✱ ♣+),❡♥❝❡ ♦✉ ❛❜,❡♥❝❡ ❞✬✉♥❡ ♠❛❧❛❞✐❡✳✳✳✮✳ ❈❡--❡ ✈❛+✐❛❜❧❡

♣♦,,%❞❡ ✉♥ ♥♦♠❜+❡ ✜♥✐ ❞❡ ♠♦❞❛❧✐-), g1, . . . , gm✳ ▲❡ ♣+♦❜❧%♠❡ ❝♦♥,✐,-❡ @ ❡①♣❧✐1✉❡+

❧✬❛♣♣❛+-❡♥❛♥❝❡ ❞✬✉♥ ✐♥❞✐✈✐❞✉ @ ✉♥ ❣+♦✉♣❡ @ ♣❛+-✐+ ❞❡, p ✈❛+✐❛❜❧❡, ❡①♣❧✐❝❛-✐✈❡,✱ ♦♥

♣❛+❧❡+❛ ❛❧♦+, ❞❡ ❞✐,❝+✐♠✐♥❛-✐♦♥ ❛✉ ❧✐❡✉ ❞❡ +)❣+❡,,✐♦♥✳

■❧ ❡,- ❜✐❡♥ ❡♥-❡♥❞✉ ✐♠♣♦,,✐❜❧❡ ❞❡ ♠♦❞)❧✐,❡+ ❞✐+❡❝-❡♠❡♥- ❧❛ ✈❛+✐❛❜❧❡ Y ♣❛+ ✉♥❡ +❡❧❛✲

-✐♦♥ ❧✐♥)❛✐+❡ ✭✐♠❛❣✐♥♦♥, 1✉❡ ❨ ,♦✐- ❧❡ ,❡①❡ ❞✬✉♥❡ ♣❡+,♦♥♥❡ ♦✉ ,♦♥ )-❛- ❞❡ ,❛♥-)✮✳

✷✳✶✳ ▲❡ ♠♦❞(❧❡ ❧✐♥,❛✐.❡ ✻

❆✜♥ ❞❡ ♣❛❧❧✐❡) * ❝❡,,❡ ❞✐✣❝✉❧,/✱ ♦♥ ✈❛ 3✬✐♥,/)❡33❡) ❛✉① ♣)♦❜❛❜✐❧✐,/3 P(Y =gk|❳ =x)✳

❙✉♣♣♦3♦♥3 ♣♦✉) 3✐♠♣❧✐✜❡) :✉❡ ❧❛ ✈❛)✐❛❜❧❡ Y ♣)❡♥♥❡ ✉♥✐:✉❡♠❡♥, ❞❡✉① ✈❛❧❡✉)3 ✿ 0 ✭❣)♦✉♣❡ 0✮ ♦✉ 1 ✭❣)♦✉♣❡ 1✮✳ ▲❛ ❝♦♥♥❛✐33❛♥❝❡ ❞❡ P(Y = 1|❳ = x) ✐♠♣❧✐:✉❡

❝❡❧❧❡ ❞❡ P(Y = 0|❳ = x) ✿ ✐❧ 3✉✣, ♣❛) ❝♦♥3/:✉❡♥, ❞❡ ♠♦❞/❧✐3❡) ❧❛ ♣)♦❜❛❜✐❧✐,/

p(x) =P(Y = 1|❳=x)✳ ❖♥ ♣❡✉, ♣❛) ❡①❡♠♣❧❡ ❡♥✈✐3❛❣❡) ✉♥❡ )❡❧❛,✐♦♥ ❞❡ ❧❛ ❢♦)♠❡

p(x) =β₀+β₁x₁+, . . . ,+β_px_p =β^⊤x. ✭✷✳✷✮

❈❡,,❡ ❛♣♣)♦❝❤❡ ♣♦33E❞❡ ♣❧✉3✐❡✉)3 ❝❛)❛❝,/)✐3,✐:✉❡3 ✿

✕ ❘❡♠❛):✉♦♥3 ,♦✉, ❞✬❛❜♦)❞ :✉❡ ❧❛ ✈❛)✐❛♥❝❡ ❞❡ Y|❳ = x ✈❛✉, p(x)(1−p(x))✳

❈♦♥,)❛✐)❡♠❡♥, ❛✉ ♠♦❞E❧❡ ❧✐♥/❛✐)❡ ❝❧❛33✐:✉❡✱ ❝❡,,❡ ✈❛)✐❛♥❝❡ ♥✬❡3, ♣❛3 ❝♦♥3,❛♥,❡

❡, ♣❛) ❝♦♥3/:✉❡♥, ❧✬❤②♣♦,❤E3❡ ❝❧❛33✐:✉❡ ❞✬❤♦♠♦3❝/❞❛3,✐❝✐,/ ❞❡3 )/3✐❞✉3 ♥❡ 3❡)❛

♣❛3 ✈/)✐✜/❡✳

✕ ▲❡ ❢❛✐, :✉✬❛✉❝✉♥❡ )❡3,)✐❝,✐♦♥ ♥❡ 3♦✐, ❡✛❡❝,✉/❡ 3✉) ❧❡3 β ✐♠♣❧✐:✉❡ :✉❡ β^⊤x

♣❡✉, ♣)❡♥❞)❡ ♥✬✐♠♣♦),❡ :✉❡❧❧❡ ✈❛❧❡✉) 3✉) R✳ ❈❡ :✉✐ ♣❡✉, J,)❡ ❣J♥❛♥, ♣♦✉)

❧✬❡3,✐♠❛,✐♦♥ ❞✬✉♥❡ ♣)♦❜❛❜✐❧✐,/✳

K♦✉) ❝❡3 )❛✐3♦♥3✱ ♥♦✉3 ❞❡✈♦♥3 /,❡♥❞)❡ ❧❡ ♠♦❞E❧❡ ❧✐♥/❛✐)❡ ❝❧❛33✐:✉❡ ❛✉① ❝❛3 ♦L ✿

✕ Y ♣❡✉, J,)❡ ✉♥❡ ✈❛)✐❛❜❧❡ :✉❛❧✐,❛,✐✈❡ ✭♣❛) ❡①❡♠♣❧❡✱ ♣)/3❡♥❝❡ ♦✉ ❛❜3❡♥❝❡ ❞✬✉♥❡

♠❛❧❛❞✐❡✱ ❛♣♣❛),❡♥❛♥❝❡ * ✉♥❡ ❝❛,/❣♦)✐❡✳✳✳✮ ❀

✕ ❧❡3 ❡))❡✉)3 ♣❡✉✈❡♥, ♥❡ ♣❛3 ❛✈♦✐) ❧❛ ♠J♠❡ ✈❛)✐❛♥❝❡ ✭3✬❛✛)❛♥❝❤✐) ❞❡ ❧✬❤②♣♦,❤E3❡

❞✬❤♦♠♦3❝/❞❛3,✐❝✐,/✮✳

✷✳✷✳ ▲❡ ♠♦❞'❧❡ ❞❡ )*❣)❡,,✐♦♥ ❧♦❣✐,/✐0✉❡ ✼

✷✳✷ ▲❡ ♠♦❞'❧❡ ❞❡ )*❣)❡,,✐♦♥ ❧♦❣✐,/✐0✉❡

✷✳✷✳✶ ■♥%&♦❞✉❝%✐♦♥

▲❡ ♠♦❞%❧❡ ❞❡ '(❣'❡**✐♦♥ ❧♦❣✐*-✐.✉❡ ❡*- ❣(♥('❛❧❡♠❡♥- ✉-✐❧✐*( ♣♦✉' ♠♦❞(❧✐*❡' ✉♥❡ '(✲

♣♦♥*❡ ❜✐♥❛✐'❡ ❞❛♥* ❧❡ ❝❛❞'❡ ❞❡ ❞♦♥♥(❡* ♠(❞✐❝❛❧❡*✳ ❯♥ ❡①❡♠♣❧❡ ❞❡ '(♣♦♥*❡ ❜✐♥❛✐'❡ ❡*-

❧❡ *-❛-✉- ❞✬✐♥❢❡❝-✐♦♥ ✭✐♥❢❡❝-( ✈! ♥♦♥ ✐♥❢❡❝-(✮ < ❧✬(❣❛'❞ ❞❡ ❝❡'-❛✐♥❡* ♠❛❧❛❞✐❡*✳ ❯♥ ♠♦✲

❞%❧❡ ❞❡ '(❣'❡**✐♦♥ ❧♦❣✐*-✐.✉❡ ♣❡✉- =-'❡ ✉-✐❧✐*( ♣♦✉' (-✉❞✐❡' ❧❛ '❡❧❛-✐♦♥ ❡♥-'❡ ❧❡ *-❛-✉-

❞✬✐♥❢❡❝-✐♦♥ ❡- ❧❡* ❞✐✛('❡♥-❡* ❝♦✈❛'✐❛❜❧❡* .✉✐ ♣❡✉✈❡♥- =-'❡ ♣❛' ❝♦♥-'❡ *♦✐- .✉❛❧✐-❛-✐✈❡*✱

*♦✐- .✉❛♥-✐-❛-✐✈❡* ✭✈♦✐' ✭❍♦*♠❡' ✫ ▲❡♠❡*❤♦✇ ✷✵✵✵✮ ❡- ✭❆♠✐♥♦- ✫ ❉❛♠♦♥ ✷✵✵✷✮✮✳

❙✐Yi '❡♣'(*❡♥-❡ ❧❡ *-❛-✉- ❞✬✐♥❢❡❝-✐♦♥ ♣♦✉' ❧❡ i ^♠❡ ✐♥❞✐✈✐❞✉ ❞❛♥* ❧✬(❝❤❛♥-✐❧❧♦♥ ❞❡ -❛✐❧❧❡

n ✭Y_i = 1 *✐ ❧✬✐♥❞✐✈✐❞✉ ❡*- ✐♥❢❡❝-(✱ ❡- Y_i = 0 *✐♥♦♥✮✱ ❡- X_i '❡♣'(*❡♥-❡ ❧❡ ✈❡❝-❡✉'

❞❡ ♣'(❞✐❝-❡✉' ❧✐♥(❛✐'❡ ❞❡ ❞✐♠❡♥*✐♦♥ p ❝♦''❡*♣♦♥❞❛♥-✱ ❧❡ ♠♦❞%❧❡ ❞❡ '(❣'❡**✐♦♥ ❧♦❣✐*✲

-✐.✉❡ -'❛❞✉✐- ❧❛ '❡❧❛-✐♦♥ ❡♥-'❡ Yi ❡- X_i = (X1, X2, ..., Xp) ❡♥ -❡'♠❡ ❞❡ ♣'♦❜❛❜✐❧✐-(

❝♦♥❞✐-✐♦♥♥❡❧❧❡ P(Y_i = 1|X_i) ♣♦✉' ❧❡ *-❛-✉- ❞✬✐♥❢❡❝-✐♦♥✱ ❝♦♠♠❡ *✉✐- ✿ log

P(Yi = 1|X_i) 1−P(Yi = 1|X_i)

=β^⊤X_i, ✭✷✳✸✮

♦L β ∈ R^p ❡*- ✉♥ ♣❛'❛♠%-'❡ ✐♥❝♦♥♥✉ ✭< ❡*-✐♠❡'✮✳ ❯♥❡ ❧✐--('❛-✉'❡ ❞(-❛✐❧❧(❡ ❛ (-(

❝♦♥*❛❝'(❡ ❥✉*.✉✬✐❝✐ < ❧✬✐♥❢('❡♥❝❡ *-❛-✐*-✐.✉❡ ❞❡* ♠♦❞%❧❡* ❞❡ '(❣'❡**✐♦♥ ❧♦❣✐*-✐.✉❡✳ ▲❡*

♣'♦❝(❞✉'❡* ❞✬❡*-✐♠❛-✐♦♥ ❡- ❞❡ -❡*-* ♣♦✉' ❝❡--❡ ❝❛-(❣♦'✐❡ ❞❡ ♠♦❞%❧❡* *♦♥- ♠❛✐♥-❡♥❛♥-

❜✐❡♥ (-❛❜❧✐❡* ❡- *♦♥- ❞✐*♣♦♥✐❜❧❡* ❞❛♥* ❧❡* ❧♦❣✐❝✐❡❧* *-❛♥❞❛'❞* ❞❡ *-❛-✐*-✐.✉❡✳ ❊♥ ♣❛'✲

-✐❝✉❧✐❡'✱ ❧✬❡*-✐♠❛-❡✉' ❞✉ ♠❛①✐♠✉♠ ❞❡ ✈'❛✐*❡♠❜❧❛♥❝❡ ❞❡ β ❡*- ♦❜-❡♥✉ ❡♥ '(*♦❧✈❛♥-

❧✬(.✉❛-✐♦♥ ❞✉ *❝♦'❡✳ ▲❡* '(*✉❧-❛-* ❛*②♠♣-♦-✐.✉❡* ♣♦✉' ❝❡- ❡*-✐♠❛-❡✉' *♦♥- ❞♦♥♥(*

❞❛♥* ✭●♦✉'✐('♦✉① ✫ ▼♦♥❢♦'- ✶✾✽✶✮ ❡- ✭❋❛❤'♠❡✐' ✫ ❑❛✉❢♠❛♥♥ ✶✾✽✺✮✱ ❡♥-'❡ ❛✉-'❡*✳

▲❡ ❧❡❝-❡✉' ♣♦✉''❛ *❡ '(❢('❡' < ✭❍♦*♠❡' ✫ ▲❡♠❡*❤♦✇ ✷✵✵✵✮ ❡- ✭❍✐❧❜❡ ✷✵✵✾✮ ♣♦✉' ❞❡*

(-✉❞❡* ❞(-❛✐❧❧(❡* ❡- ❞❡ ♥♦♠❜'❡✉① ❡①❡♠♣❧❡*✳

▲❛ ❢♦♥❝-✐♦♥ ❞❡ ❧✐❡♥ ❧♦❣✐$ ❡*- ❧❛ ♣❧✉* ❣(♥('❛❧❡♠❡♥- ✉-✐❧✐*(❡✳ ▲✬✐♥-('=- ❞❡ ❝❡--❡ ❢♦♥❝-✐♦♥

'(*✐❞❡ ❞❛♥* ❧❛ *✐♠♣❧✐❝✐-( ❞❡ ♣❛**❛❣❡ < ❧✬❡*-✐♠❛-✐♦♥ ❞✬✉♥ ♦❞❞*✲'❛-✐♦ ✭❖❘✮ ♦✉ '❛♣♣♦'-

✷✳✷✳ ▲❡ ♠♦❞'❧❡ ❞❡ )*❣)❡,,✐♦♥ ❧♦❣✐,/✐0✉❡ ✽

❞❡" ❝♦%❡" &✉✐ ♠❡"✉*❡ ❧❛ ❢♦*❝❡ ❞❡ ❧✬❛""♦❝✐❛%✐♦♥ ❡♥%*❡ ❧❛ ✈❛*✐❛❜❧❡ ❡♥❞♦❣3♥❡ ❡% ✉♥❡ ✈❛✲

*✐❛❜❧❡ ❡①♦❣3♥❡✳ ❊♥ ♣❛*%✐❝✉❧✐❡*✱ ❡♥ :♣✐❞:♠✐♦❧♦❣✐❡✱ ❧❡" *:"✉❧%❛%" ♣❡✉✈❡♥% ;%*❡ ❛✐":♠❡♥%

✐♥%❡*♣*:%:"✳ ▲❡" ❝♦❡✣❝✐❡♥%" ❡"%✐♠:" ♣❛* ❧❡ ♠♦❞3❧❡ "♦♥% ❡♥ ❡✛❡% ❧✐:" ♠❛%❤:♠❛%✐&✉❡✲

♠❡♥% @ ❧✬♦❞❞"✲*❛%✐♦✱ ❜✐❡♥ &✉✬✐❧ ♥❡ "♦✐% &✉✬✉♥❡ ❛♣♣*♦①✐♠❛%✐♦♥ ❞✉ *✐"&✉❡ *❡❧❛%✐❢✳ ▲❛

♠:%❤♦❞❡ ❞❡ *:❣*❡""✐♦♥ ❧♦❣✐"%✐&✉❡ ❡"% ❞♦♥❝ ❧❛ ♠:%❤♦❞❡ ♠✉❧%✐✈❛*✐❛❜❧❡ ❞❡ ❝❤♦✐① ♣♦✉*

*❡❝❤❡*❝❤❡* ❞❡" ❢❛❝%❡✉*" ❞❡ *✐"&✉❡ ♦✉ ❞❡" ❢❛❝%❡✉*" ♣*♦%❡❝%❡✉*" ❞❡ ♠❛❧❛❞✐❡✳

−4 −2 0 2 4

0.00.20.40.60.81.0

x

Fonction de régression logistique

❋✐❣✳ ✷✳✶ ✕ ❋♦♥❝%✐♦♥ ❞❡ *:❣*❡""✐♦♥ ❧♦❣✐"%✐&✉❡

✷✳✷✳✷ ■❞❡♥&✐✜❛❜✐❧✐&, ❞✉ ♠♦❞0❧❡

❖♥ *❛♣♣❡❧❧❡ &✉❡ ❧❡ ♠♦❞3❧❡ ❡"% ✐❞❡♥%✐✜❛❜❧❡ "✐ ♣♦✉* β 6= ˜β✱ ❧❡" ❧♦✐" ❞❡ {(Yi|xi), i= 1, . . . , n} ❛""♦❝✐:❡" @ β ❡% β˜ "♦♥% ❞✐✛:*❡♥%❡"✳ ❚♦✉% ❝♦♠♠❡ ♣♦✉* ❧❡ ♠♦❞3❧❡ ❧✐♥:❛✐*❡

✭●✉②♦♥ ✷✵✵✶✮✱ ✉♥❡ ❝♦♥❞✐%✐♦♥ ♥:❝❡""❛✐*❡ ♣♦✉* ♣♦✉✈♦✐* ❡"%✐♠❡* ❧❡" ♣❛*❛♠3%*❡" ❡"% &✉❡

❧✬:❝❤❛♥%✐❧❧♦♥ *❡♥❞❡ ❧❡ ♠♦❞3❧❡ ✐❞❡♥%✐✜❛❜❧❡ ✿ ❧❡" ❧♦✐" ❞❡ {(Yi|X_i = xi), i = 1, . . . , n}

❛""♦❝✐:❡" @ β ❡% β˜ "♦♥% ❞✐✛:*❡♥%❡"✳ ❈♦♠♠❡ Yi|❳i = xi "✉✐% ✉♥❡ ❧♦✐ ❞❡ ❇❡*♥♦✉❧❧✐ ❞❡

♣❛*❛♠3%*❡ p(xi)❡% &✉❡ ❧❛ ❢♦♥❝%✐♦♥ ❧♦❣✐$ ❡"% "%*✐❝%❡♠❡♥% ❝*♦✐""❛♥%❡✱ ❝❡%%❡ ❝♦♥❞✐%✐♦♥