Minato
March 25, 2023
690

# 勉強会資料 / “Asymptotic Statistics” Section 6.1

A.W. van der Vaart著 “Asymptotic Statistics”の勉強会資料を公開します。

6.1節 Likelihood Ratios

March 25, 2023

## Transcript

1. ୈ 10 ճ ษڧձ
A.W. van der Vaart ”Asymptotic Statistics”
Section 6.1 ໬౓ൺ
Minato
2023 ೥ 3 ݄ 25 ೔

2. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
〤」〶〠
2 / 28
ࠓճ〣಺༰ Chapter 6ɿContiguity [1]
Section 6.1ɿLikelihood Ratios (p.85 ʙ 87)
ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ (ಛҟੑ)
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
Lemma 6.2ɿ໬౓ൺ〣ੑ࣭
A.W. van der Vaart
”Asymptotic Statistics”

3. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
ه๏〣४උ
Table: ओ〟ه๏
N ࣗવ਺શମ〣ू߹
R ࣮਺શମ〣ू߹
Rk ू߹ {x = (x1
, . . . , xk
)⊤ : xi
∈ R (i = 1, . . . , k)}
E[·] ظ଴஋
Q ≪ P Q ⿿ P 〠ؔ「〛ઈର࿈ଓ
P ⊥ Q P 〝 Q 〤௚ަ
A
ࢦࣔؔ਺
Ac ू߹ A 〣ิू߹
ɹ
จষத〣৭〣۠ผ
ࠇ৭ɿจݙ [1] 〠ॻ⿾ぁ〛⿶぀಺༰ぇ࿨༁「〔〷〣
ࢵ৭ɿߦؒぇิ〘〔〷〣 (〉ࢦఠɾぢゐアぷ௖々〳『〝޾⿶〜『)
〤〟〕
៬ ৭ 1ɿจݙ [1] 〣ओுぇಋग़『぀ࡍ〠༻⿶〔໋୊
1ʰ៬৭ʢ〤〟〕⿶あʣ〝〤ɺݹ。⿾〾஌〾ぁ〔ཟછ〶〣৭໊〜ɺཟ৭〽〿〷ബ。ઙೢ৭〽〿〷ೱ⿶
৭〣〈〝〜『ɻ
3 / 28

4. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
໨࣍
1 ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
2 ゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
3 ໬౓ൺ〣ੑ࣭
3 / 28

5. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
໨࣍
1 ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
2 ゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
3 ໬౓ൺ〣ੑ࣭
4 / 28

6. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
ଌ౓〣ઈର࿈ଓੑ
Def: ଌ౓〣ઈର࿈ଓੑ
Ωɿඇۭू߹
AɿΩ ্〣 σ-ू߹ମ
P, QɿՄଌۭؒ (Ω, A) ্〣ଌ౓
ɹ
A 〣೚ҙ〣ݩ A 〠ର「ɺ
P(A) = 0 ɹ〟〾〥ɹɹ Q(A) = 0
⿿੒〿ཱ〙〝　ɺQ 〤 P 〠ؔ「〛ઈର࿈ଓ (absolutely continuous) 〜⿴぀〝
⿶⿶ɺQ ≪ P 〝ද『ɻ
5 / 28

7. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
ଌ౓〣௚ަੑ
Def: ଌ౓〣௚ަੑ
Ωɿඇۭू߹
AɿΩ ্〣 σ-ू߹ମ
P, QɿՄଌۭؒ (Ω, A) ্〣ଌ౓
ɹ
⿴぀෦෼ू߹ ΩP
, ΩQ
(⊂ Ω) ⿿ଘࡏ「〛ɺ
Ω = ΩP
∪ ΩQ
, ΩP
∩ ΩQ
= ∅ ⿾〙 P(ΩQ
) = 0 = Q(ΩP
)
⿿੒〿ཱ〙〝　ɺP 〝 Q 〤௚ަ (orthogonal) 〜⿴぀〝⿶⿶ɺP ⊥ Q 〝ද『ɻ
”௚ަ”ぇ”ಛҟ”(singular) 〝ݺ〫〈〝〷⿴぀ɻ
”P charges only ΩP
and Q lives on the set ΩQ
, which is disjoint with the
support of P.” 2
2[1] p.85 〽〿Ҿ༻ɻ
6 / 28

8. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
໨࣍
1 ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
2 ゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
3 ໬౓ൺ〣ੑ࣭
7 / 28

9. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
ີ౓〣Ծఆ〝ه๏〣४උ
Ұൠ〠ɺՄଌۭؒ (Ω, A) ্〣 2 〙〣ଌ౓ P, Q 〤ઈର࿈ଓ〝〷௚ަ〝〷ݶ
〾〟⿶ɻ
µ ぇՄଌۭؒ (Ω, A) ্〣ଌ౓〝「ɺP 〝 Q 〤〒ぁ〓ぁ µ 〠ؔ『぀ີ౓
p, q ぇ〷〙〷〣〝Ծఆ「ɺू߹ ΩP
, ΩQ

Ωp
= {p > 0}, Ωq
= {q > 0}
〠〽〿ఆٛ『぀ɻ345
P ⿿ଌ౓ µ 〠ؔ「〛ີ౓ p ぇ〷〙〝〤ɺ
P(A) =

A
p dµ A ∈ A (1)
⿿੒〿ཱ〙〽⿸〟ඇෛՄଌؔ਺ p : Ω → [0, ∞) ⿿ଘࡏ『぀〈〝ぇҙຯ『぀ɻ
3{p > 0} 〤ू߹ {ω ∈ Ω : p(ω) > 0} 〣ུه〜⿴぀ɻ
4໋୊ (Մଌؔ਺〝Մଌू߹〣ؔ܎)ɿ(Ω, A) ぇՄଌۭؒ〝『぀ɻؔ਺ f : Ω → R ⿿Մଌؔ਺〜⿴
぀〝　ɺ೚ҙ〣࣮਺ a ∈ R 〠ର「〛 {f > a}, {f < a}, {f = a}, {f ≥ a}, {f ≤ a} 〤Մଌू߹
〠〟぀ɻ
5্ه〣໋୊〽〿 {p > 0}, {q > 0} 〟〞〤Մଌू߹〜⿴぀ɻ
8 / 28

10. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
ଌ౓〝つれがぷ〣ؔ܎ਤ
9 / 28

ΩP
ΩQ
p > 0
q > 0
p > 0
q = 0
p = 0
q > 0
p = q = 0
Figure 6.1. Supports of measures.

11. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
ີ౓〝つれがぷ〣ؔ܎
P(Ωc
p
) = P({p = 0}) =

{p=0}
p dµ =

0 dµ = 0
〽〿ɺP 〣つれがぷ〤ू߹ Ωp
্〠⿴぀ɻ〳〔ɺ6 7
µ(Ωp
∩ Ωq
) > 0 〣〝　 P(Ωp
∩ Ωq
) > 0, Q(Ωp
∩ Ωq
) > 0. (2)
ࣜ (2) 〣આ໌
P(Ωp
∩ Ωq
) = 0 ⇒

Ωp∩Ωq
p dµ = 0

Ωp∩Ωq
p dµ = 0
⇒ Ωp∩Ωq
p = 0 µ-a.e.
⇒ µ ({p > 0} ∩ (Ωp
∩ Ωq
)) = 0
⇒ µ (Ωp
∩ Ωq
) = 0.
6໋୊ (ඇෛ஋Մଌؔ਺〣ੵ෼〝ྵू߹)ɿ(Ω, A, µ) ぇଌ౓ۭؒ〝『぀ɻඇෛ஋Մଌؔ਺
f : Ω → [0, ∞) 〠ର「ɺ

f dµ = 0 ⇐⇒ f = 0 µ-a.e.
7ิ଍ (”〰〝え〞”)ɿੑ࣭ P ぇຬ〔《〟⿶఺〣ू߹⿿⿴぀ଌ౓ 0 〣ू߹〠ؚ〳ぁ぀〝　ɺ〰〝え〞
ࢸ぀〝〈あ〜 (almost everywhere) P ぇຬ〔『〝⿶⿸ɻ
10 / 28

12. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
゚よがそ෼ղ
〈〈〜ɺଌ౓ Q 〤
Q = Qa + Q⊥ (3)
〝ॻ々぀ɻ〔〕「ɺQa, Q⊥ 〤ҎԼ〜ఆٛ《ぁ぀ଌ౓ 8ɿ
Qa(A) = Q(A ∩ {p > 0}), Q⊥(A) = Q(A ∩ {p = 0}) A ∈ A. (4)
ࣜ (3) 〤 P 〠ؔ『぀ Q 〣゚よがそ෼ղ (Lebesgue decomposition) 〝ݺ〥ぁ぀ɻ
Qa 〤 absolutely partɺQ⊥ 〤 singular part 〝ݺ〥ぁ぀ɻ
ɹ
ࣜ (3) 〣આ໌
ଌ౓〣ՄࢉՃ๏ੑ⿾〾ɺ
Qa(A) + Q⊥(A) = Q(A ∩ {p > 0}) + Q(A ∩ {p = 0})) = Q(A).
Lemma 6.2 〜ࣔ《ぁ぀〽⿸〠ɺQa ≪ P, Q⊥ ⊥ P ⿿੒〿ཱ〙ɻ《〾〠ɺ೚ҙ
〣Մଌू߹ A ∈ A 〠ର「ɺQa(A) =

A
q
p
dP ⿿੒〿ཱ〙ɻ
8໋୊ (ଌ౓〣੍ݶ)ɿ(Ω, A, µ) ぇଌ౓ۭؒ〝『぀ɻB ぇ Ω 〣෦෼ू߹〝「〔〝　ɺ
µB(A) = µ(B ∩ A) (A ∈ A) 〠〽〿ؔ਺ µB
ぇఆ〶぀〝ɺµB
〷〳〔 (Ω, A) ্〣ଌ౓〠〟぀ɻ〈
ぁ〽〿ɺQa, Q⊥ 〷 (Ω, A) ্〣ଌ౓〠〟぀ɻ
11 / 28

13. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
Qa(A) =

A
q
p
dP A ∈ A (5)
ࣜ (5) 〽〿ɺq/p 〤 P 〠ؔ『぀ Qa 〣ີ౓〝〟぀⿾〾ɺ〒ぁぇ
dQ
dP
〝ॻ。ɻ֬
཰ଌ౓ P 〠ؔ『぀ q/p 〣ੑ࣭〠〣〴ؔ৺⿿⿴぀〔〶ɺp = 0 〠⿼々぀ q/p 〤ٞ
࿦「〟⿶ɻఆٛ〽〿ɺີ౓
dQ
dP
〤 P-a.s. 〣ҙຯ〜Ұҙ〜⿴぀ɻ
্ه〣આ໌
೚ҙ〣Մଌू߹ A 〠ର「〛
Qa(A) =

A
q
p
dP =

A
r dP
〟぀ີ౓ r ⿿ଘࡏ「〔〝『぀〝ɺੵ෼〣ઢܗੑ〽〿

A
(
q
p
− r
)
dP = 0 〜⿴
぀⿾〾ɺq/p = r ⿿ P-a.e. 〜੒〿ཱ〙ɻ9
9໋୊ (ੵ෼〣ੑ࣭)ɿ(Ω, A, P ) ぇଌ౓ۭؒ〝「ɺf ぇՄଌؔ਺〝『぀ɻ೚ҙ〣Մଌू߹ A 〠ؔ「

A
f dP = 0 ⿿੒〿ཱ〙〝　ɺf = 0 ⿿ P -a.e. 〜੒〿ཱ〙ɻ
12 / 28

14. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
《〾〠ɺ
dQ
dP
〹゚よがそ෼ղ〤ଌ౓ µ 〝ಠཱɿଌ౓ µ 〣औ〿ํ〠〽〘〛 P 〹
Q 〣ີ౓ࣗମ〤มい〿⿸぀⿿ɺ
dQ
dP
〹゚よがそ෼ղ〤 µ 〣औ〿ํ〠〽〾〟⿶ɻ
゚よがそ෼ղ⿿ µ 〣औ〿ํ〠〽〾〟⿶〈〝 (1/2)
µ′ ぇՄଌۭؒ (Ω, A) ্〣ଌ౓ɺP 〝 Q 〣 µ′ 〠ؔ『぀ີ౓ぇ p′, q′ 〝『぀ɻ
೚ҙ〣Մଌू߹ A 〠ର「〛
Q(A ∩ {p > 0}) = Q(A ∩ {p′ > 0})
〜⿴぀〈〝ぇࣔ【〥ྑ⿶ɻP({p = 0}) = P({p′ = 0}) = 0 〽〿

{p′=0}
p dµ = 0.
〽〘〛ɺµ ({p′ = 0} ∩ {p > 0}) = 0. 〈ぁ〽〿ɺ೚ҙ〣Մଌू߹ A 〠ର「ɺ
Q(A ∩ {p > 0}) =

A∩{p>0}
q dµ
=

A∩{p>0}∩{p′=0}
q dµ +

A∩{p>0}∩{p′>0}
q dµ
13 / 28

15. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
゚よがそ෼ղ⿿ µ 〣औ〿ํ〠〽〾〟⿶〈〝 (2/2)
A ∩ {p > 0} ∩ {p′ = 0} 〤 µ-ྵू߹〕⿾〾

A∩{p>0}∩{p′=0}
q dµ = 0.
Q(A ∩ {p > 0}) =

A∩{p>0}∩{p′>0}
q dµ
= Q(A ∩ {p > 0} ∩ {p′ > 0}).
ಉ༷〠ɺ
Q(A ∩ {p′ > 0}) = Q(A ∩ {p > 0} ∩ {p′ > 0}).
〈ぁ〽〿ɺ
Q(A ∩ {p > 0}) = Q(A ∩ {p′ > 0})
্ࣜ〤೚ҙ〣Մଌू߹ A 〠ؔ「〛੒〿ཱ〙ɻ〽〘〛ɺQa(A) 〣஋〤ଌ౓ µ 〣
औ〿ํ〠〽〾〟⿶ɻ Q = Qa + Q⊥ 〽〿ɺQ⊥ 〷ଌ౓ µ 〣औ〿ํ〠〽〾〟⿶〈
〝⿿い⿾぀ɻ
14 / 28

16. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
dQ
dP
⿿ µ 〣औ〿ํ〠〽〾〟⿶〈〝
ɹ
µ′ ぇՄଌۭؒ (Ω, A) ্〣ଌ౓ɺP 〝 Q 〣 µ′ 〠ؔ『぀ີ౓ぇ p′, q′ 〝『぀ɻ
Lemma 6.2 〣݁Ռ〝ɺ゚よがそ෼ղ⿿ µ 〣औ〿ํ〠〽〾〟⿶〈〝⿾〾ɺ೚ҙ〣
Մଌू߹ A 〠ର「ɺ
Qa(A) =

A
q
p
dP =

A
q′
p′
dP.
〈ぁ〽〿ɺq/p 〷 q′/p′ 〷 P 〠ؔ『぀ Qa 〣ີ౓ɺ『〟い〖
dQ
dP
〠〟〘〛⿶぀
⿾〾ɺ〈ぁ〤 µ 〣औ〿ํ〠〽〾〟⿶〈〝⿿֬ೝ〜　〔ɻ࣮ࡍɺ೚ҙ〣Մଌू߹
A 〠ର「〛

A
(
q
p

q′
p′
)
dP = 0
⿿੒〿ཱ〙〈〝⿾〾ɺ
q
p
=
q′
p′
P-a.e.
15 / 28

17. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
໨࣍
1 ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
2 ゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
3 ໬౓ൺ〣ੑ࣭
16 / 28

18. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
໬౓ൺ〣ੑ࣭
゘へア-ぺぢぶくわඍ෼〤౷ܭֶ〜〤໬౓ൺ (likelihood ratio) 〝⿶⿸໊〜ݺ〥ぁ
぀ɻ໬౓ൺぇ
dQ
dP
: Ω → [0, ∞) 〝「ɺଌ౓ P 〠ؔ『぀໬౓ൺ〣ੑ࣭ぇࣔ『ɻ
Lemma 6.2
P, Q ぇଌ౓ µ 〠ؔ「〛ີ౓ p, q ぇ〷〙֬཰ଌ౓〝『぀ɻ〳〔ɺQa, Q⊥ ぇࣜ
(4) 〜ఆٛ「〔ଌ౓〝『぀ɻ〈〣〝　ɺ࣍⿿੒〿ཱ〙ɻ
(1) ࣍〣 3 ࣜ⿿੒〿ཱ〙ɿ
(a) Q = Qa + Q⊥.
(b) Qa ≪ P.
(c) Q⊥ ⊥ P.
(2) ೚ҙ〣Մଌू߹ A 〠ର「ɺQa(A) =

A
q
p
dP.
(3) Q ≪ P ⇐⇒ Q({p = 0}) = 0 ⇐⇒

q
p
dP = 1.
(1)(a) ⿿੒〿ཱ〙〈〝〤ط〠֬ೝ「〛⿶぀〔〶ɺ1(b) Ҏ߱ぇূ໌『぀ɻ
17 / 28

19. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Lemma 6.2 (1) (b) 〣ূ໌ (1/2)
〳』ɺA ぇՄଌू߹〝『぀〝　ɺҎԼ⿿੒〿ཱ〙ɿ
P(A) = 0 ⇐⇒ p(x) = 0 for µ-almost all x ∈ A. (6)
ࣜ (6) 〣ূ໌
P 〤 µ 〠ؔ「〛ີ౓ p ぇ〷〙〈〝⿾〾 P(A) =

A
p dµ 〜⿴〿ɺ
P(A) = 0 ⇐⇒

A
p dµ = 0
⇐⇒ A
p = 0 µ-a.e.
⇐⇒ p(x) = 0 for µ-almost all x ∈ A.
(
Remark: A
p (x) =
{
p(x) (x ∈ A)
0 (x /
∈ A)
.
)
〒〈〜ɺP(A) = 0 〜⿴぀〽⿸〟Մଌू߹ A ぇ೚ҙ〠〝぀ɻ〈〣〝　ɺࣜ (6)
〝ଌ౓ྵू߹〣ఆٛ⿾〾ɺ
µ (A ∩ {p > 0}) = 0. (7)
18 / 28

20. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Lemma 6.2 (1) (b) 〣ূ໌ (2/2)
ࣜ (7) 〝 Q 〣ଌ౓ µ 〠ؔ『぀ઈର࿈ଓੑ〽〿ɺ
Qa(A) = Q(A ∩ {p > 0}) = 0. (8)
ࣜ (8) 〣આ໌
µ(B) = 0 〜⿴぀〽⿸〟Մଌू߹ B ぇ೚ҙ〠〝぀〝ɺ
Q(B) =

B
q dµ =

B
q dµ = 0.
〽〘〛ɺQ 〤 µ 〠ؔ「〛ઈର࿈ଓɻ10 ࣜ (7) 〽〿ɺA ∩ {p > 0} 〤 µ-ྵू߹〕
⿾〾ɺઈର࿈ଓੑ〽〿 Q-ྵू߹〝〟〿ɺࣜ (8) ⿿੒〿ཱ〙ɻ
ɹ
Ҏ্〽〿ɺP(A) = 0 〜⿴぀೚ҙ〣Մଌू߹ A 〠ؔ「〛 Qa(A) = 0 ⿿੒〿ཱ〙
⿾〾ɺQa ≪ P.
10໋୊ (ྵू߹্〣ੵ෼)ɿ(Ω, A, µ) ぇଌ౓ۭؒ〝『぀ɻ〈〣〝　ɺ೚ҙ〣ඇෛ஋Մଌؔ਺ f 〠
ର「ɺ

Af dµ = 0.
19 / 28

21. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Lemma 6.2 (1) (c) 〣ূ໌
P({p = 0}) = 0 〝 Q⊥({p > 0}) = Q(∅) = 0 〽〿 Q⊥ ⊥ P ⿿੒〿ཱ〙ɻ
ɹ
্ه〣ิ଍
P({p = 0}) = 0 〤ط〠֬ೝ「〛⿼〿ɺ
Q⊥({p > 0}) = Q
(
{p > 0} ∩ {p = 0}
)
= Q(∅) = 0.
〈ぁ〽〿ɺΩ1
= {p = 0}, Ω2
= {p > 0} 〝『぀〝ɺ
Ω = Ω1
∪ Ω2
, Ω1
∩ Ω2
= ∅ 〜⿴〿ɺP(Ω1
) = Q⊥(Ω2
) = 0.
〽〘〛ɺQ⊥ ⊥ P.
20 / 28

22. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Lemma 6.2 (2) 〣ূ໌
೚ҙ〣Մଌू߹ A 〠ର「ɺ
Qa(A) = Q(A ∩ {p > 0})
=

A∩{p>0}
q dµ
=

A∩{p>0}
q
p
p dµ
=

A
q
p
dP.
໋୊ (ີ౓〣ੑ࣭)
(Ω, A, µ)ɿଌ౓ۭؒ
P, Q: Մଌۭؒ (Ω, A) ্〣ີ౓ p, q ぇ〷〙֬཰ଌ౓
P ≪ µ 〜⿴〿ɺؔ਺ g ⿿ µ-Մੵ෼〟〾

gp dµ =

g dP.
21 / 28

23. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Lemma6.2 (3) 〣ূ໌ (1/3)
〳』ɺQ ≪ P ⇐⇒ Q⊥ = 0 ⿿੒〿ཱ〙ɻ
ɹ
্ࣜ〣આ໌
ɹ
[⇒]ɿA ぇ೚ҙ〣Մଌू߹〝『぀〝ɺ
Q⊥(A) = Q(A ∩ {p = 0}) ≤ Q({p = 0}).
〈〈〜ɺP({p = 0}) = 0 〝 Q ≪ P 〽〿ɺQ({p = 0}) = 0.
〽〘〛ɺ೚ҙ〣Մଌू߹ A 〠ର「 Q⊥(A) ≤ 0 ⿿੒〿ཱ〙⿾〾ɺQ⊥ = 0.
ɹ
[⇐]ɿP(A) = 0 〜⿴぀Մଌू߹ A ぇ೚ҙ〠〝぀ɻ(1) 〣ٞ࿦〽〿ɺQa(A) = 0
⿿੒〿ཱ〙ɻ《〾〠ɺԾఆ〽〿 Q⊥ = 0 〜⿴぀〈〝⿾〾ɺ
Q(A) = Qa(A) + Q⊥(A) = 0. 〽〘〛ɺQ ≪ P.
22 / 28

24. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Lemma6.2 (3) 〣ূ໌ (2/3)
〈ぁ〽〿ɺQ ≪ P ⇐⇒ Q({p = 0}) = 0 ⿿੒〿ཱ〙ɻ
ɹ
্ࣜ〣આ໌
ɹ
[⇒]ɿԾఆ〝 Q⊥ = 0 ⇔ Q ≪ P 〽〿 Q⊥ = 0 〜⿴぀ɻ〈ぁ〽〿ɺ
Q⊥(Ω) = Q(Ω ∩ {p = 0}) = Q({p = 0}) = 0.
[⇐]ɿQ({p = 0}) = 0 〣〝　ɺ೚ҙ〣Մଌू߹ A 〠ର「ɺ
Q⊥(A) = Q(A ∩ {p = 0}) ≤ Q({p = 0}) = 0.
〽〘〛ɺQ⊥ = 0.
23 / 28

25. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Lemma6.2 (3) 〣ূ໌ (3/3)
〳〔ɺ(2) 〽〿 Qa(Ω) =

q
p
dP 〜⿴぀〈〝⿾〾
Q({p = 0}) = 0 ⇐⇒

q
p
dP = 1 (9)
੒〿ཱ〙⿾〾ɺLemma 6.2 (3) ⿿ࣔ《ぁ〔ɻ
ɹ
ࣜ (9) 〣આ໌
ɹ
[⇒]ɿQ({p = 0}) = 0 〽〿ɺQ({p > 0}) = 1. 〳〔ɺ
Q⊥(Ω) = Q(Ω ∩ {p = 0}) = Q({p = 0})
〽〘〛ɺQa(Ω) = 1 〕⿾〾ɺ

q
p
dP = 1.
ɹ
[⇐]ɿQa(Ω) = 1 〽〿ɺQ({p > 0}) = 1. 〽〘〛ɺQ({p = 0}) = 0.
24 / 28

26. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Note: ີ౓〣ੑ࣭
Ұൠ〠ɺ

f dQ =

f
dQ
dP
dP (10)
〤੒〿ཱ〔〟⿶ɻ೚ҙ〣Մଌؔ਺ f 〠〙⿶〛ࣜ (10) ⿿੒〿ཱ〙〔〶〠〤ɺQ
⿿ P 〠ؔ「〛ઈର࿈ଓ〜〟々ぁ〥〟〾〟⿶ɻ
ҰํɺҰൠ〣 (ઈର࿈ଓ〝〤ݶ〾〟⿶) ଌ౓ P, Qɺ⿼〽〨ඇෛ஋ؔ਺ f 〠ର「
〛ɺҎԼ⿿੒〿ཱ〙ɻ

f dQ ≥

{p>0}
fq dµ =

{p>0}
f
q
p
p dµ =

f
dQ
dP
dP. (11)
ෆ౳ࣜ (11) 〤ࠓޙஅ〿〟「〠༻⿶〾ぁ぀ɻ〳〔ɺ{p = 0} 〠⿼⿶〛
dQ
dP
〤ఆٛ
《ぁ〟⿶〔〶ɺෆ౳ࣜ (11) 〤౳߸⿿੒〿ཱ〔〟⿶Մೳੑ⿿⿴぀ɻ11
11”The algebraic identity dQ = (dQ/dP )dP is false, because the notation dQ/dP is used as
shorthand for dQa/dP , then we are not implicity assuming that Q ≪ P.” ([1] p.85 〽〿Ҿ༻ɻ)
25 / 28

27. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
ෆ౳ࣜ (11) 〣આ໌

f dQ =

{p>0}
f dQ +

{p=0}
f dQ

{p>0}
f dQ
=

{p>0}
fq dµ (∵ Q ≪ µ)
=

{p>0}
f
q
p
p dµ
=

{p>0}
f
q
p
dP (∵ P ≪ µ)
=

{p>0}
f
dQ
dP
dP
=

{p>0}
f
dQ
dP
dP +

{p=0}
f
dQ
dP
dP
=

f
dQ
dP
dP.
26 / 28

28. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
⿼い〿〠
ࠓճ〣ษڧձ〜〤ɺԼه〣߲໨ぇѻ⿶〳「〔ɻ
ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ (ಛҟੑ)
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
Lemma 6.2ɿ໬౓ൺ〣ੑ࣭
ɹ
࣍ճɿ֬཰ଌ౓〣 contiguity 〣ఆٛ〝ੑ࣭〣֬ೝ
27 / 28

29. ଌ౓〣ઈର࿈ଓੑ〝௚ަੑ
゚よがそ෼ղ〝゘へア-ぺぢぶくわඍ෼
໬౓ൺ〣ੑ࣭
Reference I
[1] Aad W Van der Vaart.
Asymptotic statistics, Vol. 3.
Cambridge university press, 2000.
[2] ਿӜޫ෉.
جૅ਺ֶ 2 ղੳೖ໳ 1, ୈ 2 ר.
౦ژେֶग़൛ձ.
[3] ࠤ౻ୱ.
〤」〶〛〣֬཰࿦ ଌ౓⿾〾֬཰〭.
ڞཱग़൛, 1994.
[4] Ճ౻จݩ.
਺ݚߨ࠲でがどɹେֶڭཆɹඍ෼ੵ෼.
਺ݚग़൛, 2020.
[5] ౷ܭֶ〭〣֬཰࿦, 〒〣ઌ〭: に゜⿾〾〣ଌ౓࿦తཧղ〝઴ۙཧ࿦〭〣Ս々ڮ.
಺ా࿝௽ะ, 2021.
[6] स໦௚ٱ.
֬཰࿦.
ߨ࠲਺ֶ〣ߟ⿺ํ / ൧ߴໜ [〰⿾] ฤ. ே૔ॻళ, 2004.
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