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。

担当範囲：
第6章 Contiguity
6.1節 Likelihood Ratios

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 ʱ(https://irocore.com/hanada-iro/ 〽〿Ҿ༻) 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 ぇ 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 Ω 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 = ∫ Ω 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 ⇒ ∫ Ω 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 ∫ Ω 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 ⇐⇒ ∫ Ω 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 ⇐⇒ ∫ Ω 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 = ∫ Ω 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 ∫ Ω 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෼〟〾 ∫ Ω gp dµ = ∫ Ω 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) = ∫ Ω q p dP 〜⿴぀〈〝⿾〾 Q({p = 0}) = 0 ⇐⇒ ∫ Ω 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 ∫ Ω 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 ∫ Ω f dQ = ∫ Ω 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 ∫ Ω f dQ ≥ ∫ {p>0} fq dµ = ∫ {p>0} f q p p dµ = ∫ Ω 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 ∫ Ω 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 = ∫ Ω 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࠲でがど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. 28 / 28