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勉強会資料 / “Asymptotic Statistics” Section 6.1

Minato
March 25, 2023

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

Minato

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

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  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”

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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.

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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
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  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)
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  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.
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  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.
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  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.
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  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.
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  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.
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  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.
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  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)
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  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.
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  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

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  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

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