statistician_ja_lt5.pdf

Fa101acb1230ee2d069458ff049b98f8?s=47 Takayuki Uchiba
November 15, 2020

 statistician_ja_lt5.pdf

一様最小分散不偏推定量が存在しない例を紹介しました。

Fa101acb1230ee2d069458ff049b98f8?s=128

Takayuki Uchiba

November 15, 2020
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  1. Ұ༷࠷খ෼ࢄෆภਪఆྔ͸ඞͣଘࡏ͢Δ͔ʁ !VUBLB

  2. Ұ༷࠷খ෼ࢄෆภਪఆྔ  ͷҰ༷࠷খ෼ࢄෆภਪఆྔʢ6.76&ʣɹ  ෆภੑ ͕ͲΜͳ஋Ͱ͋ͬͯ΋ɺ ͕੒Γཱͭɻ  Ұ༷࠷খ෼ࢄੑ ͕ͲΜͳ஋Ͱ͋ͬͯ΋ɺଞͷෆภਪఆྔ

    ʹൺ΂ͯ ͷ෼ࢄ͕খ͍͞ɻཁ͢Δʹɺ ͕੒Γཱͭɻ ྫαΠζ ͷಠཱඪຊΛظ଴஋ ෼ࢄ ͷਖ਼ن෼෍͔Βಘͨ৔߹ ɾظ଴஋ ͷ6.76&ඪຊฏۉ ɾ෼ࢄ ͷ6.76&ෆภ෼ࢄ T θ θ [T] = θ θ S T [T] ≤ [S] n μ σ2 μ σ2
  3. 6.76&ͷѻ͍΍͢͞ʢͦͷʣ $SBNFS3BPͷఆཧ͋Δෆภਪఆྔ͕6.76&͔൑ఆ͢Δํ๏ͷͻͱͭ ɾෆภਪఆྔͷ෼ࢄͷେ͖͞ͷԼݶ͸ܭࢉͰ͖Δɻ ɹɾ$SBNFS3BPԼݶ'JTIFS৘ใྔ ɾෆภਪఆྔ ͷ෼ࢄ͕͜ͷԼݶʹҰக͢Ε͹6.76& ʢ஫ʣ6.76&ͷ෼ࢄ͕ඞͣ͜ͷԼݶʹͳΔΘ͚Ͱ͸ͳ͍ɻ T

  4. 6.76&ͷѻ͍΍͢͞ʢͦͷʣ -FINBOO4DIF⒎Fͷఆཧ6.76&͸ಛఆͷ৚݅ͷ΋ͱͰ࡞ΕΔɻ ɾಛఆͷ৚݅׬උे෼౷ܭྔͷଘࡏ ɾ׬උे෼౷ܭྔͰද͞ΕΔ౷ܭྔ͕ෆภͳΒ͹6.76& ɾ$SBNFS3BPͷఆཧ͕༗ޮͰͳ͍έʔεͰಛʹศར ɹɾ'JTIFS৘ใྔ͕ఆٛͰ͖ͳ͍ͱ͖ʢҰ༷෼෍ͷ࠷େ஋ύϥϝʔλʣ ɹɾ6.76&ͷ෼ࢄ㱠$SBNFS3BPԼݶͷͱ͖

  5. 6.76&͸ඞͣଘࡏ͢Δ͔ʁ ໰୊ύϥϝʔλ ͷ6.76&͸ඞͣଘࡏ͢Δ͔ʁ ɾ-FINBOO4DIF⒎Fͷఆཧ׬උे෼౷ܭྔ͕ଘࡏ͢Ε͹6.76&͸࡞ΕΔɻ ɾ׬උे෼౷ܭྔ͕ଘࡏ͠ͳ͚Ε͹Ͳ͏͔ʁ ɾ)JOUҰ༷࠷খ෼ࢄੑʹݱΕΔʮ ͕ͲΜͳ஋Ͱ͋ͬͯ΋ʯ͸ڧ͍৚݅ ˠɹ ͷ஋͝ͱʹ࠷খ෼ࢄͷෆภਪఆྔ͕ҟͳΕ͹ɺ6.76&͸ଘࡏ͠ͳ͍ɻ θ

    θ θ
  6. ۩ମྫͷߏ੒ ֬཰ม਺ ͕࣍ͷ֬཰࣭ྔؔ਺  ʹै͍ͬͯΔ΋ͷͱ͠·͢ɻύϥϝʔλ ͷҰ༷࠷খ෼ࢄෆภਪఆྔ͸ଘࡏ͢Δ͔ʁ ٕज़తͳ3FNBSL Ͱද͞ΕΔͲΜͳ౷ܭྔ΋ɺ ͷܗͰද͢͜ͱ͕Ͱ͖Δɻ X

    f(x) = { p JGx = − 1 (1 − p)2px JGx = 0,1,2,⋯ p X T(X) = ∞ ∑ x=−1 tx [X = x]
  7. ෆภਪఆྔʹͳΔͨΊͷ৚݅ Λ༻͍ͯɺ౷ܭྔ ͷظ଴஋Λܭࢉ͢Δɻ  ౷ܭྔ ͕ෆภਪఆྔͳΒ͹ɺظ଴஋͸ඞͣ ʹ౳͘͠ͳΔɻ ˠɹ઴Խࣜ GPS 

      ˠɹ ͷܗͰද͞ΕΔ౷ܭྔ ͕ෆภਪఆྔʹͳΔɻ f(x) = { p JGx = − 1 (1 − p)2px JGx = 0,1,2,⋯ T(X) [T] = t−1 p + ∞ ∑ x=0 tx (1 − p)2px = t0 + ∞ ∑ x=1 (tx−2 − 2tx−1 + tx )px T(X) p tx − 2tx−1 + tx−2 = 0 x ≥ 2 t1 = 1 − t−1 t0 = 0 tx = x(1 − t−1 ) T(X)
  8. ෆภਪఆྔͷ෼ࢄΛܭࢉ͢Δʢͦͷʣ Λ༻͍ͯɺෆภਪఆྔ ͷ෼ࢄΛܭࢉ͢Δɻ ͜͏͍͏ͱ͖ʹ໾ཱͭͷ͸෼ࢄͷެࣜʂ  ͳͷͰɺ Λܭࢉ͠Α͏ɻ f(x) = {

    p JGx = − 1 (1 − p)2px JGx = 0,1,2,⋯ T(X) [T] = [T2] − [T]2 = [T2] − p2, ෆภੑ [T2]
  9. ෆภਪఆྔͷ෼ࢄΛܭࢉ͢Δʢͦͷʣ  Ώ͑ʹɺ ͷ෼ࢄ͸ ͱΘ͔Γ·͢ɻ [T2] = t2 −1 p

    + ∞ ∑ x=1 x2(1 − t−1 )2(1 − p)2px = t−1 + (1 − t−1 )2(1 − p)2 ∞ ∑ x=1 x2px = t2 −1 p + (1 − t−1 )2(1 − p)2 p(1 + p) (1 − p)3 = t2 −1 p + (1 − t−1 )2 p(1 + p) 1 − p T(X) [T] = t2 −1 p + (1 − t−1 )2 p(1 + p) 1 − p − p2
  10. ෼ࢄ͕࠷খ஋ΛͱΔͨΊͷ৚݅ʢͦͷʣ ࠷খ෼ࢄΛ༩͑Δ ʢΛ༩͑Δ ʣΛٻΊɺ ʹґଘͳΒ6.76&͸ଘࡏ͠ͳ͍ɻ Λ Ͱ੔ཧ͢Δͱɺ࣍ؔ਺͕ݱΕΔɻ  ฏํ׬੒ͯ͠ɺ࠷খ஋Λ༩͑Δ ΛٻΊͯΈΑ͏ʂ

    T(X) t−1 p [T] = t2 −1 p + (1 − t−1 )2 p(1 + p) 1 − p − p2 t−1 [T] = (p + p(1 + p) 1 − p ) t2 −1 − 2 p(1 + p) 1 − p t−1 + ( p(1 + p) 1 − p − p2 ) t−1
  11. ෼ࢄ͕࠷খ஋ΛͱΔͨΊͷ৚݅ʢͦͷʣ ฏํ׬੒͢Δͱɺ࣍ͷΑ͏ʹͳΓ·͢ɻ  ࣍ؔ਺ͷ௖఺ͷ࠲ඪΛಡΉ͜ͱͰɺ ͷͱ͖෼ࢄ࠷খͱΘ͔Γ·͢ɻ [T] = (p + p(1

    + p) 1 − p ) t−1 − 1 1 + 1 − p 1 + p 2 + const . = (p + p(1 + p) 1 − p ) {t−1 − p + 1 2 } 2 + const . t−1 = p + 1 2
  12. ݁࿦ ෼ࢄ͕࠷খʹͳΔෆภਪఆྔ͕ύϥϝʔλ ͷ஋ʹґଘ͍ͯ͠Δɻ ˠɹ ͷ6.76&͸ଘࡏ͠ͳ͍ʂʂʂʂʂ ͜ͷྫ͕ڭ͑ͯ͘Ε͍ͯΔͱࢥ͏͜ͱʢࢲײʣ ɾඞͣ͠΋ʮ͍ͭͰ΋҆ఆͯ͠ਫ਼౓͕ྑ͍ਪఆྔʯ͕ଘࡏ͢Δͱ͸ݶΒͳ͍ɻ ɾԾઆ͕͋ΔͳΒਪఆྔʹ൓өͤͯ͞ΈΔͷ΋େࣄɻ ɹɾࠓճͷྫͰ΋ɺ ͷ஋ʹԠͨ͡ਪఆྔͷબ୒ͷ༨஍͸࢒͞Ε͍ͯΔɻ

    ɹɾDMJDL཰͸খ͘͞ͳΓ͕͔ͪͩΒɺͪΐͬͱॖখͨ͠΋ͷΛ࢖͓͏ͱ͔ɻ p p p
  13. ͝ਗ਼ௌ͋Γ͕ͱ͏͍͟͝·ͨ͠ʂ ࣗݾ঺հͰ΋΍ͬͱ͖·͢ɻɻɻ ɾ!VUBLB ɾגࣜձࣾ͢͏͕͘ͿΜ͔ ڭ຿෦ ෦௕ ɾڵຯ਺ཧ౷ܭֶͷσʔλϚΠχϯά΁ͷԠ༻ ୅਺زԿֶ ɾ࿦จ ɹओஶ(MVJOH4UBCJMJUZ$POEJUJPOTPO3VMFE4VSGBDFXJUI1PTJUJWF(FOVT

     ɹɹɹɹ0TBLB+PVSOBMPG.BUIFNBUJDT BDDFQUFE   ɹڞஶࠓ೥͸ຊɺ͍ͣΕ΋ػցֶशͷ࿦จɻ