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statistician_ja_lt5.pdf

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。

Takayuki Uchiba

November 15, 2020
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  1. 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
  2. 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)
  3. 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
  4. ෼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
  5. ෼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