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statistician_ja_lt5.pdf
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Takayuki Uchiba
November 15, 2020
Science
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680
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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Transcript
Ұ༷࠷খࢄෆภਪఆྔඞͣଘࡏ͢Δ͔ʁ !VUBLB
Ұ༷࠷খࢄෆภਪఆྔ ͷҰ༷࠷খࢄෆภਪఆྔʢ6.76&ʣɹ ෆภੑ ͕ͲΜͳͰ͋ͬͯɺ ͕Γཱͭɻ Ұ༷࠷খࢄੑ ͕ͲΜͳͰ͋ͬͯɺଞͷෆภਪఆྔ
ʹൺͯ ͷࢄ͕খ͍͞ɻཁ͢Δʹɺ ͕Γཱͭɻ ྫαΠζ ͷಠཱඪຊΛظ ࢄ ͷਖ਼ن͔Βಘͨ߹ ɾظ ͷ6.76&ඪຊฏۉ ɾࢄ ͷ6.76&ෆภࢄ T θ θ [T] = θ θ S T [T] ≤ [S] n μ σ2 μ σ2
6.76&ͷѻ͍͢͞ʢͦͷʣ $SBNFS3BPͷఆཧ͋Δෆภਪఆྔ͕6.76&͔ఆ͢Δํ๏ͷͻͱͭ ɾෆภਪఆྔͷࢄͷେ͖͞ͷԼݶܭࢉͰ͖Δɻ ɹɾ$SBNFS3BPԼݶ'JTIFSใྔ ɾෆภਪఆྔ ͷࢄ͕͜ͷԼݶʹҰக͢Ε6.76& ʢʣ6.76&ͷࢄ͕ඞͣ͜ͷԼݶʹͳΔΘ͚Ͱͳ͍ɻ T
6.76&ͷѻ͍͢͞ʢͦͷʣ -FINBOO4DIF⒎Fͷఆཧ6.76&ಛఆͷ݅ͷͱͰ࡞ΕΔɻ ɾಛఆͷ݅උे౷ܭྔͷଘࡏ ɾඋे౷ܭྔͰද͞ΕΔ౷ܭྔ͕ෆภͳΒ6.76& ɾ$SBNFS3BPͷఆཧ͕༗ޮͰͳ͍έʔεͰಛʹศར ɹɾ'JTIFSใྔ͕ఆٛͰ͖ͳ͍ͱ͖ʢҰ༷ͷ࠷େύϥϝʔλʣ ɹɾ6.76&ͷࢄ㱠$SBNFS3BPԼݶͷͱ͖
6.76&ඞͣଘࡏ͢Δ͔ʁ ύϥϝʔλ ͷ6.76&ඞͣଘࡏ͢Δ͔ʁ ɾ-FINBOO4DIF⒎Fͷఆཧඋे౷ܭྔ͕ଘࡏ͢Ε6.76&࡞ΕΔɻ ɾඋे౷ܭྔ͕ଘࡏ͠ͳ͚ΕͲ͏͔ʁ ɾ)JOUҰ༷࠷খࢄੑʹݱΕΔʮ ͕ͲΜͳͰ͋ͬͯʯڧ͍݅ ˠɹ ͷ͝ͱʹ࠷খࢄͷෆภਪఆྔ͕ҟͳΕɺ6.76&ଘࡏ͠ͳ͍ɻ θ
θ θ
۩ମྫͷߏ ֬ม ͕࣍ͷ࣭֬ྔؔ ʹै͍ͬͯΔͷͱ͠·͢ɻύϥϝʔλ ͷҰ༷࠷খࢄෆภਪఆྔଘࡏ͢Δ͔ʁ ٕज़తͳ3FNBSL Ͱද͞ΕΔͲΜͳ౷ܭྔɺ ͷܗͰද͢͜ͱ͕Ͱ͖Δɻ X
f(x) = { p JGx = − 1 (1 − p)2px JGx = 0,1,2,⋯ p X T(X) = ∞ ∑ x=−1 tx [X = x]
ෆภਪఆྔʹͳΔͨΊͷ݅ Λ༻͍ͯɺ౷ܭྔ ͷظΛܭࢉ͢Δɻ ౷ܭྔ ͕ෆภਪఆྔͳΒɺظඞͣ ʹ͘͠ͳΔɻ ˠɹԽࣜ 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)
ෆภਪఆྔͷࢄΛܭࢉ͢Δʢͦͷʣ Λ༻͍ͯɺෆภਪఆྔ ͷࢄΛܭࢉ͢Δɻ ͜͏͍͏ͱ͖ʹཱͭͷࢄͷެࣜʂ ͳͷͰɺ Λܭࢉ͠Α͏ɻ f(x) = {
p JGx = − 1 (1 − p)2px JGx = 0,1,2,⋯ T(X) [T] = [T2] − [T]2 = [T2] − p2, ෆภੑ [T2]
ෆภਪఆྔͷࢄΛܭࢉ͢Δʢͦͷʣ Ώ͑ʹɺ ͷࢄ 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
ࢄ͕࠷খΛͱΔͨΊͷ݅ʢͦͷʣ ࠷খࢄΛ༩͑Δ ʢΛ༩͑Δ ʣΛٻΊɺ ʹґଘͳΒ6.76&ଘࡏ͠ͳ͍ɻ Λ Ͱཧ͢Δͱɺ͕࣍ؔݱΕΔɻ ฏํͯ͠ɺ࠷খΛ༩͑Δ 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
ࢄ͕࠷খΛͱΔͨΊͷ݅ʢͦͷʣ ฏํ͢Δͱɺ࣍ͷΑ͏ʹͳΓ·͢ɻ ࣍ؔͷͷ࠲ඪΛಡΉ͜ͱͰɺ ͷͱ͖ࢄ࠷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
݁ ࢄ͕࠷খʹͳΔෆภਪఆྔ͕ύϥϝʔλ ͷʹґଘ͍ͯ͠Δɻ ˠɹ ͷ6.76&ଘࡏ͠ͳ͍ʂʂʂʂʂ ͜ͷྫ͕ڭ͑ͯ͘Ε͍ͯΔͱࢥ͏͜ͱʢࢲײʣ ɾඞͣ͠ʮ͍ͭͰ҆ఆͯ͠ਫ਼͕ྑ͍ਪఆྔʯ͕ଘࡏ͢ΔͱݶΒͳ͍ɻ ɾԾઆ͕͋ΔͳΒਪఆྔʹөͤͯ͞ΈΔͷେࣄɻ ɹɾࠓճͷྫͰɺ ͷʹԠͨ͡ਪఆྔͷબͷ༨͞Ε͍ͯΔɻ
ɹɾDMJDLখ͘͞ͳΓ͕͔ͪͩΒɺͪΐͬͱॖখͨ͠ͷΛ͓͏ͱ͔ɻ p p p
͝ਗ਼ௌ͋Γ͕ͱ͏͍͟͝·ͨ͠ʂ ࣗݾհͰͬͱ͖·͢ɻɻɻ ɾ!VUBLB ɾגࣜձࣾ͢͏͕͘ͿΜ͔ ڭ෦ ෦ ɾڵຯཧ౷ܭֶͷσʔλϚΠχϯάͷԠ༻ زԿֶ ɾจ ɹओஶ(MVJOH4UBCJMJUZ$POEJUJPOTPO3VMFE4VSGBDFXJUI1PTJUJWF(FOVT
ɹɹɹɹ0TBLB+PVSOBMPG.BUIFNBUJDT BDDFQUFE ɹڞஶࠓຊɺ͍ͣΕػցֶशͷจɻ