06 统计学基础——点估计（三）

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王木木

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1976

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#信息技术

主要介绍了一致最小方差无偏估计、零无偏估计法、充分完全统计量法、Cranmer-Rao不等式、有效估计和估计的概率等相关知识和定理。

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1 . Lec6: : O(n) Ü•² 2011 c 3 24 F 1 ˜—• • Ã O ˜!Úó9½Â k ˜ ë ê © Ù xF = {Fθ , θ ∈ Θ},Ù ¥Θ• ë ê ˜ m. g(θ)´ ½ Â 3Θþ ¼ ê, X = (X1 , · · · , Xn )•goNFθ ¥Ä {ü , gˆ(X) = gˆ(X1 , · · · , Xn )•g(θ) ˜‡ Oþ, XÛµdˆ g (X) ` ? ˜„^ˆ g (X) − g(θ) Š•Ù , •žØˆ g (X) − g(θ) ŠÑy/+, −0Œ U-ž K•, ˜„^(ˆ g (X) − g(θ)) 5“O. duù‡þ´‘Å , òÙ²þ, =OŽÙþŠ, ± 2 ˜‡ N5 •IEθ (ˆ g (X) − g(θ))2 , ùÒ´ Oþˆ g (X) þ•Ø . ½ Â 1. gˆ(X)•g(θ) O, K¡Eθ (ˆ g (X)−g(θ))2 •ˆ g (X) þ•Ø (Mean Square Error,{ PMSE) g2 (X)•g(θ) ü‡ØÓ gˆ1 (X)Úˆ O, e 2 2 Eθ gˆ1 (X) − g(θ) ≤ Eθ gˆ2 (X) − g(θ) , é˜ƒ θ ∈ Θ, K¡3MSEOKeˆ g1 (X)`uˆ g2 (X). g ∗ (X),¦ e•3ˆ ég(θ) ?˜ Oþˆ g (X),Ñk 2 2 Eθ gˆ∗ (X) − g(θ) ≤ Eθ gˆ(X) − g(θ) , é˜ƒ θ ∈ Θ, g ∗ (X)•g(θ) ˜—• þ•Ø K¡ˆ O. ŒJ ´˜—• þ•Ø O~Ø•3. )ûù‡¯K •{ƒ˜, ´r•`5OK˜ °˜ , ¦·Üù«•`5OK O˜„U•3. l†*þŽ, 3˜‡Œ Oþ a¥é ˜—•` OØ•3, r Oþ a , ÒkŒU•3˜—•` Oþ. Ïd·‚r Oa •Ã Oa5•Ä. 3Ã Oa¥, Oþ þ•Ø ÒC•Ù• . = gˆ(X) •g(θ) Ã Ož, M SE(ˆ g (X)), d?Dθ (ˆ g (X)) = Dθ (ˆ g (X) • . g (X))L«ˆ •3ù œ/, éëêg(θ)§ Ã OØ•3. žwe~: ~1. X∼ ‘©Ù b(n, p), n®• p™•. -g(p) = 1/p,Këêg(p) Ã OØ•3. y æ^‡y{: eØ,, g(p)kÃ Oˆ g (X).duX• 0, 1, · · · , nù Š, -ˆ g (X) Š ^ˆ g (i) = ai L«, i = 0, 1, · · · , n.dˆ g (X) Ã 5,Ak n n i Ep (ˆ g (X)) = ai p (1 − p)n−i = 1/p, 0 < p < 1. i=0 i 1

2 .u´k n n i+1 ai p (1 − p)n−i − 1 = 0, 0 < p < 1. i=0 i þª†à´p n + 1gõ‘ª, §•õ3(0, 1)«mkn + 1‡¢Š, ŒÃ 5‡¦é(0, 1)¥ ?˜¢êpþªÑ¤á. ù‡gñ`²g(p) = 1/pÃ OØ•3. 8 ·‚rØ•3Ã O ëêØ . ëê Ã Oe•3, Kdëê•Œ ëê; e ëê¼ê Ã O•3, K¡d¼ê•Œ ¼ê (Estimable function). ÏdŒ ¼ê Ã Oa´š˜ . bXŒ ¼ê Ã Oa¥ Ã OØŽ˜‡, N ' §‚ ` ? Ú\e ½Â. ½ Â 2. F = {Fθ , θ ∈ Θ}´˜‡ëê©Ùx, Ù¥Θ•ëê˜m,g(θ)•½Â3Θþ Œ ∗ ∗ ¼ê. gˆ (X) = gˆ (X1 , · · · , Xn )•g(θ) ˜‡Ã O, eég(θ) ?˜Ã Oˆ g (X) = gˆ(X1 , · · · , Xn ),Ñk g ∗ (X)) ≤ Dθ (ˆ Dθ (ˆ g (X)), é˜ƒ θ ∈ Θ, g ∗ (X)´g(θ) K¡ˆ ˜—• • Ã O (Uniformly Minimum Variance Unbiased Estima- tion, {P•UMVUE). é‰½ëê©Ùx, XÛÏéŒ ëê UMVUEQ? !±eò0 ü«•{: "Ã O{Ú¿© ÚOþ{, e˜! Cramer-RaoØ ª{•´ÏéUMVUE ˜«•{. 3c¡·‚Q0 LRao-Blackwell½n, ù˜½nJø ˜‡U?Ã O •{, §3 !±eÏéUMVUE¥,å {z¯K Š^. -#Lãd½nXe ½ n 1 (Rao-Blackwell). T = T (X)´˜‡¿©ÚOþ, gˆ(X)´g(θ) ˜‡Ã O, K h(T ) = E(ˆ g (X)|T ) ´g(θ) Ã O, ¿… Dθ (h(T )) ≤ Dθ (ˆ g (X)), ˜ƒ θ ∈ Θ, (1.1) Ù¥ Ò …= Pθ gˆ(X) = h(T ) = 1,=ˆ g (X) = h(T ), a.e. Pθ ¤á. ù‡ÚnJø ˜‡U?Ã O •{, =˜‡Ã Oˆ g (X) é¿©ÚOþT (X) ^ ‡Ï"E{ˆ g (X)|T } òU Ñ˜‡# Ã O, …§ • Ø¬‡L Oþˆ g (X) • . e Oˆ g (X)Ø´T (X) ¼ê, K# Ã OE(ˆ g (X)|T )˜½' Oˆ g (X)äk• • . ù‡½n„L²˜—• • Ã O˜½´¿©ÚOþ ¼ê, ÄKŒ±ÏL¿©ÚOþ EÑ˜‡äk• • Ã O5. ~2. X = (X1 , · · · , Xn )´ l ü : © Ù x{b(1, p) : 0 < p < 1}¥ Ä {ü . w n ,,X1 ´p ˜‡Ã O, T (X) = i=1 Xi ´p ¿©ÚOþ, Á|^T = T (X) E˜‡äk 'X1 • • Ã O. ) dÚn3.4.1Œ•,N´ Ep ˜‡Ã OXeµ h(T ) = E(X1 |T = t) = 1 · P (X1 = 1|T = t) + 0 · P (X1 = 0|T = t) 2

3 . P (X1 = 1, T = t) P (X1 = 1, X2 + · · · + Xn = t − 1) = = P (T = t) P (T = t) n−1 t−1 p· t−1 p (1 − p)n−t t = n t n−t = =x ¯. t p (1 − p) n w, ¯ þŠh(T ) = X • •p(1 − p)/n, X1 • ¯ •p(1 − p), n ≥ 2žX • • . !"Ã O{ ã0 ˜‡˜„5 ½n, ^± ä,˜ Oþ´Ä•UMVUE. ½ n 2. gˆ(X)´g(θ) ˜‡Ã O, Dθ (ˆ g (X)) < ∞,é?Ûθ ∈ Θ. …é?Û÷v^‡ /Eθ l(X) = 0, é˜ƒ θ ∈ Θ0 ÚOþl(X),7k Cov θ gˆ(X), l(X) = Eθ gˆ(X) · l(X) = 0, ˜ƒ θ ∈ Θ, (1.2) Kˆ g (X)´g(θ) UMVUE. l/ªþw, ^‡/Eθ l(X) = 0, é˜ƒ θ ∈ Θ0Œ)º•/l(X) ´" Ã O0, dd ¦UMVUE •{ƒ˜ ¶¡/"Ã O{0. d½n„Œ?˜Ú\r•: gˆ(X)•g(θ) ˜‡Ã O, Dθ (ˆ g (X)) < ∞, ˜ƒ θ ∈ Θ, Kˆ g (X)´g(θ) UMVUE ¿©7‡^‡´: é?Û÷v^‡/Eθ l(X) = 0, é˜ƒ θ ∈ Θ0 ÚOþl(X), 7k(1.2)¤á. y gˆ1 (X)•g(θ) ?˜Ã O. Pl(X) = gˆ1 (X) − gˆ(X) •" Ã O, du(1.2)ª ¤á, Ï Dθ gˆ1 (X) = Dθ gˆ(X) + l(X) = Dθ gˆ(X) + Dθ l(X) + 2Cov θ gˆ(X), l(X) = Dθ gˆ(X) + Dθ l(X) ≥ Dθ gˆ(X) . ,˜•¡, XJδ = gˆ(X)•g(θ) UMVUE, Ké?¿ "Ã Oþl(x), - δ = δ + λl(X) Kδ E•g(θ) Ã O, … V ar(δ ) = V ar(δ + λl(X)) = λ2 V ar(l(X)) + 2λCov(δ, l(X)) + V ar(δ) ≥ V ar(δ). é¤k λ¤á. du λ2 V ar(l(X)) + 2λCov(δ, l(X)) ≥ 0, ∀ λ. †>kü‡Šλ = 0Úλ = −Cov(δ, l(X))/V ar(l(X)), íÑCov(δ, l(X)) = 0. ùÒy² ¤‡ (J. l½n SNw, §´˜‡ y,‡A½ Oþˆ g (X)•UMVUE •{. –uù‡A ½ Oˆ g (X)lÛ 5, ½n3.4.1ØUJø?Û•Ï, §Ø´UMVUE E5½n. gˆ(X)Œ ±l†* Ž{JÑ, XdÝ O½4Œq, O •{¼ , , |^d½n y§´Ä 3

4 .•g(θ) UMVUE. ^‡(1.2) y•ØN´, Ï•"Ã Oéõ. e¡ A‡~f`², Ù c¡˜ ~f¥J A‡~^ O, ÑŒ±^d{ yÙ•UMVUE. ~ ¦þ~¥g(p) = p UMVUE. n ) dþ~®•T = i=1 Xi •¿©ÚOþ, gˆ(X) = T /n, •‡ y§÷v½n ^‡. w,ˆ g (X)´p Ã O. …Dθ (ˆ g (X)) = p(1 − p)/n < ∞,é˜ƒ0 < p < 1.y l = l(T )•?˜ "Ã O, ¿Pai = l(i), i = 0, 1, 2, · · · , n, KÏT ∼ b(n, p), k n n i Ep l(T ) = ai p (1 − p)n−i = 0, 0 < p < 1. i=0 i Ïf(1 − p)n ,¿Pϕ = p/(1 − p) (ϕ Š(0, ∞)), òþªU • n n i ai ϕ = 0, ˜ƒ 0 < ϕ < ∞. i=0 i n þª†>´ϕ õ‘ª, ‡¦Ù•0, 7kai i = 0,=ai = 0, i = 1, 2, · · · , n. l(T )3Ù½Â• ¥??•0, Ï kl(T ) ≡ 0.l k Cov p gˆ, l(T ) = E gˆ · l(T ) = 0. =½n^‡¤á, ¯ = T /n•p gˆ(X) = X UMVUE. ~ X = (X1 , · · · , Xn )•l•ê©ÙEP (λ) ¥Ä {ü , ¦oNþŠg(λ) = 1/λ UMVUE. n ) du3•ê©Ùx¥T = i=1 Xi •λ ¿©ÚOþ, KT ∼ Γ(n, λ),Ù—Ý¼ê• λn n−1 −λt (n−1)! t e t>0 φ(t, λ) = 0 t ≤ 0, Ù¥λ > 0. gˆ(X) = T /n, w,E(ˆ g (X) •g(λ) = 1/λ Ã g (X)) = 1/λ,=ˆ O, …Dλ (ˆ g (X)) = 2 1 (nλ ) < ∞.y l = l(T )•?˜"Ã O, k ∞ λn El(T ) = l(t) tn−1 e−λt dt = 0, 0 (n − 1)! ∞ = 0 l(t)tn−1 e−λt dt = 0.ü>éλ¦ ∞ l(t)tn e−λt dt = 0, 0 dª duEλ T /n · l(T ) = Eλ (ˆ g · l) = 0,=½n^‡¤á. Ïdˆ g · l(T )) = Cov λ (ˆ g (X) = T /n •g(λ) = 1/λ UMVUE. ~ X = (X1 , · · · , Xn )•lþ!©ÙU (0, θ)¥Ä {ü , ¦θ UMVUE. ) dT = T (X) = X(n) ´ ë êθ ¿ © Ú O þ, q •ˆ g (X) = n+1 n T ´θ Ã O, 1 …Dθ (ˆ g (T )) = n(n+2) θ 2 < ∞. y l(T )•?˜"Ã O, T —Ý¼êX(??)¤«, Ïdk θ ntn−1 Eθ l(T ) = l(t) · dt = 0, ˜ƒ θ > 0, 0 θn 4

5 .u´k θ l(t)tn−1 dt = 0, ˜ƒ θ > 0. 0 òþªü>éθ ¦ l(θ)θn−1 = 0, ˜ƒ θ > 0. kl(θ) ≡ 0 é˜ƒθ > 0.Œ„Cov(ˆ g , l(T )) = n+1 g · l(T )) = 0,=½n^‡¤á. Ïdˆ E(ˆ g (X) = n X(n) •g(θ) =θ UMVUE. ~ X = (X1 , · · · , Xn )•l ©ÙN (a, σ 2 )¥Ä {ü‘Å , ¦aÚσ 2 UMVUE. ¯ T2 = n ¯ 2, ) dT = (T1 , T2 )•θ = (a, σ 2 ) ¿©ÚOþ.Ù¥T1 = X, i=1 (Xi − X) qT1 ÚT2 Õá, …T1 ∼ N a, σ /n , T2 /σ ∼ 2 2 χ2n−1 . Ïd(T1 , T2 ) éÜ—Ý• √ −1 n − n(t1 −a)2 n−1 n − 1 n−1 n−3 t2 f (t1 , t2 ) = √ e 2σ 2 · 2 2 Γ σ t2 2 e− 2σ2 , 2πσ 2 2 − ∞ < t1 < +∞, t2 > 0; Ù§?• 0. (1.3) k•Äa UMVUE, -ˆ g1 (X) •g1 (θ) = a g1 (X) = T1 ,w,ˆ Ã O, …Dθ (ˆ g1 (X)) = 2 σ /n < ∞.y l(T ) = l(T1 , T2 )•?˜"Ã O, Kk ∞ ∞ Eθ (l(T )) = l(t1 , t2 )fθ (t1 , t2 )dt1 dt2 = 0, 0 −∞ d?−∞ < a < ∞, σ > 0.òþªü>éa¦ ê, ∞ ∞ n−3 1 l(t1 , t2 )(t1 − a) · t2 2 exp − n(t1 − a)2 + t2 dt1 dt2 = 0, 0 −∞ 2σ 2 U(1.3),d= Covθ gˆ1 , l(T ) = Eθ gˆ1 · l(T1 , T2 ) = 0, −∞ < a < +∞, σ > 0, ½n3.4.1 ^‡÷v. ¤±ˆ g1 (X) = T1 •g1 (θ) = a UMVUE. ÓnŒ yT2 = S •g2 (θ) = σ 2 2 UMVUE, ù˜ y3•öS. n!¿© ÚOþ{ e ½n‰Ñ ¦UMVUE •{, =¿© ÚOþ{´dE.L. LehmannÚH. Scheffe‰ Ñ , ÚOþ Vg•´d¦‚31950cJÑ . ½ n (Lehmann-Scheffe½ ½ n) T (X)•˜‡¿© ÚOþ, eˆ g (T (X))•g(θ) ˜‡Ã O, Kˆ g (T (X))´g(θ) •˜ UMVUE (•˜5´3ù ¿Âe: eˆ g Úˆ g1 Ñ ´g(θ) UMVUE, KPθ (ˆ g = gˆ1 ) = 0,é˜ƒθ ∈ Θ). y ky•˜5. gˆ1 (T (X))•g(θ) ?˜Ã O, -δ(T (X)) = gˆ(T (X)) − gˆ1 (T (X)), KEθ δ(T (X)) = Eθ gˆ(T (X)) − Eθ gˆ1 (T (X)) = 0,é ˜ ƒθ ∈ Θ. dT (X)• Ú O þ, Œ •δ(T (X)) = 0, a.e. Pθ .=ˆ g (T (X)) = gˆ1 (T (X)), a.e. Pθ , •˜5¤á. ϕ(X)•g(θ) ?˜Ã O. -h(T (X)) = E(ϕ(X)|T ),dT (X)•¿©ÚOþ, •h(T (X))†θÃ ', Ïd´ÚOþ, Œ• Eθ (h(T (X))) = g(θ), ˜ƒ θ ∈ Θ, Dθ (h(T (X))) ≤ Dθ (ϕ(X)), ˜ƒ θ ∈ Θ. 5

6 .d•˜5 gˆ(T (X)) = h(T (X)), a.e. Pθ k g (T (X)) ≤ Dθ (ϕ(X)), Dθ (ˆ ˜ƒ θ ∈ Θ, ¤±ˆ g (T (X)) •g(θ) UMVUE, …•˜. íØ X = (X1 , · · · , Xn ) ©Ù••êx k f (x, θ) = C(θ) exp θj Tj (x) h(x), θ = (θ1 , · · · , θk ) ∈ Θ. j=1 -T (X) = (T1 (X), · · · , Tk (X)), eg,ëê˜mΘŠ•Rk f8kS:, …h(T (X))•g(θ) Ã O, Kh(T (X))•g(θ) •˜ UMVUE. y 3íØ ^‡e, d•êx 5ŸŒ•T (X)•¿© ÚOþ. dLehmann- Scheffe½n ({P•L-S½n), •h(T (X))•g(θ) •˜ UMVUE. ~ y²ü:©Ù p Ã Oˆ ¯ •p g (X) = T /n = X UMVUE. n y d Ï f © ) ½ n Œ •T (X) = Xi • ü : © Ùb(1, p)¥ ë êp ¿ © Ú O þ, i=1 d ~2.8.1•T (X)• ´ Ú O þ, ¯ ¿© gˆ(X) = T /n = X´ Ú O þT (X) ¼ ê, …Ep gˆ(X) = p,é0 < p < 1. ÏddL-S½nŒ•ˆ g (X)•p •˜ UMVUE. n ~ 3þ~¥, ®•T = Xi Ñl ‘©Ùb(n, p), …T (X)•¿© ÚOþ, ¦g(p) = i=1 p(1 − p) UMVUE. ) δ(T )•g(p) = p(1 − p) ˜‡Ã O, ‡ Ñδ(T ) Lˆª. UÃ O ½Â 9T ∼ b(n, p),Œ n n δ(t)pt (1 − p)n−t = p(1 − p), ˜ƒ 0 < p < 1. t=0 t -ρ = p/(1 − p), kp = ρ/(1 + ρ), 1 − p = 1/(1 + ρ),ò§‚“\þª n n δ(t)ρt = ρ(1 + ρ)n−2 , 0 < ρ < ∞. t=0 t n−2 n−1 n−2 n−2 òρ(1 + ρ)n−2 Ðm ρ(1 + ρ) = l ρl+1 = t−1 ρt ,òÙ“\þªm> l=0 t=1 n n−1 n n−2 t δ(t)ρt = ρ, 0 < ρ < ∞. t=0 t t=1 t−1 þªü>•ρ õ‘ª, ' ÙXê δ(t) = 0, t = 0, n; n−2 n t(n − t) δ(t) = = , t = 1, 2, · · · , n − 1. t−1 t n(n − 1) nÜþãüª T (n − T ) δ(T ) = , t = 0, 1, · · · , n n(n − 1) 6

7 . n •g(p) = p(1 − p) Ã O, § q ´ ¿ © Ú O þT = i=1 Xi ¼ ê, dL-S½ n Œ •δ(T )•g(p) UMVUE. ~ X = (X1 , · · · , Xn )•lPoisson©ÙP (λ) ¥Ä {ü‘Å , ¦ (1) g1 (λ) = λ; (2) g2 (λ) = λ , r > 0 •g,ê; (3) g3 (λ) = Pλ (X1 = x) UMVUE. r n ) d§2.7Ú§2.8Œ•T (X) = i=1 Xi •'uPoisson©Ù ¿© ÚOþ. (1) -ˆ g1 (T ) = T (X)/n, E(ˆ ¯ = λ, g1 (T )) = E(X) gˆ1 (T )´© ÚOþT ¼ê§… ´λ Ã O, dL-S½nŒ•ˆ g1 (T )´λ UMVUE. n (2) d uT (X) = i=1 Xi ∼ P (nλ),-δ(T )•g2 (λ) = λr Ã O, kEλ δ(T ) = g2 (λ),= ∞ e−nλ (nλ)t δ(t) = λr . t=0 t! dª du ∞ nt λt δ(t) = λr enλ . t=0 t! òþªm>ŠÐm ∞ ∞ nl λl+r nt−r λt λr enλ = = . l! t=r (t − r)! l=0 òÙ“\þªm> ∞ ∞ nt λt nt−r λt δ(t) = . t=0 t! t=r (t − r)! þã ªü>´λ ˜?ê, ' ÙXê δ(t) = 0, t = 0, 1, · · · , r − 1, t−r t! n t(t − 1) · · · (t − r + 1) δ(t) = = , t = r, r + 1, · · · (t − r)!nt nr nÜþãüª T (T − 1) · · · (T − r + 1) δ(T ) = , T = 0, 1, 2, · · · nr •g2 (λ) = λr Ã O, δ(T )´¿© ÚOþT ¼ê, dL-S½nŒ•δ(T )•g2 (λ) UMVUE. (3) dPλ (X1 = x) = e−λ λx x! ,Œ„§´ëêλ ¼ê, Œ^g3 (λ)L«. -ϕ(X1 ) = n I[X1 =x] , KEλ [ϕ(X1 )] = Pλ (X1 = x).Ïdϕ(X1 ) •g3 (λ) Ã O, 5¿ T = i=1 Xi ∼ n P (nλ)Ú i=2 Xi ∼ P ((n − 1)λ), k Pλ (X1 = x, T = t) δ1 (T ) = δ1 (T (X)) = E(ϕ(X1 )|T = t) = Pλ (T = t) Pλ (X1 = x)Pλ (X2 + · · · + Xn = t − x) (n − 1)t−x t! = = t Pλ (X1 + · · · + Xn = t) n (t − x)!x! t−x t (n − 1) t 1 x 1 t−x = = 1− , t≥x x nt x n n δ1 (T )•g3 (λ) Ã O, §q´¿© ÚOþT (X) ¼ê, ¤± T −x T 1 1 δ1 (T (X)) = 1− x n n 7

8 .•g3 (λ) UMVUE. ~ X = (X1 , · · · , Xn )•l•ê©ÙEP (λ)¥Ä {ü‘Å , ¦g(λ) = λ UMVUE. n ) dT (X) = Xi ¿© ÚOþ, ±9T (X) ∼ Γ(n, λ),=ëê•nÚλ Gamma© i=1 Ù, k ∞ 1 1 λn λ E = · tn−1 e−λt dt = . T 0 t (n − 1)! n−1 Ïdˆ g (T (X)) = (n − 1)/T (X) •λ Ã O, dL-S½nŒ•§´λ UMVUE. ~ X = (X1 , · · · , Xn )•l ©ÙN (a, σ 2 )¥Ä {ü‘Å , Pθ = (a, σ 2 ). (1)¦aÚσ 2 UMVUE, (2)¦g(θ) = σ r UMVUE. n ) ¯ T2 (X) = dT (X) = (T1 (X), T2 (X)), Ù¥T1 (X) = X, ¯ 2 ,•¿© (Xi − X) ÚO i=1 þ. (1)duˆ ¯ = T1 Úˆ g1 (X) = X g2 (X) = T2 (n − 1)©O•aÚσ 2 Ã O, §‚q´¿© ÚOþ, dL-S½nŒ•§‚©O´aÚσ 2 UMVUE. (2)duT2 /σ 2 ∼ χ2n−1 , σ r Ã O†T2 ˜¼êk'. kOŽeª: r/2 ∞ T2 1 r r 1 n−1 t E = E T22 = t2 · t 2 −1 e− 2 dt n−1 σ2 σ r 0 2 2 Γ n−1 2 2r/2 Γ n+r−1 2 1 = . Γ n−1 2 Kn−1,r dþªŒ• r E(Kn−1,r · T22 ) = σ r . Ïd Oþ r/2 Γ n−1 gˆ3 (T (X)) = Kn−1,r T2 = 2 T r/2 2r/2 Γ n+r−1 2 ´σ r Ã O, q ´ ¿ © Ú O þT = (T1 , T2 ) ¼ ê, dL-S½ n Œ • § ´g(θ) = r θ UMVUE. ~ ^Lehmann-Scheffe½n2•Äþ!©ÙU (0, θ), θ > 0, ¥ ëêθ UMVUE. ) dT (X) = max(X1 , · · · , Xn ) = X(n) •¿© ÚOþ, Úˆ g (T (X)) = n+1 n T (X) •θ Ã O, dL-S½ná gˆ(T (X)) •θ UMVUE.Œ„d?y²‡{ü õ. ~ •Äþ!©ÙU (0, θ), θ > 1, ¥ ëêθ UMVUE. ) dÏf©)½n• T (X) = max(X1 , · · · , Xn ) = X(n) •¿©ÚOþ§ §¿Ø• ÚOþ"¢Sþé?¿÷vEl(T ) = 0,= 1 θ l(t)tn−1 dt + l(t)tn−1 dt = 0 (∗) 0 1 ÏdŒ (n + 1)t − n, 0<t<1 l0 (t) = 0, 1 < t < θ 8

9 .Ï •3šð•" ¼êl(T ) = l0 (T )÷vEl(T ) = 0.¤±T Ø´ ÚOþ, l ØU| ^Lehmann-Scheffe½n yÄuT Ã O•UMVUE, ´·‚Œ±¦^"Ã •{. 5 ¿ –¦Eg(T )l(T ) = 0, = 1 θ g(t)l(t)tn−1 dt + g(t)l(t)tn−1 dt = 0 0 1 (Ü(∗)ªl Œ c, 0<t<1 g(t) = bt, 1 < t < θ dÃ 5k 1 θ c b n Eg(T ) = cfT (t)dt + btfT (t)dt = + n (θn+1 − 1) = θ 0 1 θn θ n+1 l Œ n+1 c = 1, b= n Ïd 1, 0<T <1 g(T ) = n+1 n T, 1 <T <θ •ëêθ UMVUE. 5 Œ±y²ÚOþ 1, 0<T <1 T˜ = T, 1 < T < θ • ÚOþ. 2 Cramer-RaoØ ª ˜!Úó Cramer-RaoØ ª({¡C-RØ ª)´ O˜‡Ã Oþ´Ä•UMVUE •{ƒ˜. ù˜•{ gŽXe: Ug ´g(θ) ˜ƒÃ O ¤ a. Ug ¥ Oþ • k˜‡e., ù‡e.¡•C-Re.. Ïd, XJg(θ) ˜‡Ã Oˆ g • ˆ ù‡e., Kˆ g Ò´g(θ) ˜‡UMVUE, , ©ÙxÚˆ g ‡÷v˜½ K^‡. ù‡Ø ª´dC.R. RaoÚH. Cramer31945Ú1946c©Oy² . ± ˜ ÚOÆöò^‡Š ˜ U?Ú°(z, (J Ä /ª¿Ã-ŒCz. ù˜•{ " € ´: d uC-RØ ª(½ e.~'ýe.• . 3˜ | Ü, • ,g(θ) UMVUE gˆ• 3, Ù• Œ uC-Re .. 3 ù ˜ œ ¹ e, ^C-RØ ªÒÃ{ ½g(θ) UMVUE•3. Ïdù˜•{ ·^‰ŒØ2. C-RØ ªØ ^u Og(θ) UMVUEƒ , §3ênÚOnØþ„kÙ§ ^?, X O ÇÚk O Vg±9Fisher&EþÑ †ƒk'. C-RØ ª¤áI‡ ©Ùx÷v˜ K^‡, ·Üù ^‡ ©Ùx¡•C-R K© Ùx, e¡‰ÑÙ½Â. 9

10 . ½Â eüëêVÇ¼êxF = {f (x, θ), θ ∈ Θ}÷ve ^‡: (i)ëê˜mΘ´†‚þ ,‡m«m; (ii) ê ∂f ∂θ (x,θ) é˜ƒθ ∈ Θ•3; (iii)VÇ¼ê | 8{x : f (x, θ) > 0}†θÃ'; (iv)VÇ¼êf (x, θ) È©†‡©$ŽŒ †, = ∂ ∂ f (x, θ)dx = f (x, θ)dx. ∂θ ∂θ ef (x, θ)•lÑ‘ÅCþ VÇ©Ù, þã^‡U•Ã¡?êÚ‡©$ŽŒ †; (v)e êÆÏ"•3, … 2 ∂ log f (X, θ) 0 < I(θ) = Eθ < ∞, ∂θ K¡T©Ùx•C-R K©Ùx. Ù¥(i)-(v)¡•C-R K^‡. I(θ)¡•T©Ù Fisher&Eþ (½¡•Fisher&E¼ê). !üëêC-RØ ª 1. C-RØ ª9~ ½n F = {f (x, θ), θ ∈ Θ}´C-R K©Ùx, g(θ) ´½Âuëê˜mΘþ Œ‡¼ ê. X = (X1 , · · · , Xn ) ´doNf (x, θ) ∈ F ¥Ä {ü‘Å , gˆ(X) ´g(θ) ?˜Ã O, …÷ve ^‡: (vi) È© ··· gˆ(x)f (x, θ)dx Œ3È©Òeéθ¦ ê, d?dx = dx1 · · · dxn , Kk (g (θ))2 g (X)] ≥ Dθ [ˆ , ˜ƒ θ ∈ Θ. (2.1) nI(θ) AO g(θ) = θž(2.1)=• 1 g (X)] ≥ Dθ [ˆ , ˜ƒ θ ∈ Θ. (2.2) nI(θ) f (x, θ)•lÑr.v. X VÇ©Ùž, (2.1)C• [g (θ)]2 g (X)] ≥ Dθ [ˆ ∂ log f (xi ,θ) 2 , ˜ƒ θ ∈ Θ. (2.3) n ∂θ f (xi , θ) i n y duX1 , · · · , Xn •i.i.d. , kf (x, θ) = f (xi , θ). P i=1 n ∂ log f (x, θ) ∂ log f (xi , θ) S(x, θ) = = , ∂θ i=1 ∂θ 10

11 .Ïdd K^‡(iii)Ú(iv)Œ• n n ∂ log f (Xi , θ) 1 ∂f (xi , θ) E{S(X, θ)} = E = · f (xi , θ)dxi i=1 ∂θ i=1 f (xi , θ) ∂θ n n ∂f (xi , θ) ∂ = dxi = f (xi , θ)dx = 0. i=1 ∂θ i=1 ∂θ dˆ g (x)•g(θ) Ã OÚ K^‡(v)Ú(vi)Œ• Cov(ˆ g (X), S(X, θ)) = E{ˆ g (X) · S(X, θ)} ∂f (x, θ) = · · · gˆ(x) dx ∂θ ∂ ∂g(θ) = · · · gˆ(x)f (x, θ)dx = = g (θ), ∂θ ∂θ n ∂ log f (Xi , θ) Dθ (S(X, θ)) = Dθ i=1 ∂θ n 2 ∂ log f (Xi , θ) = E = nI(θ). (2.4) i=1 ∂θ dCauchy-SchwartzØ ª, Dθ {ˆ g (X), S(X, θ))]2 = [g (θ)]2 g (X)} · Dθ {S(X, θ)} ≥ [Cov(ˆ ò(2.4)“\þª [g (θ)]2 g (X)] ≥ Dθ [ˆ , nI(θ) ½n y. Ø ª(2.1)¡•Cramer-RaoØ ª, {¡C-RØ ª. ÏdC-RØ ªŒÀ• y,˜Ã O´Ä•UMVUE •{. ^C-RØ ªÏég(θ) UMVUEž, Äk‡ y ©Ùx´ Ä÷v K^‡(i)-(v)Ú(vi), , 2OŽFisher&Eþ I(θ)ÚÃ Oˆ g (X) • Dθ (ˆ g (X)),w Ù´Äˆ C-Re.. y K^‡(i)-(v)Ú^‡(vi)›©æ†. 3$ ´é•êxþã K ^‡(i)-(v) ¤á. ‡5¿˜: ´: eDθ (ˆ g (X))ˆØ C-Re., ¿ØU Ñ(Ø`g(θ) UMVUEÒØ• 3, •U`^d{Ã{ O. •3ù ~f, gˆ(X)´g(θ) UMVUE, Ù• ŒuC-Re.. ¡ò‰Ñù˜~f. ~ X = (X1 , · · · , Xn )•lü:©Ùb(1, p)¥Ä {ü , y² ¯ = þŠX n 1 n Xi •p UMVUE. i=1 y ‘ÅCþX ∼ b(1, p),KÙVÇ©Ù•f (x, p) = px (1 − p)x , x = 0, 1, 0 < p < 1. d uü:©Ùx´•êx, C-R K^‡¤á. Fisher&E¼ê• 2 2 ∂ log f (X, p) X −p V arp (X) 1 I(p) = E =E = = , ∂p p(1 − p) p2 (1 − p)2 p(1 − p) ÏdC-Re.•1 [nI(p)] = p(1 − p)/n. ¯ X•p Ã ¯ = p(1 − p)/nˆ C-Re.. O, Ù• Dp (X) ¯ X•p UMVUE. 11

12 . ~ X = (X1 , · · · , Xn )•lPoisson©ÙP (λ)¥Ä {ü , ^C-RØ ª y ¯= 1 n þŠX n i=1 Xi •λ UMVUE. y ‘ÅCþX ∼ P (λ),KÙVÇ©Ù•f (x, λ) = e−λ λx x!, x = 0, 1, 2, · · · , λ > 0. d uPoisson©Ùx••êx, C-R K^‡¤á. Fisher&E¼ê• 2 2 ∂ log f (X, λ) X −λ 1 1 I(λ) = E =E = 2 Dλ (X) = , ∂λ λ λ λ C-Re.•1 [nI(λ)] = λ/n. ¯ gˆ(X) = X•g(λ) =λ Ã ¯ = λ/nˆ O, …• Dλ (X) C-Re., ¯ X•λ UMVUE. ~ X = (X1 , · · · , Xn )• l • ê © ÙEP (λ)¥ Ä {ü , ^C-RØ ª yˆ ¯ g (X) = X•g(λ) = 1/λ UMVUE. y •ê©ÙEP (λ) —Ý¼ê•f (x, λ) = λe−λx I[x>0] , λ > 0. •ê©Ùx´•êx, C-R K^‡¤á. Fisher&E¼ê• 2 2 ∂ log f (X, λ) 1 1 I(λ) = E =E −X = Dλ (X) = , ∂λ λ λ2 C-Re.•(g (λ))2 [nI(λ)] = 1 (nλ2 ). ¯ gˆ(X) = X•g(λ) = 1/λ Ã ¯ = 1 (nλ2 )ˆ O, Ù• Dλ (X) C-Re., ¯ X•g(λ) = 1/λ UMVUE. 2. C-RØ ª Ò¤á ^‡∗ •êxØ U¦½n ^‡(i)-(vi)¤á, §„´U¦C-RØ ª Ò¤á •˜©Ùx. e ½n•ÑØ •êx , g(θ) Ã O • ØU??ˆ C-Re.. ½n eC-R K ^ ‡(i)-(v)¤ á, gˆ(X)•g(θ) Ã O. g(θ)3Θþ Ø ð u ~ ê, g (X)) < ∞é˜ƒθ ∈ Θ, KC-RØ Dθ (ˆ ª¥ Òé˜ƒθ ∈ Θ¤á ¿‡^‡´ f (x, θ) = C(θ) exp{Q(θ)T (X)}h(x), (2.5) Ù¥C(θ)ÚQ(θ)•θ ëYŒ‡¼ê, …Q (θ) = 03Θþ??¤á. 53.5.1 ½n î‚Qã‡òC-RØ ªé˜ƒθ ∈ Θ¤á ¿‡^‡(2.5)U•Xe/ ª Pθ (f (x, θ) = C(θ) exp{Q(θ)T (X)}h(x)) = 1, (2.6) Ù{ØC. =‡¦(2.5)ª'uÿÝPθ VÇ•1¤á. ùé½n y²¿Ã¢ŸK•. y ½ny²¥ Ñ(2.1) '…Ú½´^ Cauchy-SchwartzØ ª: Cov 2 (ξ, η) ≤ D(ξ) · D(η). ù˜Ø ª¥ Ò¤á ¿‡^‡´ξÚηk‚5'X, =•3~êa, b ¦ ξ = aη + b. 3C-RØ ª½n¥ξ = S(x, θ), η = gˆ(X), Ïd(2.1)ª Ò¤á ¿‡^‡• S(x, θ) = a(θ)ˆ g (X) + b(θ), (2.7) ùpa(θ) = 0, b(θ) †xÃ'. ¯K=z•y²: /(2.5)é˜ƒθ ∈ Θ¤á ¿‡^‡•(2.7)ª ¤á0. e¡5y²ƒ. 12

13 . e(2.7)ª¤á, ü>éθ¦È© θ θ θ ∂ log f (x, u) du = a(u)du · gˆ(x) + b(u)du. θ0 ∂u θ0 θ0 = log f (x, θ) = Q(θ)ˆ g (x) + R(θ) + l(x), (2.8) d? θ θ Q(θ) = a(u)du, R(θ) = b(u)du, l(x) = log f (x, θ0 ). θ0 θ0 PC(θ) = eR(θ) , h(x) = el(x) , d(2.8) f (x, θ) = C(θ) exp{Q(θ)ˆ g (x)}h(x), ˜ƒ θ ∈ Θ, =(2.5)ª¤á. ‡ƒ, e(2.5)ª¤á, òÙü> éê log f (x, θ) = log C(θ) + Q(θ)ˆ g (x) + log h(x), ü>éθ¦ C (θ) S(x, θ) = + Q (θ)ˆ g (x). (2.9) C(θ) dEθ S(x, θ) = 0Œ C (θ) C(θ) + Q (θ)g(θ) = 0, =C (θ) C(θ) = −Q (θ)g(θ),òdª“\ (2.9)Œ g (x) − g(θ)) = a(θ)ˆ S(x, θ) = Q (θ)(ˆ g (x) + b(θ), d?a(θ) = Q (θ) = 0, b(θ) = Q (θ)g(θ).Ïd(2.7)¤á, ½ny.. e¡ ½n•Ñ, =¦3•êx œ/, g(θ) Ã O • U??ˆ C-Re. œ/ •Øõ. ½n X ©Ùx•Xe üëê•êx f (x, θ) = C(θ) exp{Q(θ)T (x)}h(x), K …= g(θ) = Eθ (aT (x) + b) ž, âkÙ• ??ˆ C-Re. Ã Oˆ g (X) = aT (X) + b,Ù¥a, b•†θÃ' ü‡~ê. y „ë•©z[1] P117 ½n2.2.3. ~ X = (X1 , · · · , Xn )•gPoisson©ÙP (λ)¥Ä {ü‘Å ,y²•kg(λ)´λ ‚5¼êž,â•3g(λ) Ã O,Ù• U??ˆ C-Re.. y ò X VÇ©Ù n xi n e−nλ i=1 f (x, λ) = λ xi ! i=1 L¤•êx /ª n n f (x, λ) = e−nλ exp log λ · xi xi ! i=1 i=1 13

14 . = C(λ) exp{Q(λ)T (x)}h(x). n n d?C(λ) = e−nλ , Q(λ) = log λ, T (x) = Xi , h(x) = 1 xi ! . i=1 i=1 d½nŒ• …= g(λ) = E(aT (x) + b) = a · nλ + b = cλ + b, c, b•~ê ž, Ã Oˆ g (X) = aT (X) + b • ??ˆ C-Re., …´g(λ) UMVUE. Ù§/ª ëê ÑØ•3ÙUMVUE. AO g(λ) = λ,=a = 1/n, b = 0, Kˆ ¯ g (X) = T /n = X • ??ˆ C-Re., Ïd§ ´λ UMVUE. ù†~3.5.2‰Ñ (JƒÓ. 3. Fisher&E¼ê òC-RØ ª¥ 2 ∂ log f (X, θ) I(θ) = Eθ ∂θ ¡•Fisher&E¼ê (½¡•Fisher&Eþ . •)ºI(θ),Ø”-g(θ) = θ¿b½C-RØ ªe .1/(nI(θ))Œˆ . ùžnI(θ) Œ, g(θ) Ã Oˆ g (X) • , L²g(θ) = θ Œ± O °. nI(θ)†nÚI(θ)¤ ', n´ Nþ, ùL²e± Oþ • êŠ• Oþ° Ý •I, K°Ý†n¤ '. '~Ïf, =I(θ),‡NoN©Ù ˜«5Ÿ. Ò´`, oN©Ù I(θ) Œ, ¿›XoN ëê N´ O, ½ö`, ToN . Jø &Eþ õ. k ndrI(θ)À•ïþoN .¤¹&Eõ þ—&Eþ. I(θ)•Œ±)º¤ü‡ Jø &Eþ. duX1 , · · · , Xn ´i.i.d. , §‚ / ´² , z‡ ¬JøÓ õ &EI(θ), = ‡ (X1 , · · · , Xn )¤¹&Eþ•nI(θ). Fisher&Eþ I(θ) -‡¿Â„3u, 3: OŒ nØ ïÄ¥, §åƒ Š^. X3 Ï¦θ 4Œq, Oθˆ∗ ìC©Ù, θˆ∗ ìC ©Ù • ÒŒ±^Fisher&EþL«, = −1 ∂ log f (x, θ) 1 nEθ = . ∂θ nI(θ) ¤±3½n3.3.2 ^‡e, ëêθ 4Œq, Oθˆ∗ ìC©ÙŒL«•N θ, 1 [nI(θ)] . ùL ²ÙìC• † Fisher&Eþ¤‡'. Ïd, I(θ) Œž, ìC• , ^θ 4Œq , Oθˆ∗ 5 OθÒ °. ddw5, FisheròI(θ)¡•&Eþ, (k˜½ Šâ. n!õëêC-RØ ª{0∗ ±þ?Ø Ñ´ëêθ•˜‘ œ/, éθ•p‘ œ/•Œ±ïáaq (J. •dkÚ? ˜ PÒ. A = (aij )ÚB = (bij )´Ó šK½• , eA − B´šK½ , KP•A ≥ B. ù ž7kaii ≥ bii ,é˜ƒi. y θ = (θ1 , · · · , θk ),oNVÇ¼êP•f (x, θ), X = (X1 , · · · , Xn )•loN¥Ä {ü ‘Å . ˆ ˆ ˆ ˆ θ = θ(X) = (θ1 , · · · , θk )´θ ˜‡Ã ˆ O. ±Cov θ (θ) Pθ • , §´˜ ‡k šK½• , Ù(i, j) •Eθ (θˆi − θi )(θˆj − θj ) , K3aquθ•˜‘ K^‡e, Œ± y² ˆ ≥ (nI(θ))−1 , Cov θ (θ) (2.10) 14

15 .ùpI(θ) = (Iij (θ))´˜‡k ½• , … ∂ log f (x, θ) ∂ log f (x, θ) Iij (θ) = Eθ , i, j = 1, 2, · · · , k, (2.11) ∂θi ∂θj K(2.10)Ò´õ‘ C-RØ ª. eP(I(θ))−1 = I ∗ (θ) = (Iij ∗ (θ)), d(2.10)Œ Dθ (θˆi ) ≥ Iii∗ (θ)/n, i = 1, 2, · · · , k. (2.12) ù‰Ñ θ z‡©þθi Ã Oθˆi • e•. ~ X = (X1 , · · · , Xn )•l oNN (a, σ 2 )¥Ä {ü , Pθ = (a, σ 2 ), Ù ¥θ1 = a, θ2 = σ 2 .¦θ C-Re.,¿òÙ†θ1 Úθ2 Ã ¯ OXÚS 2 • ?1' " ) ‘ÅCþ —Ý¼ê• 1 1 f (x, θ) = (2πθ2 )− 2 exp − (x − θ1 )2 , 2θ2 Œ• ∂ log f (x, θ) x − θ1 ∂ log f (x, θ) −θ2 + (x − θ1 )2 = , = . ∂θ1 θ2 ∂θ2 2θ22 ddŽÑ 1 1 I11 (θ) = , I22 (θ) = , I12 (θ) = I21 (θ) = 0, σ2 2σ 4 k n/σ 2 0 σ 2 /n 0 nI(θ) = 4 , (nI(θ))−1 = , 0 n/2σ 0 2σ 4 /n eP n n ¯= 1 θˆ1 = X Xi , θˆ2 = S 2 = 1 ¯ 2, (Xi − X) n i=1 n−1 i=1 Kdõ‘C-RØ ªŒ• θˆ1 σ 2 /n 0 Covθ ≥ . θˆ2 0 2σ 4 /n Dθ (θˆ1 ) = σ 2 /nˆ C-Re., §´θ1 = a UMVUE. |^(n − 1)S 2 /σ 2 ∼ χ2n−1 9Dθ [(n − 1)S 2 /σ 2 ] = 2(n − 1)Œ 2σ 4 2σ 4 Dθ (θˆ2 ) = > . n−1 n Ïd, S 2 = θˆ2 • ŒuC-Re.. 3~3.4.11¥·‚®y² ¯ XÚS 2 ©O´aÚσ 2 UMVUE. d ~wÑ, =¦3 oN• ù {ü |Ü, • σ 2 UMVUE S •ˆØ 2 C-Re .. Ïd3˜ ¯K¥C-Re.~'ýe.• . ùL²±C-RØ ªŠ•ÏéUMVUE •{ ´Ø nŽ . õc5k˜ Æö˜†3ïÄU?ù‡Ø ª ¯K. o!k OÚ O Ç ½Â 3.5.2 gˆ(X)•g(θ) Ã O, 'Š [g (θ)]2/nI(θ) egˆ (θ) = Dθ [ˆ g (X)] 15

16 .¡•Ã Oˆ g (X) Ç(Efficiency). w,0 < egˆ (θ) ≤ 1, egˆ (θ) = 1ž, ¡ˆ g (X)´g(θ) k O (Effective estimation). e lim egˆ (θ) = 1, K¡ˆ g (X)´g(θ) ìCk O (Asymptotically n→∞ effective estimation). ù˜VgkÙØvƒ?: k O´Ã Oa¥•Ð O, <‚ ,F"¦^§. ŒJ, k O´Øõ , ìCk O%Ø . lk O ½Âw, k O˜½´UMVUE, éõUMVUEØ´k O. ù´Ï•C-Re. , 3éõ|ÜUMVUE • ˆØ C-Re .. , C-RØ ª¤á cJ‡¦ ©Ùx÷vC-R K^‡. ù ^‡Ø¤áž, C-RØ ªŒ±Øé, ùž•â§¤Jø C-Re. ½Â O ÇÒØÜn . ½ Â3.5.3 gˆ•g(θ) ˜‡CAN O, ÙìC• •σ 2 (θ), K¡ 1 −1 aegˆ (θ) = σ 2 (θ) = nI(θ)σ 2 (θ) nI(θ) •ˆ g ìC Ç (Asymptotic efficiency). âd, 4Œq, OkìC Ç1. –uÝ O, ·‚3§3.2 ®y²§‚3é˜„^‡e •CAN O, ØLØ ~„ A‡~f(3Ù¥Ý O†MLE-Ü)ƒ ,Ý OìC Ç˜„ Ñ$u1, Ï~<‚`Ý OØX4Œq, O, ŒVÒ´•ù˜:. ~3 X = (X1 , · · · , Xn )•lN (a, σ 2 )¥Ä {ü‘Å , (i) a™•ž, y² • S Ø´σ2 2 k O§ ´ìCk O. (ii) a®•ž, ¦σ 2 k O. n ¯ 2 ) (i) a™•ž, S 2 = 1 n−1 i=1 (Xi − X) ƒ• •2σ 4 /(n − 1)ˆØ C-Re.2σ 4 /n, §Ø´σ 2 k O. O Ç•eS2 (σ ) = (n − 1)/n < 1, ´ 2 n−1 lim eS2 (σ 2 ) = lim = 1, n→∞ n→∞ n ÏdS 2 ´σ 2 ìCk O. n 1 (ii) dua®•, -Sa2 = i=1 (Xi − a) , dunSa /σ n 2 2 2 ∼ χ2n , D nSa2 /σ 2 = 2n, Ïd kD(Sa2 ) = 2σ 4 /n,§ˆ C-Re., Sa2 •σ 2 k O. a®•ž, |^Lehmann-Scheffe½ n•N´y²Sa2 •σ 2 UMVUE. ~3 X = (X1 , · · · , Xn )´le ¹k ˜ëê •ê©Ùx¥Ä {ü , f (x, a) = e−(x−a) I[a,∞) (x), −∞ < a < +∞, ¦a UMVUE. ) þã©ÙxØ´C-R Kx, =C-R K^‡Ø¤á. Ï•3a = xž ∂f ∂a (x,a) Ø•3, …—Ý¼ê | 8{x : f (x, a) > 0} = {x : x ≥ a}†™•ëêak', Ïd§Ø÷vC-R K^ ‡.Ïd·‚ØU^C-RØ ª5¦a UMVUE. w,, • gSÚOþX(1) ´a ¿© ÚOþ. X(1) —Ý¼ê• f (y, a) = ne−n(y−a) I[a,∞) (y). u´ ∞ 1 E(X(1) ) = n ye−n(y−a) dy = a + , a n 1 dL-S½nŒ•ˆ a(X) = X(1) − n ´a UMVUE; ØŒ±?Øˆ a(X)´Ä•k O, Ï •C-R K^‡Ø¤á, ?ØC-Re.Ò” ¿Â. 16

17 . e¡´˜‡(½ìC Ç ~f: ~ X = (X1 , · · · , Xn )•l ©ÙN (a, σ 2 )¥Ä {ü‘Å . Pθ = (a, σ 2 ), gˆ(X)• ¥ ê, §´a ˜‡ O. ¦§ ìC Ç" √ ) d© ê O ìC©Ù, ¿5¿ d?f (ξ1/2 ) = f (a) = 1 2πσ, k √ 1 L 2 n· √ g (X) − a) −→ N (0, 1), (ˆ 2πσ ½U • √ π 2 L g (X) − a) −→ N 0, n(ˆ σ , 2 =ˆ g (X) ìC• •πσ 2 (2n).d~3.5.4Œ•I(θ) = 1/σ 2 , U½Â3.5.3Œ•ˆ g (X) ìC Ç ´ 1 σ2 π 2 2 aegˆ (θ) = σ 2 (θ) = σ = . nI(θ) n 2n π ~ |^k O VgÚ½nŒ±y²e 4Œq, O ˜^-‡5Ÿ, QãXe: X = (X1 , · · · , Xn )• g © Ù x{f (x, θ), θ ∈ Θ} ¥ Ä {ü‘Å , Θ•R1 þ m « m,P •A. eg(θ) k g (X) = gˆ(X1 , · · · , Xn )• 3, Kg(θ)ƒMLE gˆ∗ (X) = Oˆ ∗ ∗ gˆ (X1 , · · · , Xn ) 7†ˆ g (X)-Ü, =ˆ g (X) = gˆ(X). y eˆ g (X)•g(θ) k O, =Ù• ˆ C-Re., d½n3.5.2Œ•ù˜¯¢¤á ¿‡^‡•(2.7)ª¤á, =S(x, θ) = a(θ)ˆ g (x) + b(θ).yòdªU ˜e, Ï•Eθ {S(X, θ)} = a(θ)g(θ) + b(θ) = 0,=b(θ) = −a(θ)g(θ),òÙ“\þª g (x) − a(θ)g(θ) = a(θ) (ˆ S(x, θ) = a(θ)ˆ g (x) − g(θ)) , (2.13) Ù¥a(θ) = 0,é˜ƒθ ∈ Θ,…3ΘþëY, a(θ)3Θ½ Œu0, ½ u0. Ã”-a(θ) > 0, é˜ƒθ ∈ Θ,Kd(2.13)Œ•, gˆ(x) > g(θ)ž, S(x, θ) = ∂ log f (x, θ) ∂θ > 0, L«éêq, ¼ê3(−∞, gˆ(x)) ∩ ASî‚üNþ,; gˆ(x) < g(θ)ž, ∂ log f (x, θ) ∂θ < 0L«éêq,¼ ê3(ˆ g (x), ∞) ∩ Aþî‚üNeü, ¤±ˆ g (x)´q,•§ •˜), …´4Œz , gˆ(X)= ∗ •g(θ) MLE, Ïdˆ g (X) = gˆ (X),y.. 17

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