03 统计学基础--指数族和充分完备统计量

1 . Lec3: •êxÚ¿© ÚOþ Ü•² 2011 c 9 12 F 1 •êx 3ÚOnدK¥, NõÚOíä•{ `û5, é˜a‰Œ2• ÚO . (½¡•©Ù x), k ÷¿ (J. ùa©Ùx¡••êx. ~„ ©Ù, X ©Ù! ‘©Ù! Poisson ©Ù! K ‘©Ù! •ê©ÙÚGamma ©Ù Ñáuùa©Ùx, ù L¡þw5ˆØƒÓ ©Ù, Ù¢§‚ÑŒ±Ú˜3˜«•Û•2 ˜a¡••êx ª¥. ,Ú?ù«©Ù x nd, ̇Ø3u*¦/ªþ Ú˜, 3uù«Ú˜84 §‚ 5, ÏdNõÚOn دK, é•êx¼ ”. )û. !0 •êx ½Â9{ü5Ÿ. ˜!½Â†~f ½  1. F = {f (x, θ) : θ ∈ Θ} ´½Â3 ˜mX þ ©Ùx, Ù¥Θ •ëê˜m. eÙVǼêf (x, θ) ŒL«¤Xe/ª k f (x, θ) = C(θ) exp Qi (θ)Ti (x) h(x), i=1 K¡d ©Ùx••ê.©Ùx({¡•êx (Exponential family). Ù¥k•g,ê, C(θ) > 0 ÚQi (θ) (i = 1, 2 · · · , k) Ñ´½Â3ëê˜mΘ þ (Œÿ) ¼ê, h(x) > 0 ÚTi (x) (i = 1, 2, · · · , k) Ñ´½Â3X þ (Œÿ) ¼ê. •êx ˜‡5Ÿ´x¥ ¤k©Ùäk Ó | 8( G(x) ¡•VǼêp(x) | 8, eG(x) = {x : p(x) > 0} ). d½ÂŒ„•êx| 8{x : f (x, θ) > 0} = {x : h(x) > 0} †θ à '. ?˜©ÙxeÙ| 8†θ k', Kx¥©ÙØ2äk Ó| 8, Ï 7Ø´•êx. ~1. ©Ùx{N (µ, σ 2 ) : −∞ < µ < ∞, σ 2 > 0}´•êx. Proof. X = (X1 , · · · , Xn )•l ©ÙN (µ, σ 2 )¥Ä {ü ,X éÜ—Ý• n 2 √ −n 1 f (x; µ, σ ) = 2πσ exp − 2 (xi − µ)2 . (1.1) 2σ i=1 Pθ = (µ, σ 2 ), Këê˜m•Θ = {θ = (µ, σ 2 ) : −∞ < µ < +∞, σ 2 > 0}.ò(1.1) U • n n √ −n nµ2 µ 1 f (x, θ) = 2πσ e− 2σ2 exp xi − x2i σ2 i=1 2σ 2 i=1 1 

2 . = C(θ) exp{Q1 (θ)T1 (x) + Q2 (θ)T2 (x)}h(x), (1.2) √ nµ2 n d ?C(θ) = ( 2πσ)−n e− 2σ2 , Q1 (θ) = µ/σ 2 , Q2 (θ) = − 2σ1 2 , T1 (x) = i=1 xi , T2 (x) = n i=1 x2i , h(x) ≡ 1 . Ïdd½ÂŒ• ©ÙxN (µ, σ ) ´•êx. 2 ~2. ‘©Ùx{b(n, θ) : 0 < θ < 1}´•êx. Proof. X∼ ‘©Ù b(n, θ), ÙVÇ¼ê• n x p(x, θ) = Pθ (X = x) = θ (1 − θ)n−x x n θ x = (1 − θ)n , x = 0, 1, 2, · · · , n. (1.3) x 1−θ d? ˜mX = {0, 1, 2, · · · , n}, ëê˜mΘ = {θ : 0 < θ < 1} = (0, 1). òþªU • θ n p(x, θ) = (1 − θ)n exp x log · 1−θ x = C(θ)exp{Q1 (θ)T1 (x)}h(x). (1.4) θ n d?C(θ) = (1 − θ)n , Q1 (θ) = log 1−θ , T1 (x) = x, h(x) = x , U½Â ‘©Ùxb(n, θ) •´• êx. ~3. þ!©Ùx{U (0, θ), θ > 0} Ø´•êx. Proof. d•êx ½ÂŒ•§Ù| 8•{x : p(x, θ) > 0} = {x : h(x) > 0}, §†θ Ã'. þ!©Ùx{U (0, θ), θ > 0} | 8•{x : p(x, θ) > 0} = (0, θ) †θ k', Ïd§Ø´•ê x. !•êx g,/ª9g,ëê˜m k 3•êx ½ÂC(θ) exp Qi (θ)Ti (x) h(x) ¥, e^ϕi “OQi (θ), òC(θ) L¤ϕ i=1 k ¼êC ∗ (ϕ), ϕ = (ϕ1 , ϕ2 , · · · , ϕk ), ÙLˆªC•C ∗ (ϕ) exp ϕi Ti (x) h(x). 2Uϕi i=1 k •θi , i = 1, 2, · · · , k, Kþª=•: C(θ) exp θi Ti (x) h(x), dª¡••êx g,/ª(½ i=1 ¡•IO/ª). ke ½Â ½  2. XJ•êxke /ª n f (x, θ) = C(θ)exp θi Ti (x) h(x), (1.5) i=1 K¡••êx g,/ª(Natural form). dž8Ü k Θ∗ = (θ1 , θ2 , · · · , θk ) : exp θi Ti (x) h(x)dx < ∞ (1.6) X i=1 ¡•g,ëê˜m (Natural parametric space). ~4. ò ©ÙxL«••êx g,/ª, ¿¦ÑÙg,ëê˜m. 2 

3 .Proof. d n n n 1 nµ2 µ 1 f (x; µ, σ 2 ) = √ e− 2σ2 exp xi − x2i , 2πσ σ2 i=1 2σ 2 i=1 ëê˜mΘ = {(µ, σ ) : −∞ < µ < ∞, 0 < σ < ∞}.-ϕ1 = µ/σ 2 , ϕ2 = − 2σ1 2 , )Ñσ = 2 2 n nµ2 n nϕ2 1 − 2ϕ1 2 , µ2 /σ 2 = ϕ21 (− 2ϕ1 2 ), Ïdk √1 2πσ e− 2σ2 = −2ϕ2 2π e 4ϕ2 = C ∗ (ϕ), ϕ = (ϕ1 , ϕ2 ), n n f (x, ϕ) = C ∗ (ϕ) exp ϕ1 xi + ϕ2 x2i h(x) i=1 i=1 = C ∗ (ϕ) exp{ϕ1 T1 (x) + ϕ2 T2 (x)}h(x). 2Uϕi •θi (i = 1, 2), þªC• f (x, θ) = C ∗ (θ) exp{θ1 T1 (x) + θ2 T2 (x)}h(x). (1.7) d•Ùg,/ª. Ùg,ëê˜m• Θ∗ = {(θ1 , θ2 ) : −∞ < θ1 < +∞, −∞ < θ2 < 0}. n!•êx 5Ÿ ½ n 1. 3•êx g,/ªe, g,ëê˜m•à8. y² •{Xeµ ?‰θ(1) = (θ11 , · · · , θk1 ), θ(0) = (θ10 , · · · , θk0 ) áug,ëê˜mΘ∗ , 0 < α < 1, -θ = αθ (1) + (1 − α)θ (0) (=θi = αθ + (1 − α)θ , i = 1, 2, · · · , k),eUy²θ ∈ Θ∗ , 1 i 0 i KUà8 ½Â, ½n y. ½ n 2. •êx g,/ª¥, g,ëê˜mkS:, g(x) ´?˜k.ŒÈ¼ê, Ké k G(θ) = g(x) exp θj Tj (x) h(x)dx, X j=1 k k ∂ m G(θ) ∂m = g(x) exp θj Tj (x) h(x) dx, ∂θ1 · · · ∂θkmk m1 X ∂θ1 · · · ∂θkmk m1 j=1 k Ù¥ j=1 mj = m, =éG(θ) 'uθ ?¿ êŒ3È©e¦ . d½n •˜„ /ª9Ùy² wë•©z[1] P21 ½n1.2.1. 2 ¿©ÚOþ ·‚• , ÚOþ´é {z, F"ˆ : (i) {z §Ýp; (ii) &E ›” . ˜‡Ú OþU8¥ ¥&E õ , †ÚOþ äN/ªk', ••6u¯K ÚO .. •Ð œ 3 

4 .¹´ÚOþr ¥ Ü&EÑ8¥å5, •Ò´`&EÛ”, ·‚¡ù ÚOþ•¿© ÚOþ. 'u X = (X1 , X2 , · · · , Xn ) &EŒ± Ž¤Xe úª: X¥•¹ëê &E = ÚOþT (X)¥¤¹ëê &E + 3• T (X) Xÿ¹k'uëê •{&E T (X)• ¿ © Ú O þ ‡ ¦ 8 ( •: ‡ ¦ ˜ ‘ & E •0. ^ Ú O Š ó 5 £ ã, = ‡ ¦Pθ (X|T = t)†θÃ'. Ïd·‚ Xe ½Â: ½  1. X ©Ùx{Fθ (x), θ ∈ Θ}, θ•©Ù ëê. T = T (X)•˜ÚOþ, e3® •T ^‡e, X ^‡©Ù†θÃ', K¡T (X)•¿©ÚOþ (Sufficient statistic). ¢SA^ž^‡©Ù^^‡VÇ(lÑœ/) ½^‡—Ý(ëYœ/) 5“O. n ~1. X = (X1 , X2 , · · · , Xn )•l0 − 1©Ù¥Ä {ü , KT (X) = i=1 Xi •¿©Ú Oþ. Proof. U½Â·‚•‡y²e ^‡VdžθÃ'. P (X1 = x1 , · · · , Xn = xn |T (x) = t0 )  n  P (X1 =x1 ,··· ,Xn =xn ,T =t0 ) = 1 n ,  xi = t 0  P (T (x)=t0 ) t0 i=1 = n  0, xi = t 0 .   i=1 n þã^‡VdžθÃ',ÏdT (X) = i=1 Xi •¿©ÚOþ. 1 n ~2. X = (X1 , X2 , · · · , Xn )•l oNN (θ, 1)¥Ä {ü , KT (X) = n i=1 Xi = ¯ •¿©ÚOþ. X Proof. 2‰ C† n y1 = √1 xi , n i=1 n yj = ajk xk , j = 2, · · · , n. k=1 n n d oNe þŠÚ • ©Ù ÑL§Œ• i=1 Yi 2 = i=1 Xi 2 ,…Y1 , Y2 , · · · , Yn ´ √ ƒpÕá , Y1 ∼ N ( nθ, 1), Yi ∼ N (0, 1), i = 2, · · · , n. Ïd(Y1 , · · · Yn ) éÜ—Ý• n √ − 21 yi2 − 12 (y1 − nθ)2 −n f (y1 , · · · yn ) = (2π) 2 e i=2 . 2dY1 —Ý¼ê• 1 1 √ 2 fY1 (y1 ) = √ e− 2 (y1 − nθ) 2π •3‰½Y1 ž, (Y1 , · · · , Yn ) ^‡—Ý´ n f (y1 , · · · , yn ) 1 n−1 − 2 yi2 f (y1 , · · · , yn |y1 ) = = (2π)− 2 e i=2 (2.1) fY1 (y1 ) †θÃ'. 4 

5 . √ ùp|^ e ¯¢: -¡{(Y1 · · · Yn ) : Y1 = nt = y1 }´d-¡{(X1 , · · · , Xn ) : T (X) = t}² ^= 5,-¡ ±ØC. Ïd3-¡{(X1 , · · · , Xn ) : T (X) = t}þ ^‡Vdž3- ¡{Y1 , · · · , Yn ) : Y1 = y1 } þ ^‡VǃÓ. k n − 21 yi2 − n−1 f (x1 , · · · , xn |T = t) = f (y1 , · · · , yn |Y1 = y1 ) = (2π) 2 e i=2 ¯ †θÃ', ¤±T (X) = X´¿©ÚOþ. !¿©5 OOK—Ïf©)½n Ïf©)½n´dR.A. Fisher 3 ›-V ›c“JÑ5,§ •˜„/ªÚî‚êÆy ², ´Halmos ÚSavage31949cŠÑ5 . ½ n 3. ( Ïf©)½n) X = (X1 , · · · , Xn ) VǼêf (x, θ) = f (x1 , · · · , xn ; θ) • 6uëêθ, T = T (X) = (T1 (X), · · · , Tk (X)´˜‡ÚOþ, KT•¿©ÚOþ ¿‡^‡ ´f (x, θ)Œ±©)• f (x, θ) = g(T (x), θ)h(x) (2.2) /G.5¿d?¼êh(x) = h(x1 , · · · , xn )Ø•6uθ. ùpVǼ괕: eX•ëY., Kf (x, θ)´Ù—ݼê; eX´lÑ., Kf (x, θ) = Pθ (X1 = x1 , · · · , Xn = xn ), = X VÇ©Ù. í Ø 1. T = T (X)•θ ¿©ÚOþ, S = ϕ(T)´üŠŒ_¼ê, KS = ϕ(T)•´θ ¿©Ú Oþ. ~3. X = (X1 , · · · , Xn )•l oNN (a, σ 2 )¥Ä {ü , -θ = (a, σ 2 ), KT (X) = n n ( i=1 Xi , i=1 Xi2 )•¿©ÚOþ. Proof. X éÜ—Ý• n n 1 1 f (x, θ) = √ exp − (xi − a)2 2πσ 2σ 2 i=1 n n n 1 1 = √ exp − 2 x2i − 2a xi + na2 2πσ 2σ i=1 i=1 = g(T (x), θ) · h(x). n n d?h(x) ≡ 1, dÏf©)½nŒ•T (X) = ( i=1 Xi , i=1 Xi2 )•¿©ÚOþ. n n ¯ S 2 )•˜˜éA C†, díØŒ•(X, ¯ S 2 )•´¿©ÚOþ. du( i=1 Xi , i=1 Xi2 )†(X, n ~4. X = (X1 , · · · , Xn )•loNb(1, θ)¥Ä {ü , KT (X) = i=1 Xi ´¿©ÚO þ. Proof. X éÜ©Ù´ f (x, θ) = Pθ (X1 = x1 , · · · , Xn = xn ) n n xi n− xi = θ i=1 (1 − θ) i=1 = g(T (x), θ)h(x). n d?h(x) ≡ 1, dÏf©)½nŒ•T (X) = i=1 Xi •¿©ÚOþ. 5 

6 .n!4 ¿©ÚOþ∗ ˜‡©ÙxF ¿©ÚOþ ØŽ˜‡, @o3¦^¥ATXÛ]ÀQ? ·‚• ,Ú Oþ´d \ó 5 , X ÙÚó¤ã, é \ów,Œ±JÑü^‡¦: (1) 3\ ó¥, ¤¹ëêθ &E›” Ð. e\ó¥d«&EÎÛ”, @Ò´¿©5 ‡¦. (2) \ó¥, ¤ ÚOþ•{z Ð, {z §ÝŒ±^ÚOþ ‘ê5ïþ, •Œ±^¼ê' m n X5L«. ~X阇 ‘ÚOþT1 (X) = ( i=1 Xi , i=m+1 Xi ),2?˜Ú\ó ˜‘Ú n OþT2 = i=1 Xi . †*þN´wÑ, T2 'T1 {z. …Œ±wÑ, T2 ´T1 ¼ê. ˜„5`, eT †s´ü‡ÚOþ, …T ´S ¼ê, =T = q(S), @od¼ê ½ÂŒ•, T 'S{z. ½  2. T ´©ÙxF ¿©ÚOþ, eéF ?˜¿©ÚOþS(X), •3˜‡¼êqS (·)¦ T (X) = qS (S(X)), K¡T (X)´d©Ùx 4 ¿©ÚOþ. 3 ÚOþ∗ ½  1. F = {Fθ (x), θ ∈ Θ}•˜©Ùx, Θ´ëê˜m. T = T (X)•˜ÚOþ, eé?˜ ¢¼êϕ(·),d Eθ ϕ(T (X)) = 0, ˜ƒ θ ∈ Θ, (3.1) oŒíÑ Pθ (ϕ(T (X)) = 0) = 1, ˜ƒ θ ∈ Θ, (3.2) K¡T (X)´˜ ÚOþ (Complete Statistic). d½ÂŒ„, eT (X)´ ÚOþ, K§ ?˜¢¼êg(T )•´ ÚOþ. 51. ÚOþT (X) 5Ø= ûuT /G, „ ûu X ©Ùx. 5(½¡ 5) ù‡¶¡, ´5 u ¼ênØ¥ ˜‡aqVg. •{üO, ÚOþT (X)k—ݼ êgθ (t),K(3.1)ªŒ • ϕ(t)gθ (t)dt = 0, ˜ƒ θ ∈ Θ. (3.3) È © ϕ(t)gθ (t)dt = 0/ ª þ Œ w ¤/ϕ†gθ 0. u ´, ^ ‡(3.1)Œ ` ¤ ´/ϕ† ¼ ê X{gθ , θ ∈ Θ} 0. 3 ¼êØ, eM L«˜ ¼êX, …Ø•3†M š"¼ê, K¡M • X. d(3.3) wÑ, ·‚ùp 5 Іdƒ . ØL·‚Ø¡—ݼê X{gθ , θ ∈ Θ} , ¡ÚOþT . du{gθ , θ ∈ Θ}´dÚOþT û½ , ù«¡ ØK•¢ Ÿ. n ~1. X = (X1 , · · · , Xn )•loNb(1, θ)¥Ä {ü , KT (X) = i=1 Xi ´ ÚO þ. Proof. w,, T (X) ∼ b(n, θ), k n k P (T (X) = k) = θ (1 − θ)n−k , k = 0, 1, 2, · · · , n. k ϕ(t)•?˜¢¼ê, ÷vEθ ϕ(T ) = 0, ˜ƒ0 < θ < 1,d= n n k ϕ(k) θ (1 − θ)n−k = 0 ⇐⇒ k k=0 6 

7 . n k n θ ϕ(k) = 0, 0 < θ < 1. k 1−θ k=0 -θ/(1 − θ) = δ, Kþª du ∞ n ϕ(k) δ k = 0, 0 < δ < ∞. k k=0 þª†>´δ õ‘ª, 7k n ϕ(k) = 0, k = 0, 1, 2, · · · , n. k n =ϕ(k) = 0, k = 0, 1, 2, · · · , n. ùÒy² T (X) = i=1 Xi ´ ÚOþ. ~2. X = (X1 , X2 , · · · , Xn )•l oNN (θ, 1)¥Ä {ü ¯ , KT (X) = X• Ú Oþ. ¯ = 1 n Proof. w,T (X) = X n i=1 Xi ∼ N (θ, 1/n), ϕ(t)•t ?˜¢¼ê, ÷vEθ ϕ(T ) = 0, 阃−∞ < θ < ∞. d= ∞ ∞ n n(t−θ)2 n nt2 nθ 2 ϕ(t)e− 2 dy = ϕ(t)e− 2 · e− 2 · entθ dt = 0. 2π −∞ 2π −∞ ¤± ∞ nt2 ϕ(t)e− 2 · entθ dt = 0, −∞ < θ < ∞ −∞ -z = nθ,K ∞ nt2 G(z) = ϕ(t)e− 2 etz dt. −∞ òzÀ•Eê, G(z)• ²¡þ )Û¼ê, …G(z) z ¢êž•0, d)Û¼ê •˜5½n, G(z)3 ‡E²¡þ•0, AO z = iµ,K ∞ nt2 G(µ) = ϕ(t)e− 2 · e−iµt dt = 0. −∞ dF ourierC† _C†úª, Œ• 2 ϕ(t)e−nt /2 = 0. ¯ kϕ(t) = 0, |t| < ∞, ÏdT (X) = X• ÚOþ. !•êx¥ÚOþ 5 ½ n 4. X = (X1 , X2 , · · · , Xn ) VǼê k f (x, θ) = C(θ)exp Qi (θ)Ti (x) h(x), θ∈Θ i=1 ••êx. -T (X) = (T1 (X), · · · , Tk (X)), eg,ëê˜mΘ∗ Š•Rk f8kS:, KT (X)´ ÚOþ. 7 

8 .~3. X = (X1 , · · · , Xn )´lþ!©ÙU (θ − 1/2, θ + 1/2)¥Ä {ü , KT (X) = (X(1) , X(n) )´¿©ÚOþ, Ø´ ÚOþ. Proof. T (X) = (X(1) , X(n) ) ¿©53~2.7.9¥®y. e¡5y²§Ø´ . ‡ y ² ˜ ‡ Ú O þT (X)Ø ´ , •‡é ˜ ‡ ¢ ¼ êϕ(t)¦ Eθ ϕ(T ) = 0, /ϕ(T ) = 0, a.e. Pθ 0´Ø¤á =Œ. -Z = X(n) − X(1) , Yi = Xi − (θ − 1/2), i = 1, 2, · · · , n,KY1 , · · · , Yn i.i.d. ∼ U (0, 1),†θà '. džZ = X(n) − X(1) = Y(n) − Y(1) ©Ù•†θÃ'. é~êa < b¦ P (Z < a) = P (Z > b) > 0. ½Â     1, Z < a, ϕ(t) = −1, Z > b,  0, Ù§.   K´„ϕ(t)÷v:Eθ ϕ(T ) = 0, ϕ(t) ≡ 0 (=P (ϕ(t) = 0) > 0). U½ÂT (X) = (X(1) , X(n) )Ø´ ÚOþ. n!k. ÚOþ9Ù5Ÿ ½  2. eé?Û÷v Eθ ϕ(T (X)) = 0, 阃 θ ∈ Θ k.(½a.e.k.) ¼êϕ(·) Ñk Pθ ϕ(T (X) = 0) = 1, 阃 θ ∈ Θ, K¡T (X)•k. ÚOþ. d½ÂŒ„: ˜‡/ ÚOþ07•/k. ÚOþ0, ‡ƒØ7é. ½n 5. ( Basu½n F = {Fθ (x), θ ∈ Θ}•˜©Ùx, Θ´ëê˜m. X = (X1 , · · · , Xn )´ l©ÙxF ¥Ä {ü , T (X)´˜k. ÚOþ,…´¿©ÚOþ. er.v. V (X) ©Ù†θÃ', Ké?Ûθ ∈ Θ, V (X)†T (X)Õá. ~4. X = (X1 , · · · , Xn )´ lN (θ, 1)¥ Ä {ü , R(X) = X(n) − X(1) ¡ • 4 , ¯ = 1 n Xi †R(X) Õá. KT (X) = X n i=1 Proof. du ©ÙN (θ, 1)••êx, g,ëê˜mΘ∗ = {θ : −∞ < θ < ∞}Š•R1 f8k S:. T (X)•¿© ÚOþ. -Yi = Xi − θ,KYi ∼ N (0, 1), i = 1, 2, · · · , n. ÏdY1 , · · · , Yn i.i.d. ∼ N (0, 1),†θÃ'.l Y(n) − Y(1) ©Ù•†θÃ'. R(X) = X(n) − X(1) = Y(n) − Y(1) ƒ©Ù†θ Ã', díØ2.8.1Œ•T (X)†R(X)Õá. 8