11 统计学基础--假设检验（三）

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王木木

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主要介绍了一致最优检验与无偏检验，其中具体介绍了Neyman-Pearson引理、利用NP引理求UMP检验，还介绍了似然比检验的定义、若干例子以及似然比的渐近分布，最后简单介绍了Wald检验和Score检验。

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1 . Lec11: b u (n) Ü•² 2011 c 4 28 F 1 ˜—•`u †Ã u ˜!Úó9½Â k X, § Šu ˜m X , X ©Ùáu©Ùx{Fθ , θ ∈ Θ}, Θ•ëê˜m. X§5.1¤ã, ëêθ b u ¯KŒ±L¤Xe ˜„/ª H0 : θ ∈ Θ0 ←→ H1 : θ ∈ Θ1 , (1.1) d?Θ0 •ëê˜mΘ š˜ýf8, Θ1 = Θ − Θ0 . éu ¯K(1.1)Œ^««ØÓ•{ u , ùÒ )ØÓu ' ¯K, ±93««¿Â e¦/•`0u ¯K. ù†·‚31nÙëê O¯K¥, 3Ã O¥é˜—• • Ã O ¯K ƒq. e¡Äk‰Ñ˜—•`u ½Â. ½ Â5.4.1 ku ¯K(1.1),-0 < α < 1, PΦα •(1.1) ˜ƒY²•α u 8Ü. eϕ ∈ Φα , …é?Ûu ϕ1 ∈ Φα k βϕ (θ) ≥ βϕ1 (θ), ∀ θ ∈ Θ1 (1.2) K¡ϕ•(1.1) ˜‡Y²•α ˜—•`u (Uniformly Most Powerful Test), {¡Y² •α UMPu . ϕ•Y²α UMPu ž, §3•›1˜a†ØVÇØ‡Lα ^‡e, o ¦1 a†ØVÇˆ • . Ïde±†ØVÇ•ïþu ` •˜þÝ, … É•›1˜ a†ØVÇ K, KUMPu ´•Ð u . ØL, UMPu •3˜„´~ Ø~„ . ndXe: eΘ1 ØŽ•¹˜‡:, K 3Ù¥ ü‡ØÓ:θ1 Úθ2 ž, •¦βϕ (θ1 ) ¦ŒUŒ @ «u ϕ, Ø„ Óž•U¦βϕ (θ2 )Œ. •k3Θ0 ÚΘ1 Ñ••¹˜:ž, UMPu â•3. ùÒ ´e¡Neyman-Pearson({¡NP)Ún SN. !Neyman–PearsonÚn ½ n 5.4.1 (NPÄ Ún) X ©ÙkVÇ¼êf (x, θ), ëêθ•kü‡ŒU Šθ0 Úθ1 , •Äe u ¯K H0 : θ = θ0 ←→ H1 : θ = θ1 , (1.3) Ké?‰ 0 < α < 1k

2 . (i) •35. éu ¯K(1.7)7•3˜‡u ¼êϕ(x) 9šK~êcÚ0 ≤ r ≤ 1, ÷v^‡ (a)  1,   f (x, θ1 )/f (x, θ0 ) > c  ϕ(x) = r, f (x, θ1 )/f (x, θ0 ) = c (1.4)    0, f (x, θ1 )/f (x, θ0 ) < c (b) Eθ0 [ϕ(X)] = α (1.5) (ii) ?Û÷v(1.4)Ú(1.5) u ϕ(x)´u ¯K(1.7) UMPu . 55.4.1 1. r.v. X•ëY.©Ùž(1.4)ª¥ ‘Åz´Ø7‡ . ùž r = 0, =(1.4)ªC• 1, f (x, θ1 )/f (x, θ0 ) > c ϕ(x) = 0, f (x, θ1 )/f (x, θ0 ) ≤ c Ù¥cdEθ0 ϕ(X) = P (f (X, θ1 )/f (X, θ0 ) > c |H0 ) = α5(½. 2. l/q,50 *: wNPÄ Ún´é˜Ù : éz‡ X, θ0 Úθ1 /q,Ý0 ©O•f (x, θ0 )Úf (x, θ1 ). 'Šf (x, θ1 )/f (x, θ0 )•Œ, Ò‡N3 Xž, θ•”θ1 šθ0 , ù XÒ•–•uÄ½/H0 : θ = θ0 0 b . y (i)ky²•35: P‘ÅCþf (X, θ1 )/f (X, θ0 ) ©Ù¼ê• G(y) = P (f (X, θ1 )/f (X, θ0 ) ≤ y), −∞ < y < ∞ KG(y)äk©Ù¼ê 5Ÿ: üN! šü! mëY… lim G(y) = 0 , lim G(y) = 1. l y→−∞ y→∞ d0 < α < 1ÚG(y) üN5Œ•:7•3c¦ G(c − 0) ≤ 1 − α ≤ G(c). XÛ(½r, ©e n«œ/?Ø: (a) eG(c) = 1 − α, K r = 0, ùžd(1.4)(½ ϕ(x)÷v Eθ0 [ϕ(X)] = Pθ0 f (X, θ1 ) f (X, θ0 ) > c = 1 − Pθ0 f (X, θ1 ) f (X, θ0 ) ≤ c = 1 − G(c) = α. (b) eG(c − 0) = 1 − α, K r = 1, džd(1.4)½Â ϕ(x)÷v Eθ0 [ϕ(X)] = 1 − P f (X, θ1 ) f (X, θ0 ) < c = 1 − G(c − 0) = α. 2

3 . (c) eG(c − 0) < 1 − α < G(c), K r = [(1 − α) − G(c − 0)] [G(c) − G(c − 0)], w,, dž éd(1.4)½Â ϕ(x)k Eθ0 [ϕ(X)] = P f (X, θ1 ) f (X, θ0 ) > c + r · P f (X, θ1 ) f (X, θ0 ) = c (1 − α) − G(c − 0) = 1 − G(c − 0) − (G(c) − G(c − 0)) + · (G(c) − G(c − 0)) G(c) − G(c − 0) = 1 − (1 − α) = α. •35y.. (ii)2yd(1.4)Ú(1.5)½Â ϕ(x)kUMP5Ÿ. ϕ1 (x) •u ¯K(1.7) ?˜Y²•α u , ·‚‡y²Eθ1 [ϕ(X)] ≥ Eθ1 [ϕ1 (X)]. •d½Â ˜m X þ f8: S + = {x : ϕ(x) > ϕ1 (x)}, S − = {x : ϕ(x) < ϕ1 (x)}. K3S + þk: ϕ(x) > ϕ1 (x) ≥ 0, d(1.4)Œ•dž f (x, θ1 ) ≥ c; f (x, θ0 ) x ∈ S − žk: ϕ(x) < ϕ1 (x) ≤ 1, Ïdkϕ(x) < 1, d(1.7)Œ•dž7k f (x, θ1 ) ≤ c. f (x, θ0 ) 3S = S + ∪ S − þ7k (ϕ(x) − ϕ1 (x))(f (x, θ1 ) − cf (x, θ0 )) ≥ 0 (Ï•3S + þ, üÏf , 3S − þüÏf K) . Ïd ϕ(x) − ϕ1 (x) f (x, θ1 ) − cf (x, θ0 ) dx X = ϕ(x) − ϕ1 (x) f (x, θ1 ) − cf (x, θ0 ) dx ≥ 0, S + ∪S − = ϕ(x)f (x, θ1 ) dx − ϕ1 (x)f (x, θ1 )dx X X ≥c ϕ(x)f (x, θ0 )dx − ϕ1 (x)f (x, θ0 )dx . (1.6) X X d(1.5)•Eθ0 [ϕ(X)] = X ϕ(x)f (x, θ0 )dx = α, ϕ1 (x)•u ¯K(1.7) Y²•α ?˜ u , kEθ1 [ϕ1 (X)] ≤ α, •(1.6)ªm>šK, l †>•šK, Ïdk βϕ (θ1 ) = ϕ(x)f (x, θ1 )dx ≥ ϕ1 (x)f (x, θ1 )dx = βϕ1 (θ1 ). X X ùÒy² ϕ(x)•(1.7) Y²•α UMPu . ½ny.. ~5.4.1 X = (X1 , · · · , Xn )•g oNN (µ, 1)¥Ä ‘Å , Ù¥µ™•, ¦b u ¯K H0 : µ = 0 ←→ H1 : µ = µ1 (µ1 > 0) 3

4 . Y²•α UMPu . d?µ1 Úα‰½. ) dNPÚn, k¦f0 (x)Úf1 (x) Lˆª: n 1 f0 (x) = (2π)−n/2 exp − x2i , 2 i=1 n 1 f1 (x) = (2π)−n/2 exp − (xi − µ1 )2 . 2 i=1 q,'ŒL«• f1 (x) 1 λ(x) = = exp − nµ21 + nµ1 x ¯ f0 (x) 2 w, µ1 > 0ž, λ(x)•¯ x O¼ê, UMPu Ä½•• ¯ > A} D = {X : λ(X) > c)} = {X : X ¯ ∼ N (0, 1 ), H0 ¤áž, X dNPÚnŒ•: n √ √ ¯ > A|H0 ) = P ( nX P (X ¯ > nA|H0 ) = α, √ ¯ √ √ Ù¥ nX ∼ N (0, 1). d nA = uα =⇒ A = uα / n, =u Y²•α UMPu u ¼ê• √ 1, x ¯ > uα n, ϕ(x) = √ 0, ¯ ≤ uα x n. Œ„ϕ(x)†µ1 Ã', Œ„þãu ¼êϕ(x)•´u ¯K H0 : µ = 0 ←→ H1 : µ > 0 Y²•α UMPu . 55.4.2 d~wŠ·‚: 3, œ¹e, XJdNPÚn UMPu Ø•6uéáb äNŠ, KŒdd ˜‡*Œ , éáb •EÜb u ¯K Y²•α UMPu . aq ~Œ±¦ u ¯KH0 : µ = 0 ←→ H1 : µ < 0 u Y²•α UMPu . ~5.4.2 X = (X1 , · · · , Xn )•lü:©Ùb(1, p)¥Ä ‘Å , Ù¥p•™•ëê. ¦u ¯K H0 : p = p0 ←→ H1 : p = p1 (p1 > P0 ) Y²•α UMPu . d?p0 , p1 Úα‰½. ) dNPÚn, k¦f0 Úf1 Lˆª: n n xi f0 (x) = p(x, p0 ) = p0 i=1 (1 − p0 )n− i=1 xi n n xi f1 (x) = p(x, p1 ) = p1 i=1 (1 − p1 )n− i=1 xi n PT (x) = i=1 xi , q,' n T (x) p(x, p1 ) 1 − p1 p1 (1 − p0 ) λ(x) = = . p(x, p0 ) 1 − p0 p0 (1 − p1 ) 4

5 .d up1 > p0 , 1 − p0 > 1 − p1 ,Ï dp1 (1 − p0 )/p0 (1 − p1 ) > 1, λ(x)' uT (x)ü N O. d ur.v. T (X)ÑllÑ.©Ù, I‡‘Åz. dNPÚnŒ•u ¼ê•     1, T (x) > c ϕ(x) = r, T (x) = c    c, T (x) < c n H0 ¤ážT (X) = i=1 Xi Ñl ‘©Ùb(n, p0 ), α‰½ž, cde Ø ª(½: n n n k n k p (1 − p0 )n−k > α > p (1 − p0 )n−k = α1 . k 0 k 0 k=c k=c+1 α − α1 r= n , c pc0 (1 − p0 )n−c K7k Ep0 [ϕ(X)] = Pp0 (T (X) > c) + r · Pp0 (T (X) = c) = α. Ïdϕ(X)•Y²•α UMPu . duþãu ¼êϕ(x)†p1 Ã', §•´u ¯K H0 : p = p0 ←→ H1 : p > p0 Y²•α UMPu . n 55.4.3 'u‘Åzu ¯K. ~¥ ÑyT (x) = i=1 xi = cž, k‰˜‡äk¤õ Ç•r BenoulliÁ . eTÁ ¤õ, KÄ½H0 ; eØ,K ÉH0 . Xr = 12 KŒÏL•˜þ! M1, 5½Ñy ¡•¤õ. e•Ñ ¡KÄ½H0 ; eØ,K ÉH0 . X·‚3§5.1¥¤ã, é‘Åzu ©üÚr: (i)ÄkÏLÁ ¼ * , (ii)k n ¯ u , ÑyAÏŠ(X ~¥ i=1 xi = c) I‘Åzž2Š˜gÁ . Á (J•A½A, )VÇ•P (A) = r, eAu), KáýH0 ; ÄK ÉH0 . ~5.4.3 X = (X1 , · · · , Xn )´5gþ!©ÙU (0, θ), θ > 0 ‘Å , ¦e u ¯ K H0 : θ = θ0 ←→ H1 : θ = θ1 (θ1 > θ0 > 0) Y²•α UMPu . ) Ñlþ!©Ù X —Ý¼êÚq,'©O• 1 f (x, θ) = I[0<x(1) ≤x(n) <θ] , θn n  f (x, θ1 )  θθ10 , 0 < x(n) < θ0 λ(x) = = f (x, θ0 )  ∞, θ0 < x(n) < θ1 . dNPÚn, Œ•Y²•α UMPu ¼êk/ª 1, x(n) > c; ϕ(x) = 0, x(n) ≤ c. 5

6 .T = X(n) —Ý¼ê• ntn−1 gθ (t) = I[0<t<θ] θn ntn−1 H0 ¤áž, T (X) —Ý¼ê•gθ0 (t) = θ0n I[0<t<θ0 ] , Ïdk θ0 ntn−1 cn Eθ0 [ϕ(X)] = ϕ(t)gθ0 (t)dt = n dt = 1 − n = α 0 c θ0 θ0 √ c = θ0 n 1 − α, Ïd √ n 1, X(n) > θ0 1 − α; ϕ(X) = √ n 0, X(n) ≤ θ0 1 − α. •˜‡Y²•α UMPu . dudu ϕ(X)†θ1 Ã', §•´ H0 : θ = θ0 ←→ H1 : θ > θ0 Y²•α UMPu . 55.4.4 dþ¡n‡~fŒ„UMPu ¼êϕ(x) •¿©ÚOþ ¼ê, ù´ÄäkÊ H¿ÂQ?·‚ke (Ø: r.v. X —Ý¼ê•f (x, θ), θ ∈ Θ •™•ëê, X = (X1 , · · · , Xn )•goNX¥Ä ‘Å , T = T (X)•θ ¿©ÚOþ, Kd(1.4)Ú(1.5) ½Â u ¼êϕ(x)´¿©ÚOþT ¼ê. ù˜(J y²¿ØJ, •‡|^¿©ÚOþ Ïf©)½n=Œy . ½ n 5.4.1’ (NPÄ Ún _) X ©ÙkVÇ¼êf (x, θ), ëêθ•kü‡ŒU Šθ0 Úθ1 , •Äe u ¯K H0 : θ = θ0 ←→ H1 : θ = θ1 , (1.7) Ké?‰ 0 < α < 1, b H0 ↔ H1 •3˜‡Y²α UMPu ϕ(x), K (a)7•3˜‡šK~êc¦  1, f (x, θ1 )/f (x, θ0 ) > c ϕ(x) = (1.8) 0, f (x, θ1 )/f (x, θ0 ) < c (b) e?˜Ú„kEθ1 [ϕ(X)] = ϕ(X)f (x, θ1 )dx < 1, K7k Eθ0 [ϕ(X)] = α (1.9) y² dNPÚn••3˜‡Y²α UMPu ϕ÷v ˜  1, f (x, θ1 )/f (x, θ0 ) > c ϕ(x) ˜ = (1.10) 0, f (x, θ )/f (x, θ ) < c 1 0 6

7 . PS + = {x : ϕ(x) ˜ > ϕ(x)} ,S − = {x : ϕ(x) ˜ < ϕ(x)}± 9S = (S + S − ) {x : f (x, θ1 )/f (x, θ0 ) = c}, K3Sþk (ϕ(x) ˜ − ϕ(x))(f (x, θ1 ) − cf (x, θ0 )) > 0 l XJP (S) > 0, òk (ϕ(x) ˜ − ϕ(x))(f (x, θ1 ) − cf (x, θ0 ))dx = (ϕ(x) ˜ − ϕ(x))(f (x, θ1 ) − cf (x, θ0 ))dx > 0 χ S Ïd (ϕ(x) ˜ − ϕ(x))f (x, θ1 ) > c[α − Eθ0 ϕ(x) ≥ 0. χ d=βϕ˜ (θ1 ) > βϕ (θ1 ), ù†ϕ•Y²α UMPgñ. ÏdkP (S) = 0, =ϕÚϕ3{x ˜ : f (x, θ1 )/f (x, θ0 ) = c}þ±VÇ1ƒ . u´(a) y. é(b), XJEθ0 ϕ(X) < α,K- φ(x) = min{1, ϕ(x) + α − Eθ0 ϕ(X)} KEθ0 φ(X) ≤ α, =φ•Y²αu . ,˜•¡é¤kx, kφ(x) ≥ ϕ(x) … ª¤á …= ϕ(x) = 1. duEθ1 ϕ(X) < 1, Pθ1 (ϕ(X) = 1) < 1, ùò ÑEθ1 ϕ(X) < Eθ1 φ(X)†ϕ•Y ˜ ²αUMPgñ, Ïd7kEθ0 ϕ(X) = α. ~5.4.4 X = (X1 , · · · , Xn )•g oNN (µ, 1)¥Ä ‘Å , Ù¥µ™•, y² b u ¯K H0 : µ = 0 ←→ H1 : µ = 0 Ø•3Y²•α UMPu . n!|^NPÚn¦UMPu NPÚn Š^Ì‡Ø3u¦–u ¯K(1.2)@ UMPu , Ï•¢SA^¥–(1.2) @ u ¯K´Ø~„ . ˜„œ/´"b Úéáb Ñ´EÜ œ/. NPÚn Ì‡ Š^´3u§´¦•E,œ/eUMPu óä. 3c¡ ~5.4.1! ~5.4.2Ú~5.4.3ùn‡ ~f¥®²òu ¯Kí2 éáb ´EÜ œ/. •˜„ b u ¯KX(1.1)¤«, =H0 : θ ∈ Θ0 ←→ H1 : θ ∈ Θ1 , Ù¥Θ0 ÚΘ1 •EÜœ/(=Ù¥•¹ëê˜mΘ ¥ :Ø Ž˜‡) . Ïéùau ¯K UMPu ˜„Ž{´: 3Θ0 ¥]˜‡θ0 ¦ŒU†Θ1 C, 2 3Θ1 ¥]˜‡θ1 , ^NPÚn‰ÑX(1.4)Ú(1.5) UMPu ϕθ1 . ˜„ θ1 3Θ1 ¥CÄž, ϕθ1 Ø ‘θ1 Cz Cz, =ØØθ1 3Θ1 ¥XÛCz, ϕθ1 = ϕ†θ1 Ã', Kϕ•´H0 : θ = θ0 ←→ H1 : θ ∈ Θ1 UMPu . Ïd, •?˜ÚeUy²: du é?Ûθ ∈ Θ0 ku Y²α, Kϕ• ´H0 : θ ∈ Θ0 ←→ H1 : θ ∈ Θ1 Y²•α UMPu . d{‡1 Ï•ØN´. •k3ëê˜m•˜‘î¼˜mR1 ½Ù˜«m, u b ´ ü> , =•H0 : θ ≤ θ0 ←→ H1 : θ > θ0 ½öH0 : θ ≥ θ0 ←→ H1 : θ < θ0 ž, …é ©Ù k˜½‡¦ž, þã•{âŒ1. AO ©ÙäküNq,'5Ÿž, þãüaü>u UMPu ´•3 . e¡Ò5?Øƒ. 7

8 . ½ Â (MLR) ¡‘ÅCþX ©Ùf (x, θ)äküNq,'5Ÿ, XJ•3˜‡¢Š¼ êT (x), ¦ é?¿ θ < θ , (1) ©Ùf (x, θ)†f (x, θ )´ØƒÓ ; (2) 'Šf (x, θ )/f (x, θ)•T (x) šü¼ê. ©ÙäkMLR5Ÿž,éXeü>u ¯K H0 : θ ≤ θ0 ←→ H1 : θ > θ0 (1.11) ke (Ø: ½n 5.4.2 X = (X1 , · · · , Xn ) ©ÙäkMLR5Ÿ, ëê˜mΘ•R1 = (−∞, +∞) ˜k•½Ã•«m, θ0 •Θ ˜‡S:, Ku ¯K(1.11) Y²•α UMPu •3(0 < α < 1), …k/ª   1, T (x) > c   ϕα (x) = r, T (x) = c (1.12)    0, T (x) < c Ù¥cÚr (0 ≤ r ≤ 1)÷v^‡: Pθ0 (T (X) > c) + r · Pθ0 (T (X) = c) = α (1.13) y ? θ1 > θ0 , Äk•Äu ¯K Ho : θ = θ0 ←→ H1 : θ = θ1 (1.14) kq,' f (x, θ1 ) λ(x) = . f (x, θ0 ) dMLR5Ÿ,λ(x)•T (x) šü¼êÏddNPÚnŒ•u ¯K(1.14) UMPu ¼ê•      1, λ(x) > c  1,   T (x) > c ϕ(x) = r, λ(x) = c ⇐⇒ ϕ(x) = r, T (x) = c      0, λ(x) < c  0, T (x) < c, Ù¥~êcÚr÷veª Eθ0 [ϕ(X)] = Pθ0 (T (X) > c) + rPθ0 (T (X) = c) = α ducÚr†θ1 Ã', d(1.12)Ú(1.13)(½ u ¼êϕ(x) •´eãu ¯K H0 : θ = θ0 ←→ H1 : θ > θ0 Y²•α UMPu . ·‚•‡y²ϕ(x)Š•u ¯K(1.11) u , äkY²α, =Œ ¤y². •d·‚•Iy ²ϕ(x) õ ¼êβϕ (θ)´θ üNO¼ê=Œ. e¡·‚5y²ù˜¯¢. 8

9 . ? θ < θ , PA = {x : f (x, θ ) < f (x, θ )} ,B = {x : f (x, θ ) > f (x, θ )} ±9sup ϕ(x) = A a, inf ϕ(x) = b, 5¿ ϕ(x)•T (x) šü¼ê, u´b − a ≥ 0. ¤± B βϕ (θ ) − βϕ (θ ) = ϕ(x)[f (x, θ ) − f (x, θ )]dx X ≥ a [f (x, θ ) − f (x, θ )]dx + b [f (x, θ ) − f (x, θ )]dx A B = (b − a) [f (x, θ ) − f (x, θ )]dx ≥ 0, B =βϕ (θ ) > βϕ (θ ), é?‰ θ > θ ¤á, ùÒy² βϕ (θ)•θ üNO¼ê, k sup βϕ (θ) ≤ βϕ (θ0 ) = Eθ0 [ϕ(X)] = α θ≤θ0 Ïdd(1.12)Ú(1.13)(½ ϕ(X)•u ¯K(1.11) Y²•α UMPu . ½ny.. 55.4.5 1. 3½n5.4.2¥e ©Ù´ëY.©Ù, KUMPu ØI‡‘Åz. u ¯K(1.11) Y²•α UMPu , ÏL(1.12)Ú(1.13)¥-r = 0¼ . 2. AO ©Ù•Xe•êxž, f (x, θ) = c(θ) exp{Q(θ)T (x)}h(x), (1.15) Ù¥c(θ) > 0ÚQ(θ)•θ î‚O¼ê, T (x)Úh(x) ´ x ¼ê. Kf (x, θ)äküNq ,'5Ÿ, u´½n5.4.2¤á. eQ(θ)•θ î‚ü¼ê, Ù{ØC, Ku ¯K(1.11) Y ²•α UMPu , I‡ÏLò(1.12)Ú(1.13)ª¥ Ø Ò‡•( ÒØC) , =Œ . •Ä†(1.11)ƒ‡ ü>u ¯K H0 : θ ≥ θ0 ←→ H1 : θ < θ0 (1.16) 'uù˜u ¯K Y²•α UMPu ke ½n. ½ n 5.4.3 e½n5.4.2 ^‡¤á, Ku ¯K(1.16) Y²•α UMPu •3, …k /ª   1, T (x) < c;   ϕ(x) = r, T (x) = c; (1.17)    0, T (x) > c; Ù¥cÚr (0 ≤ r ≤ 1)÷v^‡: Pθ0 (T (X) < c) + r · Pθ0 (T (X) = c) = α, (1.18) d½n y²•{†½n5.4.2aq, lÑ. ~ 5.4.4 ¯K†~5.4.1ƒÓ, = X = (X1 , · · · , Xn )•l oNN (θ, 1)¥Ä {ü §¦u ¯KH0 : θ ≤ θ0 ←→ H1 : θ > θ0 UMPu , d?θ0 Úu Y²α‰½. 9

10 . ) ©Ù••êx©Ù, —Ý• n n f (x1 , · · · , θ) = (2π)− 2 exp{−nθ2 /2} exp{nθ¯ x} exp − x2i /2 , i=1 = c(θ)exp{Q(θ)T (X)}h(x) n ¯ Q(θ) = nθ•θ d?c(θ) = (2π)n/2 exp{−nθ2 /2}, h(x) = exp{− i=1 x2i /2}, T (X) = X, î ‚ O ¼ ê, d ½ n5.4.2 (d u © Ù • ë Y © Ù, u ¼ ê Ø I ‡ ‘ Å z) Œ • Y ² •α UMPu deª‰Ñ:  1, T (x) > c; ϕ(x) = . 0, T (x) ≤ c; ¯ ∼ N (θ, 1/n), √ ¯ − θ) ∼ N (0, 1), - duT (X) = X n(X √ √ ¯ − θ0 ) > n(c − θ0 )), α = Eθ0 [ϕ(X)] = Pθ0 (T (X) > c) = Pθ0 ( n(X √ Œ• n(c − θ0 ) = uα , =c = θ0 + √1 uα . •Y²•α UMPu • n √  1, T (x) > θ0 + uα / n; ϕ(x) = √ . 0, T (x) ≤ θ0 + uα / n; AO θ0 = 0Ò†~5.4.1¥ u (JƒÓ. ~ 5.4.5 l˜Œ1 ¬¥Ä n‡u Ù(J, X = (X1 , · · · , Xn ), Ù¥Xi = 1,e 1i‡ ¬•¢¬,ÄK•0, i = 1, · · · , n.¦ H0 : p ≤ p0 ←→ H1 : p > p0 Y²•α UMPu . Ù¥p0 Úα‰½. n ) -T (X) = Xi •n‡ ¬¥ ¢¬ê, KT ∼ ‘©Ù b(n, p). ‘©Ù••êx, i=1 ÙVÇ©Ù• n t n p f (t, p) = p (1 − p)n−t = (1 − p)n exp log T (x) t x 1−p n n p Ù¥c(p) = (1 − p)n , T (x) = Xi , h(x) = x , Q(p) = log 1−p •p î‚üNO¼ê, d½ i=1 n5.4.2Œ•   1, T (x) > c;   ϕ(x) = r, T (x) = c;    0, T (x) < c; Ù¥cke Ø ªû½ n n n i n i α1 = p (1 − p0 )n−i < α < p (1 − p0 )n−i , i=c+1 i 0 i=c i 0 r• α − α1 r= n , c pc0 (1 − p0 )n−c 10

11 .K7k Eϕ0 [ϕ(X)] = Pϕ0 (T (X) > c) + r · P (T (X) = c) = α, Ïdþãu ϕ(x)•Y²•α UMPu .ù´é~5.4.2 Ö¿" ~ 5.4.6 X = (X1 , · · · , Xn )•gPoissonoNP (λ)¥Ä ‘Å , λ > 0•™•ë ê. ¦ H0 : λ ≤ λ0 ←→ H1 : λ > λ0 Y²•α UMPu . Ù¥λ0 Úα‰½. ) Poisson©Ù••êx©Ù. X —Ý¼ê• n xi −nλ λ i=1 e e−nλ f (x, λ) = n = n exp (log λ)T (x) , i=1 xi ! i=1 xi ! n n d?c(λ) = e−nλ , T (x) = i=1 Xi , h(x) = 1 i=1 xi !, Q(λ) = log λ•λ î‚O¼ê, d½ n5.4.2Œ•   1, T (x) > c;   ϕ(x) = r, T (x) = c;    0, T (x) < c; n Ù¥cde Ø ª(½(5¿u ^‡þT (X) = Xi ∼ P (nλ)) : i=1 ∞ ∞ (nλ0 )k e−nλ0 (nλ0 )k e−nλ0 α1 = ≤α< . k! k! k=c+1 k=c r• (α − α0 )c! r= , (nλ0 )c e−nλ0 K7k Eλ0 [ϕ(X)] = Pλ0 (T (X) > c) + r · Pθ0 (T (X) = c) = α, þãu ϕ(x)•Y²•α UMPu . ~5.4.7 X = (X1 , · · · , Xn )•g•ê©ÙoNEP (λ)¥Ä ‘Å , λ > 0•™• ëê. ¦ H0 : λ ≤ λ0 ←→ H1 : λ > 0 Y•α UMPu , d?λ0 Úα‰½. ) •ê©Ùáu•êx. X —Ý¼ê• n f (x, λ) = λn exp −λ xi I[xi >0, i=1,2··· ,n] , i=1 n d?c(λ) = λn , h(x) = I[xi >0,i=1,2··· ,n] , T (x) = i=1 xi , Q(λ) = −λ•λ üNü¼ê, d½ n5.4.2 55.4.5Œ• 1, T (x) < c; ϕ(x) = . 0, T (x) ≥ c; 11

12 .dín2.4.5Œ•2nT (X) ∼ χ22n , k α = Pλ0 (T (X) < c) = Pλ0 (2λ0 T (X) < 2λ0 c), 1 Ïd2λ0 c = χ22n (1 − α), =c = 2 2λ0 χ2n (1 − α). Ïd 1 2 1, T (x) < 2λ0 χ2n (1 − α); ϕ(x) = 1 2 0, T (x) ≥ 2λ0 χ2n (1 − α); •Y•α UMPu . ~ 5.4.8 X = (X1 , · · · , Xn )•g oNN (0, σ 2 )¥Ä ‘Å , σ 2 •™•ëê. ¦ H0 : σ 2 ≥ σ02 ←→ H1 : σ 2 < σ02 Y•α UMPu , d?σ02 Úα‰½. ) ©ÙN (0, σ 2 )••êx, —Ý¼ê• n n n 1 1 f (x, σ 2 ) = f (xi , σ e ) = √ exp − x2i , i=1 2πσ 2σ 2 i=1 n n d?c(σ) = √1 , h(x) ≡ 1, T (x) = x2i , Q(σ 2 ) = − 2σ1 2 •σ 2 î‚üNO¼ê, d½ 2πσ i=1 n5.4.3Œ• 1, T (x) < c; ϕ(x) = 0, T (x) ≥ c; n du Xi2 /σ 2 ∼ χ2n , - i=1 n n 2 i=1 X1 c α = Eσ02 [ϕ(X)] = Pσ02 X12 < c = Pσ02 < , i=1 σ02 σ02 kc/σ02 = χ2n (1 − α), =c = σ02 χ2n (1 − α). Ïd 1, T (x) < σ02 χ2n (1 − α); ϕ(x) = 0, T (x) ≥ σ02 χ2n (1 − α); •Y²•α UMPu . ∗ o!Ã u c¡®²`L, UMPu •3´é ~ . ÏdŠ•˜—•`u OK, § Š^´k • . • ·^‰Œ•2 u OK, Œæ e •{: ké¤•Ä u –\,«Ün ˜„5 •›, ù Ò ¤•Ä u ‰Œ, , 3ù ‰Œ¥é˜—•`u . X3: O¯K¥, ·‚k•› Oþ7L´Ã ,, 3Ã Oa¥, Ïé• ˜— • Ã O. Äuù«Ž{Ú?Ã u ½Â. 12

13 . ½ Â 5.4.2 ϕ• u ¯ K(1.1) ˜‡u , βϕ (θ)• Ù õ ¼ ê. e é ? Ûθ1 ∈ Θ0 9θ2 ∈ Θ1 , okβϕ (θ1 ) ≤ βϕ (θ2 ), K¡ϕ´(1.1) ˜‡Ã u (Unbiased Test). eÃ u kY²α, K¡ϕ•Y²α Ã u . ù‡½Â ˜‡ d / ª Q ã X e: ϕ• u ¯ K(1.1) ˜‡u , eÙõ ¼ êβϕ (θ)÷v^‡: é∀ θ1 ∈ Θ0 kβϕ (θ) ≤ α, é∀ θ2 ∈ Θ1 kβϕ (θ) ≥ α, K¡ϕ•Y²•α Ã u , ½{•Ã u . Ã u †*¿Âé˜Ù: eϕ•H0 ←→ H1 Ã u , KÙ‹1˜a†Ø VÇØA ‡LØ‹1 a†Ø VÇ. e¡‰Ñ˜—•`Ã u ½Â. P Uα = {ϕ : ϕ •Y² α Ã u } =Uα •˜ƒY²•α Ã u a. ½Â 5.4.3 eϕ ∈ Uα , …é?Ûϕ1 ∈ Uα , k βϕ (θ) ≥ βϕ1 (θ), é˜ƒ θ ∈ Θ1 , K¡ϕ´(1.1) ˜‡Y²•α ˜—•`Ã u (Uniformly Most Powerful Unbiased Test,{ P•UMPUu ). 5 5.4.7 dþã½ÂŒ•?˜UMPu 7•UMPUu . `²Xe: PUMPu ϕ õ ∗ ¼ê•βϕ (θ), dUMPu ½ÂŒ•βϕ (θ) ≤ α, é˜ƒθ ∈ Θ0 , qw„ϕ ≡ α´Y²•α u , dUMPu ½ÂŒ•βϕ (θ) ≥ βϕ∗ (θ) ≡ α, é˜ƒθ ∈ Θ1 , Œ„k βϕ (θ2 ) ≥ α ≥ βϕ (θ1 ), é?‰ θ1 ∈ Θ0 , θ2 ∈ Θ1 , u ϕ´Ã , q´UMPu , Ïd7•UMPUu . UMPUu •3 œ¹'UMP‡2˜ . ée üëê•êx f (x, θ) = c(θ) exp{Q(θ)T (x)}h(x) ·‚3c¡ ½n5.4.2Ú½n5.4.3¥®y² e u ¯K Y²•α UMPu •3, Ï •´UMPU . (1) H0 : θ ≤ θ0 ←→ H1 : θ > θ0 , (2) H0 : θ ≥ θ0 ←→ H1 : θ < θ0 . „Œ?˜Úy²e üaüë•êx Y²•α UMPUu ´•3 : (3) H0 : θ = θ0 ←→ H1 : θ = θ0 , (4) H0 : θ1 ≤ θ ≤ θ2 ←→ H1 : θ < θ1 ½ θ > θ2 , Ù¥(3)Ú(4)üau ¯K UMPUu •35®‡Ñ Ö ‰Œ, k, ÓÆŒëwë• ©z[1]P359 . 13

14 .2 q,'u q,'u ´NeymenÚPearson31928cJÑ Eb u ˜„•{. §3b u ¥ / ,ƒ u4Œq, O3: O¥ / . §ŒÀ•F isher 4Œq, n3b u ¯K¥ Ny. dù«•{ EÑ5 u , ˜„`k' ûÐ 5Ÿ, cA!J Ø - ‡u Ñ´q,'u . ù‡•{ ˜‡-‡`:Ò´·^¡2. Ò´`, §é©Ùx /ªv kŸoAÏ ‡¦. ˜. q,'u ½Â k©Ùx{f (x, θ), θ ∈ Θ}, Θ•ëê˜m. -X = (X1 , · · · , Xn )•gþã©Ùx¥Ä {ü‘Å , f (x, θ)• VÇ¼ê.‡•Äu ¯K(??). 3k x òf (x, θ)À •θ ¼ê, ¡•q,¼ê. X1nÙ0 4Œq, Ož¤ã. ef (x, θ1 ) < f (x, θ2 ),K·‚@ •ýëê•θ2 /q,50 Ù•θ1 /q,50Œ. dub u 3/θ ∈ Θ0 †θ ∈ Θ1 ”ù ö¥ÀÙ˜, ·‚g,•Ä±eü‡þ LΘ0 (x) = sup f (x, θ), θ∈Θ0 LΘ1 (x) = sup f (x, θ). θ∈Θ1 •ÄÙ'ŠLΘ1 (x)/LΘ0 (x), ed'Š Œ, K`²ýëê3Θ1 S /q,50 Œ, Ï ·‚–•uÄ½b /θ ∈ Θ0 0. ‡ƒ, ed'Š , ·‚–•u Éb /θ ∈ Θ0 0. ePλ(x) = LΘ (x)/LΘ0 (x), Ù¥LΘ (x) = supθ∈Θ f (x, θ).duλ(x) †LΘ1 (x)/LΘ0 (x) ÓO ½Ó~, ·‚Œ^λ(x)“O'ŠLΘ1 (x)/LΘ0 (x) , ù ‰ Ð?´LΘ (x) = supθ∈Θ f (x, θ) OŽ 'LΘ1 (x)‡N´. Ïd Xe½Â. ½ Â 5.5.1 XkVÇ¼êf (x, θ), θ ∈ Θ, Θ0 •ëê˜mΘ ýf8, •Äu ¯K(??),KÚOþ λ(x) = sup f (x, θ) sup f (x, θ) (2.1) θ∈ Θ θ∈ Θ0 ¡•'uTu ¯K q,'. deã½Â u ¼ê   1, λ(x) > c;   ϕ(x) = r, λ(x) = c; (2.2)    0, λ(x) < c. Ù¥c, r (0 ≤ r ≤ 1)•–½~ê, ¡•u ¯K(??) ˜‡q,'u (Likelihood Ratio Test),k ©z¥•¡•2Âq,'u . X·‚3§5.4¥¤ã, e ©Ù•ëY.©Ùž, 3(2.2)¥-r = 0.= 1, λ(x) > c; ϕ(x) = 0, λ(x) ≤ c. 3(2.2)¥~êcÚr ÀJ´‡¦u äk‰½ Y²α. 14

15 . Šâþ¡¤`, éq,'u k±eÚ½: 1.¦q,¼êf (x, θ), ¿²(ëê˜mΘÚΘ0 ´Ÿo. 2.ŽÑLΘ (x) = sup f (x, θ)ÚLΘ0 (x) = sup f (x, θ). θ∈Θ θ∈Θ0 3.¦Ñλ(X)½†Ù d ÚOþ ©Ù. 4.(½cÚr¦(2.2)äk‰½ u Y²α. Ù¥, •'… ´1nÚ. ˜„λ(x) LˆªE,, ¦Ù©ÙØ´. eλ(x) = g(T (x)) •T (x) üNþ,(½eü) ¼ê, Ku (2.2)w, du   1, T (x) > c;   ϕ(x) = r, T (x) = c;    0, T (x) < c. Ïd“O¦λ(X) ©Ù, ·‚•‡¦ÑT (X) ©Ù=Œ(eλ(x) •T (x) üNeü¼ê,K òϕ(x)¥ Ø ª‡•). XJλ(X)©ÙÃ{¦ , Œ^Ù4•©ÙCq“O, ù3 !• ˜ã0 . !eZ~f ~ 5.5.1 X = (X1 , · · · , Xn )´l ©ÙxN (µ, σ 2 ), −∞ < µ < +∞, σ 2 > 0¥Ä ‘Å , ¦e u ¯K Y²•α q,'u H0 : µ = µ0 ←→ H1 : µ = µ0 . (2.3) ) Pθ = (µ, σ 2 ),q,¼ê• n n 1 f (x, θ) = (2πσ 2 )− 2 exp{− (xi − µ)2 }, (2.4) 2σ 2 i=1 3ùp, ëê˜m• Θ = {θ = (µ, σ 2 ) : −∞ < µ < +∞, σ 2 > 0}. "b H0 éA Θ f8• Θ0 = {θ = (µ, σ 2 ) : µ ≤ µ0 , σ 2 > 0}, 3Θþ, µÚσ 2 4Œq, O(MLE)• n ¯ 1 ¯ 2; µ ˆ = X, ˆ2 = σ (Xi − X) n i=1 3Θ0 þ, σ 2 MLE• n 1 ˆ2 = σ (Xi − µ0 )2 . n i=1 15

16 . k −n/2 n −n/2 2πe sup f (x, θ) = f (x, µ ˆ2) = ˆ, σ ¯ )2 (xi − x θ∈Θ n i=1 −n/2 n −n/2 2πe ˆ2) = sup f (x, θ) = f (x, µ0 , σ (xi − µ0 )2 . (2.5) θ∈Θ0 n i=1 l k n n −n/2 2 λ(x) = ¯)2 (xi − x (xi − µ0 ) i=1 i=1 √ 2 n/2 n x − µ0 | n|¯ 1 2 = 1+ n = 1+ T2 , i=1 (xi ¯)2 −x n−1 √ ¯ − µ0 )/ n ¯ 2 ), duλ(X)•|T | Ù¥T = n(X i=1 (Xi − X) O¼ê, Ïdq,'u Ä½• •D = X = (X1 , . . . , Xn ) : λ(X) > c = {X : |T | > c}.- P (|T | > c|H0 ) = α. |^e ¯¢: H0 ¤ážT ∼ tn−1 ,KŒ•c = tn−1 ( α2 ). Ïd 1, |T | ≥ tn−1 (α/2); ϕ(x) = 0, |T | < tn−1 (α/2). ´(2.3) ˜‡Y²•α q,'u ,Œ±y²§´(2.3)Y²•α UMPUu . ~5.5.2 ¯K†~5.5.1ƒÓ, ¦e u H0 : µ ≤ µ0 ←→ H1 : µ > µ0 (2.6) Y²•α q,'u . ) džq,¼êf (x, θ)ÚΘ†~5.5.1¥ƒÓ, Θ0 = {θ = (µ, σ 2 ) : µ ≤ µ0 , σ 2 > 0}, Ïd, LΘ (x) = sup f (x, µ, σ 2 )†~5.5.1 ¥ ƒÓ. ‡5¿ θ∈Θ n 1 LΘ0 (x) = sup (2πσ)−n exp − (xi − µ)2 } θ∈Θ0 2σ 2 i=1 n 1 x − µ)2 n(¯ = sup (2πσ)−n exp − 2 ¯)2 − (xi − x . θ∈Θ0 2σ i=1 2σ 2 x−µ)2 n(¯ Pg(µ) = exp − 2σ 2 . σ2 ½,µ ≤ x ¯ž§g(µ)'uµüNO. Ïd ¯ > µ0 ž,eH0 ¤á§g(µ)3µ = µ0 ?ˆ (i) x •Œ§ k n n min (xi − µ)2 = (xi − µ0 )2 . µ≤µ0 i=1 i=1 16

17 . (ii) ¯ ≤ µ0 ž, eH0 ¤á§g(µ)3µ = x x ¯?ˆ •Œ§ k n n min (xi − µ)2 = ¯ )2 (xi − x µ≤µ0 i=1 i=1 Ïdk   LΘ (x), ¯ ≤ µ0 ; x LΘ0 (x) = n −n/2  ( 2πe )−n/2 − µ0 )2 n i=1 (xi , x ¯ > µ0 . Ïdk   1, ¯ ≤ µ0 ; x λ(x) = n 2 n n i=1 (xi −µ0 ) 2 1 2 2  n (x i −¯ x ) 2 = 1+ n−1 T , x ¯ > µ0 . i=1 √ ¯ − µ0 )/ 1 n ¯ 2 , duλ(x)•T d?T = n(X n−1 i=1 (Xi − X) O¼ê, Ïdq,'u Ä½ •• D = {X = (X1 , · · · , Xn ) : λ(X) > c } = {X : T > c}. duu Y²α‰½, cdeª(½: P (T > c | µ = µ0 ) = α µ = µ0 ž, T ∼ tn−1 , •c = tn−1 (α). aq3§5.2¥¤ã, þãu õ ¼êβϕ(µ)´µ üNO¼ê, k βϕ(µ) ≤ βϕ(µ0 ) = α, µ ≤ µ0 . Ïd 1, T (x) > tn−1 (α); ϕ(x) = 0, T (x) ≤ tn−1 (α). •u ¯K(2.6) Y²•α q,'u . Œ±y²§´(2.6) Y²•α UMPUu . aq•{Œ¦ u ¯K: H0 : µ ≥ µ0 ←→ H1 : µ < µ0 (2.7) Y²•α q,'u , ù3‰Öö‰öS. ~5.5.3 X = (X1 , · · · , Xn )•g ©ÙoNN (µ, σ 2 )¥Ä ‘Å , •ÄXe u ¯K Y²•α q,'u H0 : σ 2 = σ02 ←→ H1 : σ 2 = σ02 , (2.8) d?σ02 Úα‰½. ) džq,¼êE•(2.4), ëê˜mΘX~5.5.1, Θ0 = θ = (µ, σ 2 ) : −∞ < µ < +∞, σ 2 = σ02 , (2.9) 17

18 .LΘ (x)•†~5.5.1ƒÓ, n 1 LΘ0 (x) = sup (2πσ02 )−n/2 exp − (xi − µ)2 µ 2σ02 i=1 n 1 = (2πσ02 )−n/2 exp − ¯)2 , (xi − x 2σ02 i=1 Ïdk −n/2 n −n/2 n e 1 1 λ(x) = ¯)2 (xi − x exp ¯)2 (xi − x n σ02 i=1 2σ02 i=1 1 n −n -ξ = σ02 i=1 (xi −x 2 ¯) , g(ξ) = ξ 2 ξ/2e , K3ξ > 0žg(ξ)'uξkü ,, … ξ → 0Úξ → ∞ž, g(ξ) 4• •+∞,Ù/GXã5.5.1 . Ïdq,'u É•• n ¯ = {X : λ(X) ≤ c} = 1 ¯ 2 ≤ k2 D X : k1 ≤ (Xi − X) σ02 i=1 1 n ¯ 2 ∼ χ2 , Ï k1 Úk2 •e •§| ) du3H0 ¤áž, σ2 i=1 (Xi − X) n−1 0 g(k1 ) = g(k2 ) ⇐⇒ P (k1 ≤ ξ ≤ k2 | H0 ) = 1 − α n/2 n/2 k1 e−k1 /2 = k2 e−k2 /2 (2.10) P (ξ < k1 |H0 ) + P (ξ > k2 |H0 ) = α. d•§|(2.10) ) Ä½•(½ q,'u Ø´UMPU , †ƒƒ Ø . Œ±y²: eò(2.10) 1˜‡•§¥ nU•n − 1 ) k1 Úk2 , KdÙ(½ q,'u ´UMPU . •§(2.10) )Ø´ , ˜„ P (ξ < k1 |H0 ) = α/2, P (ξ > k2 |H0 ) = α/2 k1 = χ2n−1 (1 − α/2), k2 = χ2n−1 (α/2) dk1 , k2 (½ Ä½•†·‚3§5.2¥^†*•{¦ ü‡ oN• u (J´˜— . Ïdu ¯K(2.8) Y²•α q,'u ´ 1 n 0, χ2n−1 (1 − α/2) < σ02 i=1 (xi ¯)2 < χ2n−1 (α/2); −x ϕ(x) = 1, Ù§. ù˜Y²•α u †UMPUu ƒ Øõ. ü>u H0 : σ 2 ≥ σ02 ←→ H1 : σ 2 < σ02 , Y²•α UMPu ®d~5.4.8‰Ñ. ù˜u ¯K9H0 : σ ≤ 2 σ02 ←→ H1 : σ > 2 σ02 Y²•α q,'u 3‰ÖöŠ•öS. 'uü oNþŠ Ú• ' q,'u •{†c¡ ~fƒÓ§•´Lˆ‡ E,˜ §®òÙ˜ SK¥§øÖööS" 18

19 . ~ 5.5.4 X = (X1 , · · · , Xn )•gþ!©ÙoNU (0, θ)¥Ä ‘Å ,¦ H0 : θ ≤ θ0 ←→ H1 : θ > θ0 (2.11) Y²•α q,'u . d?αÚθ0 ‰½. ) džq,¼ê• θ−n , x(n) ≤ θ; f (x, θ) = 0, x(n) > θ. ˜m Θ = (0, θ), Θ0 = (0, θ0 ).du max Xi •θ MLE, k 1≤i≤n LΘ (x) = sup f (x, θ) = (x(n) )−n θ∈Θ Ú LΘ (x), x(n) ≤ θ0 ; LΘ0 (x) = sup f (x, θ) = θ∈Θ0 0, x(n) > θ0 . Ïdk 1, x(n) ≤ θ0 ; λ(x) = ∞, x(n) > θ0 . u Ä½•• D = {X = (X1 , . . . , Xn ) : λ(X) ≥ 1}. ò8ÜG = {x = (x1 , . . . , xn ) : λ(x) = 1}©•üÜ©G1 = {x : c < x(n) ≤ θ0 }, G2 = G − G1 . Ïdu Ä½• d/ª• D = {X : X(n) > c} n−1 nt duT = X(n) —Ý¼ê•f (t) = θn I[0<t<θ] , d ∞ ntn−1 α = P (X(n) > c|θ = θ0 ) = dt c θ0n θ0 n ntn−1 c = dt = 1 − , c θ0n θ0 √ n )Ñc = θ0 1 − α, Ä½•• √ n D = {X : X(n) > θ0 1 − α}. u õ ¼ê• √ ∞ n ntn−1 βϕ (θ) = P (X(n) > θ0 1 − α) = √ dt θ0 n 1−α θn θ ntn−1 1 = √ n dt = n (θn − θ0n (1 − α)) θ0 n 1−α θ θ = 1 − (1 − α)(θ0 /θ)n . 19

20 .§´θ üNO¼ê, k βϕ (θ) ≤ βϕ (θ0 ), é˜ƒ θ ≤ θ0 , Ïd±D•Ä½• u Y²•α. Ïd √ n 1, X(n) > θ0 1 − α; ϕ(x) = √ n 0, X(n) ≤ θ0 1 − α. •u ¯K(2.11) Y²•α q,'u . ~ 5.5.5 X = (X1 , . . . , Xn ) g•ê©ÙoN§Ù—Ý¼ê• 1 f (x, θ) = exp{−(x − θ)/2}, x ≥ θ, −∞ < θ < ∞ 2 ¦u ¯K H0 : θ = θ0 ←→ H1 : θ = θ0 (2.12) Y²•α q,'u . d?αÚθ0 ‰½. ) džq,¼ê• n 1 1 f (x, θ) = n exp − xi − nθ . 2 2 i=1 ëê˜mÚH0 éA ëê˜m f8©O• Θ = {θ : −∞ < θ < ∞}, Θ0 = {θ : θ = θ0 } 3ΘÚΘ0 þq,¼ê •ŒŠ©O• n 1 1 LΘ (x) = exp − xi − nx(1) , 2n 2 i=1 n 1 1 LΘ0 (x) = n exp − xi − nθ0 , 2 2 i=1 d?x(1) = min{X1 , . . . , Xn }.q,'• λ(x) = LΘ (x)/LΘ0 (x) = exp n(x(1) − θ0 )/2 •x(1) üNO¼ê§ k 1, X(1) > c ϕ(x) = 0, X(n) < c. -T (X) = X(1) ,´•T —Ý¼ê• n g(t) = exp{−n(t − θ)/2}. 2 Ïdk ∞ n α = P (X(1) > c|H0 ) = exp{−n(t − θ0 )/2}dt = e−n(c−θ0 )/2 . c 2 ü> éê •§: −n(c − θ0 )/2 = log α. )•§ c = θ0 − n2 log α.Ïdu Ä½•• 2 D = X : X(1) > θ0 − log α . n 20

21 .n!q,' ìC©Ù 3 !1˜Ü©q,'u ½Â5.5.1¥, • ½Ñ(2.2)ª¥ cÚr, ÒI‡• q, 'λ(X))3"b ¤áž ©Ù. 3{ü ~f¥, X !1 Ü© A‡~f¥, q,' ° (©ÙŒ±¦ . 3Nõœ¹e, q,'kéõE, /G, Ù°(©ÙÃ{¦ . 1938c, S.S. Wilksy² : eX1 , . . . , Xn ´Õá‘Å ,K n → ∞ž, 3"b ¤áƒe, q,'k ˜‡{ü 4•©Ù. |^§ 4•©ÙŒCqû½(2.2)ª¥ cÚr. W ilks½ n (ƒ•ãI‡•ã˜Œæ'uoNVÇ©Ù b ½, Ù y ² • é E ,. ·‚Ñ ù • ã, • r N Ù ^ ¥ ˜ ‡ – ' - ‡ ƒ :, = ‡ ¦ ë ê ˜ mΘ ‘ê‡pu "b ¤áž Θ0 ‘ ê. X 2 X1 , . . . , Xn i.i.d. ∼ N (µ, σ ), H0 : µ = µ0 ←→ H1 : µ = µ0 ,KΘ = {θ = (µ, σ ) : −∞ < µ < +∞, σ 2 > 0}´R2 ¥ 2 þŒ²¡, Θ ‘ê´2; Θ0 = {θ = (µ, σ 2 ) : µ = µ0 , σ 2 > 0},§´Θ¥ ˜^†‚, Ù‘ê•1.Ïdd~¥Θ‘êp uΘ0 ‘ê. qX, ¥N´n‘8, ˜m ˜‡:´"‘8. ²( ù˜:, Wilks ½nŒ—Œ Lˆ• ½ n 5.5.1 Θ ‘ê•k, Θ0 ‘ê•s, ek − s = t > 0, Kéu ¯K(2.1) 3"b H0 ¤áƒe, Œ n → ∞žk L 2 log λ(X) −→ χ2t . ½n •[•ã9y²Œëwë•©z[1]P326 .„k˜:I‡²(: "b Θ0 ¥Œ±•¹Ø Ž˜‡:, ùž½n5.5.1 ¹Â´: ØØýëêá3Θ0 ¥Û?, 2 log λ(X) 4•©Ùo´gd Ý•t χ2 ©Ù. ~ 5.5.7 Xi = (Xi1 , · · · , Xin ), i.i.d. ∼ N (µi , σi2 ), 1 ≤ i ≤ m,… Ü Õá. ‡ u b H0 : σ12 = · · · = σm 2 ←→ H1 : σ12 = · · · = σm 2 Ø ƒÓ. P ni ni 1 1 Si2 = (Xij − X¯i )2 , X¯i = Xij . ni j=1 ni j=1 m m 1 2 2 S = ni si , n= ni n i=1 i=1 KØJŽÑ m λ(X1 , X2 , · · · , Xm ) = S n / Sini . i=1 n Yn ≡ 2 log λ = n log S − ni log si . i=1 Pk•ëê˜mΘ ‘ê, r•H0 ¤ážëê˜mf8Θ0 ‘ê. â½n5.5.1, H0 ¤áž, …min{n1 , n2 , . . . , nm } → ∞ ž, k L Yn −→ χ2k−r = χ2m−1 , 21

22 .d?k − r = 2m − (m + 1) = m − 1.dd Œ u kY²Cq•α Ä½• D = {(X1 , . . . , Xm ) : Yn > χ2m−1 (α)}. 3 Wald u 5¿ ˆ eθ•ëêθ MLE, Kdq, O ìC 5•3 Kz^‡e √ d n(θˆ − θ) −→ N (0, I −1 (θ)) l é"b H0 : θ = θ0 , Wald u ÚOþ½Â• W = n(θˆ − θ0 ) I(θ0 )(θˆ − θ0 ) 3"b e, W ìCÑlχ2 ©Ù. 4 Score u Scoreu K´ÄuScore ¼ê EPScore ¼ê• ∂l(θ) S(θ) = ∂θ 3"b H0 : θ = θ0 e˜„k 1 d √ S(θ) −→ N (0, I(θ0 )) n Ké"b Scoreu ÚOþ• 1 U= S(θ0 ) I −1 (θ0 )S(θ0 ) n 3"b e, U ìCÑlχ2 ©Ù. 22

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