主要介绍了容忍区间和容忍限的提出和定义,具体介绍了正态总体的容忍区间和容忍限、非参数容忍限和容忍区间的若干定理和例子。

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1. Lec8: «m O( ): N=«m Ü•² 2011 c 3 28 F 1 N=«m†N=•* !‡?Ø ¯K, ÙJ{†«m O¿Ã Óƒ?. ˜&«m´£‰oNX ,‡ë ê(½A ) •¹3Äu E ,‡‘Å«m¥, Ã{£‰ÃXoN˜½'~ •¹3 Äu E ,‡‘Å«mƒa ¯K. ˜!¯K J{9½Â k©Ùx{Fθ , θ ∈ Θ}, Θ´ëê˜m, X = (X1 , · · · , Xn )•l©Ùx,oN¥Ä {ü . d?Ø´•Äëêθ ˜&«m, ´•ÄoN‘ÅCþX /˜&«m0, ¡ƒ•N=« m. =F"¦T1 = T1 (X)ÚT2 = T2 (X), ¦ éoN – 100(1 − β)Ü©´á3«m[T1 , T2 ]S: Pθ {X ∈ [T1 (X), T2 (X)]} ≥ 1 − β. žw±eü‡~f: ~4.5.1 ,¶«‚k˜ÜgÄzÅì, ) †»•0.25cm ¶«, #NØ •0.001cm. ) ¥‡¦99% ¬ˆ ±þ5½, =‡¦¶« †»á3[0.249, 0.251] ƒm. 8é˜1¶« Ä n‡, ÿ Ù†»•X1 , · · · , Xn , ¯ù˜1¶«´ÄÜ‚? ‘ÅCþXL«¶« †», ٩ټêFθ (x). )ûù˜¯K •{, Ò´d X= (X1 , · · · , Xn )(½ü‡ÚOþT1 = T1 (X)ÚT2 = T2 (X), ¦ Pθ (X ∈ [T1 , T2 ]) = Fθ (T2 ) − Fθ (T1 ) ≥ 0.99, e[T1 , T2 ] ⊂ [0.249, 0.251], K`²d1¶«Ü‚. ùÒ8(•¦N=«m ¯K. ~ 4.5.2 g‚) ,«gá, ‡¦Ùgá rÝØ uξ0 (X, ξ0 = 120ü rÝ) . e ) ¥gárÝk99%ÎÜþ㇦, @•d1gáÜ‚. 8é˜1gá, ÿÁ nŠ, rÝ •X1 , · · · , Xn , ¯ù1gá´ÄÜ‚? gá r Ý • ‘ Å C þX, Ù © Ù ¼ ê •Fθ (x). ) û ù ˜ ¯ K ˜«•{´d X = (X1 , · · · , Xn )(½˜‡ÚOþTL (X), ¦ Pθ (X ∈ [TL (X), ∞)) = 1 − Fθ (TL (X)) ≥ 0.99, eTL (X) ≥ ξ0 , K`²ù1gáÜ‚, ùÒ8(•¦N=e• ¯K. òþãü‡~fJÑ ¯KÚ˜3˜‡ .e, £ãXe: koNX ©Ùx{Fθ , θ ∈ Θ}. -X=(X1 , · · · , Xn )•ld©Ùx¥Ä {ü , ‡é ü‡ÚOþT1 = T1 (X)ÚT2 = T2 (X), ¦ é0 < β < 1k Pθ (T1 (X) ≤ X ≤ T2 (X)) = Fθ (T2 (X)) − Fθ (T1 (X)) ≥ 1 − β, (1.1)

2.½é˜‡ÚOþTL (X), ¦ é0 < β < 1k Pθ (TL (X) ≤ X < +∞) = 1 − Fθ (TL (X)) ≥ 1 − β, (1.2) ½é˜‡ÚOþTU (X), é0 < β < 1k Pθ (−∞ < X ≤ TU (X)) = Fθ (TU (X)) ≥ 1 − β. (1.3) 3(1.1)¥, duT1 (X)ÚT2 (X)•‘ÅCþ, Fθ (T1 (X))ÚFθ (T2 (X))•´‘ÅCþ, ¤±/F (T2 )− F (T1 ) ≥ 1 − β0´˜‡‘ů‡. w,ØU yù˜¯‡ýéu), u´•Uü$‡¦: ‰½r (Ï~0 < r < 1) , ‡¦/F (T2 ) − F (T1 ) ≥ 1 − β0ù‡¯‡– ±VÇ1 − r¤á,= Pθ F (T2 ) − F (T1 ) ≥ 1 − β ≥ 1 − r. é(1.2)Ú(1.3)¥ TL , TU •Œ±JÑaq‡¦, ùÒÚ N=«mÚN=• Vg. e¡‰ ѽÂ. ½ Â4.5.1 X = (X1 , · · · , Xn )•loNX ∼ Fθ , θ ∈ 靎 {ü . q T1 (X)ÚT2 (X)´ü‡ÚOþ, …T1 (X) ≤ T2 (X), eé?¿‰½ β, γ (Ï~ ê, Xβ = 0.05, γ = 0.01), 0 < β, γ < 1k Pθ Pθ (T1 ≤ X ≤ T2 ) ≥ 1 − β = Pθ F (T2 ) − F (T1 ) ≥ 1 − β ≥ 1 − γ, ˜ƒ θ ∈ Θ, (1.4) K¡[T1 , T2 ]´Fθ ˜‡Y²•(1 − β, 1 − γ) N=«m (Tolerance interval). TL = TL (X)ÚTU = TU (X)´ü‡ÚOþ, eé?¿‰½ β, γ, 0 < β, γ < 1,Ú˜ ƒθ ∈ Θ,©Ok Pθ Pθ (TL ≤ X) ≥ 1 − β = Pθ 1 − F (TL ) ≥ 1 − β = Pθ F (TL ) ≤ β ≥ 1 − γ, (1.5) Pθ Pθ (X ≤ TU ) ≥ 1 − β = Pθ F (TU ) ≥ 1 − β ≥ 1 − γ, (1.6) K¡TL ÚTU ©O´Fθ ˜‡Y²•(1 − β, 1 − γ)N=e• (Tolerance lower limit)ÚN=þ• (Tolerance upper limit). 5¿þã½Â¥ ü‡Pθ ¹Â´ØÓ , p¡ Pθ ´UoN©ÙFθ 5OŽ , ¡ Pθ ´U X = (X1 , · · · , Xn ) éÜ©Ù5OŽ . N=«mÚN=•ƒmke 'Xµ Ú n 4.5.1 eT2 (X)ÚT1 (X)©O´©ÙFθ Y²•(1 − β/2, 1 − γ/2) N=þ!e•, …okT2 (X) ≥ T1 (X), K[T1 (X), T2 (X)]•Fθ Y²•(1 − β, 1 − γ)N=«m. y -AL«/F (T1 ) ≤ β/20; BL«¯‡/F (T2 ) ≥ 1 − β/20; CL«¯‡/F (T2 ) − F (T1 ) ≥ 1 − β0, Kd½Â4.5.1Œ• Pθ (A) ≥ 1 − γ/2, Pθ (B) ≥ 1 − γ/2. (1.7) 2

3.·‚F"y²Pθ (C) ≥ 1 − γ. d±þ½ÂŒ•, eA!BÓž¤á, K7kF (T2 ) − F (T1 ) ≥ 1 − β, =C¤á, ÏdAB ⊂ C, kPθ (C) ≥ Pθ (AB), ddŒ Pθ (C) = Pθ (Fθ (T2 ) − Fθ (T1 ) ≥ 1 − β) ≥ P (AB) = P (A) + P (B) − P (A ∪ B) ≥ (1 − γ/2) + (1 − γ/2) − 1 = 1 − γ. ½ny.. ! oN N=«mÚN=• X = (X1 , · · · , Xn )•g oNN (µ, σ 2 )¥Ä {ü‘Å , θ = (µ, σ 2 ) ¿©ÚO ©þ• n n ¯= 1 X Xi , S2 = 1 ¯ 2. (Xi − X) n i=1 n−1 i=1 ¯ S 2 )5 ·‚òÄu¿©ÚOþ(X, E oN N=•ÚN=«m¯K. 3oNN (µ, σ 2 )¥, eµÚσ 2 ®•, KY²•(1 − β, 1 − γ) N=þe•ÚN=«m©O•µ + σuβ , µ − σuβ Ú[µ − y3µÚσ 2 ™•, ·‚• XÚS σuβ/2 , µ + σuβ/2 ]. ¯ 2 ©O´µÚσ 2 ûÐ O, ÏdòþãN ¯ =þ•¥ µÚσ^XÚS“O ¯ + Suβ . du O ‘5 ‘Å5, Y²(1 − β, 1 − γ) X N=þ•Ø„ дX¯ + Suβ , ŒU‡òXêuβ ?U•,‡λ, λQ†βk', •†γk'(5 ¿uβ †γÃ'). N=e•ÚN=«m•Xd?n. ¯ + λS•Y²(1 − β, 1 − γ) Ïd·‚Äk5¦N=þ•. =éλ¦X N=þ•. U½Â, é ‰½ βÚγ, 0 < β, γ < 1, ‡(½λ, ¦ ¯ + λS) ≥ 1 − β} ≥ 1 − γ. Pθ {Pθ (X ≤ X du(X − µ)/σ ∼ N (0, 1), ٩ټê•Φ(·), Ïdþª†>•• X −µ X¯ − µ + λS Pθ Pθ ≤ ≥1−β σ σ ¯ − µ + λS X = Pθ Φ ≥1−β σ X¯ − µ + λS = Pθ ≥ Φ−1 (1 − β) = uβ . (1.8) σ √ ¯ − µ)/σ, S ∗ = S/σ,KZ ∼ N (0, 1), S∗ ∼ -Z = n(X χ2n−1 /(n − 1).Ïd √ √ Z − nuβ Z − nuβ = ∼ tn−1, δ , S/σ S∗ √ =gdÝn − 1, š¥%ëêδ = − nuβ š¥% t©Ù. d(1.8)Œ• Pθ P θ X ≤ X¯ + λS ≥ 1 − β ≥ 1 − γ ⇐⇒ X¯ − µ + λS Pθ ≥ uβ ≥ 1 − γ ⇐⇒ σ 3

4. √ Z − nuβ √ Pθ ≥ − nλ ≥ 1 − γ. (1.9) S∗ √ √ e Pλ = λ(n, β, γ), d− nλ = tn−1, δ (1 − γ), ) λ(n, β, γ) = −tn−1, δ (1 − γ)/ n = √ ¯ + λS, Ù¥λ = tn−1, δ (γ)/√n. a tn−1, δ (γ)/ n. ÏdŒ•, Y²•(1 − β, 1 − γ) N=þ••X qŒ¦Y²•(1 − β, 1 − γ) N=e••X ¯ − λS, λÓþ. é~„ n, λ, γ®?› λ(n, β, γ)Š L, „NL6. ¦ ¯ − λS, X oNN (µ, σ 2 ) N=«m, Œ|^Ún4.5.1Œ [X ¯ + λS], d?λ(n, β, γ) = √ ∗ √ tn−1,δ∗ (γ/2)/ n. Ù¥δ = − nuβ/2 . ,, yk š¥% t©ÙL„Ø Œ, ؘ½UlLþ † š¥%t©Ù © ê Š. Ö"NL7 ‰Ñ λ(n, β, γ)Š L. ~ 4.5.3 ,‚) Wì^ qÜ7‚. ² L²: qÜ7‚ |.rÝÑl ©Ù. 8 l˜1 ¬¥‘ÅÄ 10‡ ¬, ÿ å|.rÝ•(ü : kg/mm ) 2 10512, 10623, 10668, 10554, 10776, 1071, 10557, 10581, 10666, 10670 ¦TqÜ7‚|.rÝN=e•( Y²•(0.95, 0.95)) ) d¯K¥n = 10, Y²•(0.95, 0.95)=,β = 0.05, γ = 0.05. Ïd1 − β = 0.95, 1 − γ = 0.95. dê⎠¯ = 10632.4, X S 2 = 6738.77, S = 82.09. NL6 λ = 2.91, Ïd ¯ − λS = 10632.4 − 2.91 × 82.09 = 10393.52. Ïd NNe•TL = X ù1qÜ7‚|.rÝØ$u10393.52 kg/mm2 . ~4.5.4 ² L²™ã ä KÖ(ü : z©ƒ˜Úî) Ñl ©Ù, yl˜1™ã ¥‘ÅÄ 12Š, ÿ Ùä KÖ• 228.6, 232.7, 238.8, 317.2, 315.8, 275.1, 222.2, 236.7, 224.7, 251.2, 210.4, 270.7 ¦™ãä KÖ Y²•(0.95,0.99) N=«m. ) d¯K¥n = 12, 1 − β = 0.95, 1 − γ = 0.99,dê⎠¯ = 252.0, X S 2 = 1263.4, S = 35.5 NL7 λ(12, 0.95, 0.99) = 3.87,ddŽ ¯ − λS = 252.0 − 3.87 × 35.5 = 114.6 TL = X ¯ + λS = 252.0 + 3.87 × 35.5 = 389.4 TL = X Ïd™ãä KÖ Y²(0.95, 0.99) N=«m•[114.6, 389.4]. n!šëêN=•ÚN=«m 3¢S¯K¥, „¬²~‘ ù ¯K, <‚•• oN©ÙF ´ëY. , ‡¦d©Ù N=•ÚN=«m. duùžØ• ©Ù äN/ª, !Øþ$^©Ù 5Ÿ, •U|^ ‰Ñ &E. e¡?ØÄugSÚOþXÛ‰ÑF N=•ÚN=«m. ky²˜‡ý •£. 4

5. Ú n 4.5.2 ˜‘‘ÅCþX ∼ F (x), F (x)´©Ù¼ê…??ëY, KY = F (X)Ñl þ!©ÙU (0, 1). y Ï•0 < Y < 1•Ié0 < y < 1y²P (Y < y) = y=Œ. Pt = inf{x : F (x) ≥ y},K dF (x)??mëY…šü, ´„F (t) = y, ±9F (x) < y ⇐⇒ x < t, k P (Y < y) = P (F (X) < y) = P (X < t) = F (t) = y. Úny.. y X1 , · · · , Xn •loNX ∼ F (x)¥Ä {ü‘Å , X(1) ≤ X(2) ≤ · · · ≤ X(n) •Ù gSÚOþ, ŠâÚn4.5.2Œ•, ePUi = F (Xi ), i = 1, 2, · · · , nKU1 , · · · , Un i.i.d. ∼ U (0, 1). U(1) ≤ U(2) ≤ · · · ≤ U(n) •ÙgSÚOþ. 1. ¦F Y²(1 − β, 1 − γ) N=«mÚN=• PUi = F (Xi ), i = 1, 2, · · · , n; Vij = U(j) − U(i) , 1 ≤ i < j ≤ n, KVij —Ýgnij ®dc¡ ‰Ñ. e± X(i) , X(j) Š•N=«m, U½Âk P P X(i) ≤ X ≤ X(j) ≥ 1 − β ≥ 1 − γ. (1.10) þª†>• 1 P F (X(j) ) − F (X(i) ) ≥ 1 − β = P Vij ≥ 1 − β = gnij (v)dv. (1.11) 1−β XJÀJ· i, j¦(1.10)ª ¥ È©Ø u‰½ 1 − γ, K[X(i) , X(j) ]Ò ´F ˜‡Y ²(1 − β, 1 − γ) N = « m. — Ýgnij ¡ •Beta © Ù, Ù ë ê •j − iÚn − j + i + 1, P •Be(j − i, n − j + i + 1). Beta ©Ù ëêØ7• ê, •‡Œu0Ò1. p > 0, q > 0ž, Be(p, q)L«˜©Ù, Ù©Ù¼ê• x 1 Ip,q (x) = tp−1 (1 − t)q−1 dt, (1.12) B(p, q) 0 1 p−1 Ù¥B(p, q) = 0 t (1 − t)q−1 dt,¡•Beta È©.§†Gamma¼êΓ(x)k'XµB(p, q) = Γ(p)Γ(q)/Γ(p + q),ù‡úª3‡È¥®²‰ÑL. úª(1.12) 0 < x < 1ž,¡•Ø Beta È©, Pearson Q‰§E L. ùLŒ^uÀ Ji, j ¯K. éF (x) Y²(1 − β, 1 − γ) N=þe• ¯K•Ó ?n. éN=þ•k P P X ≤ X(i) ≥ 1 − β ≥ 1 − γ, dª†>´P F (X(i) ) ≥ 1 − β = P U(i) ≥ 1 − β .|^(??)ª, 3Ù¥˜F (x) = x, f (x) = 1 U(i) —Ý n i−1 gni (x) = i x (1 − x)n−i I[0<x<1] . i Ïdk 1 P F (X(i) ) ≥ 1 − β = P U(i) ≥ 1 − β = gni (v)dv. (1.13) 1−β 5

6.ÀJi¦(1.13)ªØ u1 − γ, KŠâ½ÂX(i) Ò´F ˜‡Y²(1 − β, 1 − γ) N=þ•. ÓnÀJj, ¦ eªØ u1 − γ, = β P F (X(j) ≤ β) = P U(j) ≤ β = gnj (v)dv ≥ 1 − γ, (1.14) 0 KX(j) Ò´F ˜‡Y²(1 − β, 1 − γ) N=e•. 3(1.13)Ú(1.14)¥ÀJi, j¦È©Ø u1 − γ, Œ/ÏuØ Beta¼êL¦ . 2. A~œ/ (1) bX·‚ X(n) Š•F ˜‡Y²(1 − β, 1 − γ) N=þ•, Nþn7Lk˜½ ‡¦, ÄK§ØUŠ•Ü· N=þ•. @o Nþn– A•õ ? ùÒr¦N=þ• ¯ K=z•(½ Nþ ¯K. duX ∼ F (x), …F (x)??ëY, F (X) ∼ U (0, 1). F (X(n) ) = U(n) ´5goNU (0, 1) Nþ•n •ŒgSÚOþ, Ù—Ý¼ê• gnn (y) = ny n−1 I[0<y<1] . u´‡¦(½n, ¦ P F (X(n) ) ≥ 1 − β = P U(n) ≥ 1 − β ≥ 1 − γ, = 1 ny n−1 dy ≥ 1 − β ⇐⇒ 1 − (1 − β)n ≥ 1 − γ. 1−β Ïdk ln γ n≥ ln (1 − β) 鉽 β, γ, Œ±Ž ÷vþãØ ª • g,ên. (2) aqOŽŒ•, e X(1) Š•F ˜‡Y²(1−β, 1−γ) N=e•, Œ•F (X(1) ) = U(1) —Ý¼ê• gn1 (y) = n(1 − y)n−1 I[0<y<1] ‡¦ β P F (X(1) ) ≤ β = P U(1) ≤ β = n(1 − y)n−1 dy ≥ 1 − γ, 0 Ó )Ñn ≥ ln γ/ ln (1 − β). éÏ~^ β, γ, (½F Y²(1 − β, 1 − γ) N=þe•¤I‡• Nþn, ®?› L, •„NL8. X1 − β = 0.90, 1 − γ = 0.95, lNL8þ n = 29, e1 − β = 0.95, 1 − γ = 0.99lLþ n = 90. (3) e [X(1) , X(n) ]Š•F (x) ˜‡Y²•(1 − β, 1 − γ) N=«m, U½Â P F (X(n) ) − F (X(1) ) ≥ 1 − β = P U(n) − U(1) ≥ 1 − β ≥ 1 − γ. (1.15) 6

7.du(U(1) , U(n) ) éÜ—Ý p(y1 , y2 ) = n(n − 1)(y2 − y1 )n−2 I[0≤y1 ≤y2 ≤1] , ¤±(1.15)ªŒU • β 1 p(y1 , y2 )dy1 dy2 = n(n − 1)(y2 − y1 )n−2 dy2 dy1 0 y1 +(1−β) y2 −y1 ≥1−β ≥ 1 − γ, )ƒŒ n(1 − β)n−1 − (n − 1)(1 − β)n ≤ γ. 鉽 βÚγ (½ d ‰½1 − β, 1 − γ) Œ±lþãØ ª¥)Ñn5. é~^ 1 − βÚ1 − γ, •®?E (½F Y²(1 − β, 1 − γ) N=«m¤I• Nþ L, „NL9. ~X, 1−β = 0.90, 1−γ = 0.95, lNL9þ n = 46; 1−β = 0.95, 1−γ = 0.99,l Lþ n = 130. 7