13 统计学基础--非参数统计方法（二）

1 . Lec13: šëêÚO•{( ) Ü•² May 4, 2011 §1 [Ü`Ýu ëêb u Ñ´3b½oN´,«äN©Ù ^‡e?1 , ´ù‡b ؘ ½¤á, ·‚Œ±ÏL (X1 , · · · , Xn ) 5u §. ˜„/, u H0 : X Ñl,«©Ù Œ±æ^Karl Pearson JÑ χ2 [Ü`Ýu . §1.1 lÑoNœ/ (1) nØ©Ùع™•ëê œ/ ,oNX Ñl˜‡lÑ©Ù, …Šâ² •oNá3aOa1 , · · · , ak nتǩ O•p1 , · · · , pk , ylToNÄ ˜‡ þ•n , Ùá3aOa1 , · · · , ak *ÿê© O•n1 , · · · , nk . a, ¯K´u nØªÇ´Ä (, =e¡b ´Ä (: H0 : P (X ∈ a1 ) = p1 , · · · , P (X ∈ ak ) = pk . ùa¯K•J"b ØJéáb , ƒA u •{¡•[Ü`Ýu . w,, 3"b e, ˆaO nتê©O•np1 , · · · , npk , ònتêÚ*ÿªê ueL: aO a1 a2 ··· ak nتê np1 np2 ··· npk *ÿªê n1 n2 ··· nk dŒê½Æ•, 3"b ¤áž, ni /n •VÇÂñupi , nتênpi †*ÿªêni C. u ÚOþ • k (ni − npi )2 χ2 = . i=1 npi {ü/, Ò´ (O − E)2 χ2 = , E Ù¥O •*ÿªê, E •Ï"ªê. ù‡ÚOþ¥z‘ ©1 À k:ùÄ, ·‚Œ±ù oÑ/)º: b ni ÑlPoisson √ ©Ù, Kni þŠÚ• þ•npi , l (ni − npi )/ npi 4•©Ù•IO ©Ù, Ïdχ2 1 

2 .Cq•k ‡ÑlgdÝ•1 χ2 ©Ù ‘ÅCþƒÚ, du ki=1 (ni − npi ) = 0, ùk ‡‘ ÅCþ÷v˜‡ å, l χ2 gdÝ•k − 1. ¯¢þ, Œ±î‚/y², 3˜½ ^‡e, χ2 4•©ÙÒ´gdÝ•k − 1 χ2 ©Ù, Ùy²‡Ñ ‘§ ‡¦‰Œ. e¡‰Ñ˜‡~f5`²[Ü`Ýu A^. ~ 1. k<›E˜‡¹6 ‡¡ f, ¿(¡´þ! . y O˜‡¢ 5u d·K: ëY Ý•600 g, uyÑy8¡ ªê©O•97, 104, 82, 110, 93, 114. ¯UÄ3wÍ5Y²0.2 e@• f´þ! ? ): T¯K O oN´˜‡k6 ‡aO lÑoN, PÑy8‡¡ VÇ©O•p1 , · · · , p6 , K"b Œ±L«• H0 : pi = 1/6, i = 1, · · · , 6. 3"b e, nتêÑ´100, u ÚOþχ2 Š• (97 − 100)2 (104 − 100)2 (82 − 100)2 (110 − 100)2 (93 − 100)2 (114 − 100)2 + + + + + = 6.94, 100 100 100 100 100 100 ‹gdÝ•6 − 1 = 5 χ2 ©Ù þ0.05 © êχ25 (0.2) ≈ 7.29 ' , ØUáý"b , =Œ 3wÍ5Y²0.2 e@• f´þ! . ~ 2. Š (Mendel) Î, Á "X‘ÚXɬ«, § Ï•‘ÚéÉÚ´w5 § 3Mendel1˜½Æ(gd©l½Æ) b e§ “ Î¥ATk75 ´‘Ú §25 ´É Ú "3 ) n = 8023‡ “ Î¥§kn1 = 6022‡‘Ú§n2 = 2001‡ÉÚ"·‚ ¯K ´u ù ù1êâ´Ä|±Mendel1˜½Æ§‡u b ´ H0 : π1 = 0.75, π2 = 0.25 ): 3Mendel1˜½Æ(H0 )e§‘ÚÚÉÚ ‡êÏ"Š• µ1 = nπ1 = 8023 ∗ 0.75 = 6017.25, µ2 = nπ2 = 8023 ∗ 0.25 = 2005.75 KPearson χ2 ÚOþ• (O − E)2 Z= = (6022 − 6017.25)2 /6017.25 + (2001 − 2005.75)2 /2005.75 = 0.015 E gdÝdf = 1§p − value•0.99996. ÏdŒ±@•ù êâÑlMendel1˜½Æ"FisherÄ uMendel ù êâ§uyÙêâ†nØŠÎÜ Ð§p − value = 0.99996§ ùoÐ [Ü3AZgÁ ¥âu)˜g§Ï Fisherä½êâŒUk–E v¦" (2) nةٹeZ™•ëê œ/ nØoNo¹k™• ëêž, nتênpi ˜„•†ù ëêk', džAT^· OX4Œq, O“Où ëê± pi Opˆi , ÚOþP• k (ni − nˆpi )2 χ2 = . i=1 nˆ pi [Ü`Ýu JÑöKarl Pearson •Ð@•3"b e, u ÚOþ χ2 4•©ÙE ugdÝ•k − 1 χ2 ©Ù, R. A. Fisher uygdÝAT uk − 1 ~ O Õáëê ‡êr, =k − 1 − r. 2 

3 .~ 3. l,<+¥‘ÅÄ 100 ‡< É—, ¿ÿ½¦‚3,ÄÏ :? ÄÏ.. b T :•kü‡ ÄÏA Úa, ù100 ‡ÄÏ.¥AA, Aa Úaa ‡ê©O•30, 40, 30, KU Ä30.05 Y²e@•T+N3d :?ˆ Hardy-Weinberg ²ï ? ): "b • H0 : Hardy-Weinberg ²ï ¤á. <+¥ ÄÏA ªÇ•p, KT<+3d :?ˆ Hardy-Weinberg ²ï • ´3 <+¥3 ‡ÄÏ. ªÇ©O•P (AA) = p2 , P (Aa) = 2p(1 − p) ÚP (aa) = (1 − p)2 , ="b Œ d/ ¤ H0 : P (AA) = p2 , P (Aa) = 2p(1 − p), P (aa) = (1 − p)2 . 3H0 e, 3 ‡ÄÏ. nتê•100 × pˆ2 , 100 × 2 × pˆ2 (1 − pˆ) Ú100 × (1 − pˆ)2 , Ù¥pˆ u O ÄϪÇ0.5, “\χ2 ÚOþLˆª, ÚOþ Š u4. TÚOþ ŠŒug dÝ•3 − 1 − 1 = 1 (TИ‡gdëê O) χ2 ©Ùþ0.05 © ê3.84, Œ30.05 Y²e@•™ˆ Hardy-Weinberg ²ï . §1.2 éL Õá5Úà˜5u (1) Õá5u e¡•Äé~^ éL. éL´˜«Uü‡á5ŠV•©a L. ~X_J¾<Œ ±U¤3š (á5A) Ú´Ä•ªk (á5B) ©a. 8 ´wØÓš ´ÄØÓ. qX? ŒUž •ª(á5A, ©ü‡Y²: 1Zž †<óž ) Ú ß¸u˜G¹(á 5B, ©ü‡Y²: ~†É~) 5©a. ùü‡~f¥ü‡á5Ñ•kü‡Y², ƒA éL¡•“o‚L”, ˜„/, XJ1˜‡á5ka ‡Y², 1 ‡á5kb ‡Y², ¡•a × b L(„ áp268) . ¢SA^¥, ~„ ˜‡¯K´• ü‡á5´ÄÕá. ="b ´ H0 : á5A †á5B Õá. ù´ éL Õá5u ¯K. b þ•n, 1(i, j) ‚ ªê•nij . Ppij = P (á5A, B ©O?uY²i, j), ui = P (á5A kY²i), vi = P (á5B kY²j). K"b Ò´pij = ui vj . òui Úvj w¤ëê, Ko Õáëêka − 1 + b − 1 = a + b − 2 ‡. §‚ 4Œq, O• ni· n·j u ˆi = , vˆj = . n n b a 䧂 ªÇ(y²ëw á) . Ù¥ni· = j=1 nij , n·j = i=1 nij . 3H0 e, 1(i, j) a b ‚ nتê•nˆ pij = ni· n·j /n, Ïd3H0 e, i=1 j=1 (nij pij ) AT − nˆ . u ÚOþ• a b (nij − ni· n·j /n)2 χ2 = . i=1 j=1 (ni· n·j /n) 3"b eχ2 4•©Ù´kgdÝ•k − 1 − r = ab − 1 − (a + b − 2) = (a − 1)(b − 1) χ2 ©Ù. éuo‚L, gdÝ•1. 3 

4 .(2) à˜5u ‹ éLk' ,˜a-‡ u ´à˜5u , =u ,˜‡á5A ˆ‡Y²éA ,˜‡á5B ©Ù ܃Ó, ù«u ‹Õá5u kX Ÿ «O. Õá5¯K¥ü á5Ñ´‘Å ; à˜5¯K¥á5A ´š‘Å , ù 9 ©Ù¢Sþ´^‡©Ù. •,Xd, ¤æ^ u •{‹Õá5u ˜ . ~ 4. e¡L´`¯üš _J¾<)•œ¹. I‡Šâù êâ äüš £ J´ Ę . `!¯ü _J CÏ )• k ÜO ` 150(n11 ) 88(n12 ) 238(n1· ) ¯ 36(n21 ) 18(n22 ) 54(n2· ) ÜO 186(n·1 ) 106(n·2 ) 292(n) ): ù´˜‡à˜5u ¯K. u ÚOþχ2 *ÿŠ•0.2524, ugdÝ•1 χ2 ©Ù þ0.05 © ê, Œ± É"b , =3Y²0.05 eŒ±@•ü‡š à O . k,‡‚f ªê ž, XJ#N {Œ±Ü¿‚f´z‡‚f ªêv Œ, ¢ S¯K¥Ø#NÜ¿‚f(Ü¿ ” ¢S¿Â), džŒ±^Fisher °(u {. §1.3 ëYoNœ/ (X1 , · · · , Xn ) ´ goNX ˜‡ , PX ©Ù¼ê•F (x), I‡u @«© Ù¥¹kr ‡oNëêθ1 , · · · , θr . ·‚‡3wÍ5Y²α eu H0 : F (x) = F0 (x; θ1 , · · · , θr ), Ù¥F0 (x; θ1 , · · · , θr ) L«I‡u @«©Ù ©Ù¼ê. ~X, ·‚‡u H0 : X ∼ N (µ, σ 2 ) ž, r = 2, θ1 = µ, θ2 = σ 2 . x 1 1 F0 (x; µ, σ 2 ) = √ exp − (t − µ)2 dt. −∞ 2πσ 2 2σ 2 þãb Œ±ÏL· lÑzoN©Ù, æ^[Ü`Ý{5‰u . Äkr¢ê¶© ¤k ‡f«m(aj−1 , aj ], j = 1, · · · , k, Ù¥a0 Œ± −∞, ak Œ± ∞. ù E ˜‡lÑ oN, Ù ŠÒ´ùk ‡«m. P pj = PH0 (aj−1 < X ≤ aj ) = F0 (aj ; θ1 , · · · , θr ) − F0 (aj−1 ; θ1 , · · · , θr ), j = 1, · · · , k. XJH0 ¤á, KVÇpj AT†êâá3«m(aj−1 , aj ] ªÇfj = nj /n C, Ù¥nj L« ƒA ªê. pi ŠØ¹™•ëêž, u ÚOþ k (nj − npj )2 χ2 = , j=1 npj 4 

5 .ÄK k (nj − nˆp j )2 χ2 = , j=1 nˆ pj Ù¥pˆi ´òpi ¥ ™•ëꆤ· O pi O. áý• • χ2 > χ2k−r−1 (α) . XJpi ¥Ø¹™•ëê, Kr = 0. ¦^χ2 ?1[Ü`Ýu ž˜„‡¦n ≥ 50, nˆ pj ≥ 5, j = 1, · · · , k, XJØ÷vù‡^ ‡, •Ðr, |Š· Ü¿. ~ 5. l,ëYoN¥Ä ˜‡ þ•100 , uy þŠÚ IO ©O•−0.225 Ú1.282, á3ØÓ«m ªêXeL¤«: «m (−∞, −1) [−1, −0.5) [−0.5, 0) [0, 0.5) [0.5, 1) [1, ∞) *ÿªê 25 10 18 24 10 13 nتê 27 14 15 14 13 17 ŒÄ3wÍ5Y²0.05 e@•ToNÑl ©Ù? ): nØ ©Ù þŠÚ• ©O•µ Úσ 2 , P1i ‡«m•(ai−1 , ai , i = 1, · · · , 6, K á31i ‡‚f nØVê•100P (ai−1 < X ≤ ai ), Ù¥X ∼ N (µ, σ 2 ). òµ = −0.225 Úσ = 1.282 “\ O nتê, uþL¥. H0 : oNÑl ©Ù ddŽ u ÚOþχ2 Š •9.34, †gdÝ•5 χ2 ©Ù þ0.1 © êχ25 (0.1) ≈ 9.24 ' Œ±áý"b , =Œ±3wÍ5Y²0.1 e@•ToNØÑl ©Ù. §2 Ù¦~^u •{ ˜! … #xâÅu ¦+Pearson χ2 u é?Ûa. ©Ùu ÑŒ±^. ØLéuëY. ‘ÅCþ, … #xâÅu J•Ð . ù´Ï•Pearsonχ2 u •6ur(−∞, +∞)©¤r‡«m ä Ny©•{, •)r ÀJÚ«m ˜. c€éͶêÆ[… #xâÅ1933cJÑ ˜« # 'uoN©Ù [Ü`Ýu •{¨… #xâÅu ({¡…¼u {) . ©Ù¼êF (x)™•, X1 , · · · , Xn •lF ¥Ä r.v. X {ü‘Å , F0 (x)•‰½ ,‡©Ù¼ê. ·‚5ïÄe u ¯K: H0 : F (x) = F0 (x). (2.1) Äkl Ñu¦ÑF (x) ² ©Ù¼êXe:   0,   x ≤ X(1) ; Fn (X) = k/n, X(k) < x ≤ X(k+1) ; k = 1, 2, · · · , n − 1 (2.2)    1, x > X(n) . 5 

6 .ùpX(1) ≤ X(2) ≤ · · · ≤ X(n) ´ X1 , · · · , Xn gSÚOþ. Fn (x) 5Ÿ„§ 1.3 n. -u ÚOþ• Dn = sup |Fn (x) − F0 (x)|, (2.3) −∞<x<+∞ Dn ~¡•Fn †F0 ƒm …¼ål. dGlirenko-Cantelli½n•, XJH0 ¤á, KP ( lim Dn = n→∞ 0) = 1. †óƒ, XJH0 ¤á, nq Œ, Dn Š–•u Š. XJDn Š Œ, –•uÄ ½H0 .=u ŒQã•: Dn ≥ cžÄ½H0 , c• .Š, –½. Ù[Ü`Ý OŽúªXe: 3k äN , OŽÑDn äNŠD0 ,KVÇ p(D0 ) = P (Dn ≥ D0 |H0 ) (2.4) Ò´3…¼åle, X1 , · · · , Xn †nØ©ÙF0 (x) [Ü`Ý. e•½˜‡zŠα(½¡u Y²), KI½Ñ˜‡~êDn, α ,¦ p (Dn, α ) = P (Dn ≥ Dn, α |H0 ) = α, (2.5) K Dn > Dn, α žÄ½H0 ,Ø,Ò ÉH0 ,ùÒ´…¼[Ü`Ýu . n ž, Dn, α ®› ¤L, „NL13. Pearson χ2 u †… #xâÅu ' : ŒNþŒ±ù `: 3oNX •˜‘…nØ ©Ù• ®• ëY©Ùž, … #xâÅu `uχ2 u . ù´Ï•: (i) Pearson χ2 Ú OþƒŠ•6ur(−∞, +∞)©•r‡«m äN©{, •)r À Ú«m ˜, …¼å lsup |Fn − F0 |Kvkù‡•65. (ii) ˜„`5…¼•{•Oår. •Ò´`, 3F0 Ø´o NX ©Ùž, ^…¼u { N´uy. ,˜•¡, Pearson χ2 u •k§ `:: (i) oNX ´õ‘ž, ?n•{†˜‘˜ , 4•©Ù /ª•†‘êÃ'. (ii) cÙ-‡ ´: éunةٕ¹™•ëêž, χ2 u N´?n, …¼•{?nå5éJ. ! d’ âÅu Xi1 , · · · , Xini •Ägäk˜‘ëY©ÙoNFi {ü‘Å ,i = 1, 2,…Ü Õá. F1 (x), F2 (x)´™• ü‡ëY¼ê. •Äu ¯K H0 : F1 (x) = F2 (x), −∞ < x < +∞. (2.6) F1n1 ÚF2n2 ©OPùü| ² ©Ù¼ê, - Dn+1 , n2 = sup (F1n1 (x) − F2n2 (x)) , −∞<x<+∞ Dn1 , n2 = sup |F1n1 (x) − F2n2 (x)| . −∞<x<+∞ c€éêÆ[d’ âÅ(Smirnov)u1936cy² e (J: ½n6.6.2 b (2.6)¤á, Kk 2 n1 n2 1 − e−2λ , x > 0; lim P D+ ≤x = n1 →∞ n1 →∞ n1 + n2 n1 , n2 0, x ≤ 0, n1 n2 lim P Dn , n ≤ x = K(x), n1 →∞ n1 →∞ n1 + n2 1 2 6 

7 .Ù¥K(x)†(??)Ó. Dn+1 , n2 ÚDn1 , n2 ©O¡•ü>ÚV> d’ âÅu ÚOþ. XJ‡u b ´(2.6), Dn1 , n2 Š•u ÚOþ, K Dn1 , n2 > Dn1 , n2 ;α žÄ ½ H0 . .Š n1 n2 Dn1 , n2 ;α = λ , n1 + n2 Ù¥λ ŠŒdNL14 Ñ. ùÒ´d’ âÅu . eb u ¯K• H0 : F1 (x) ≤ F2 (x) ←→ K : F1 (x) > F2 (x), x ∈ (−∞, ∞), K^Dn+1 ,n2 Š•u ÚOþ. n! 5u ∗ 3¢SóŠ¥~~‡u ˜‡‘ÅCþ´ÄÑl ©Ù, ù ‰ 5u . c¡ 0 Pearson χ u !…¼u { 2 ,Œ±¦^. ´duþã•{´Ï^ , ·^¡ 2, k é5Ør ":. ù •{Ñvk¿©|^ b ¤áž &E, u õ Øp. é ©Ù Œ±é éùaA½©Ùõ p u . e¡0 ü«ÄugSÚ Oþ 5u : ( Œ 33–50ƒm) W u ÚŒ ( Œ 350–1000ƒ m) Du Œ±ŽÑþã":, Jpu õ . ùü‡•{® \·IÚO•{ I[ IOGB4882-85ƒ¥, „ë•©z[9]. 1. W u (Wilku ) •Äu ¯K: H0 : X Ñl ©Ù ←→ H1 : X ØÑl ©Ù. (2.7) X1 , · · · , Xn •5g oNX ∼ N (µ, σ 2 ) , X(1) ≤ · · · ≤ X(n) •ÙgSÚOþ. Yi = (Xi − µ)/σ, i = 1, · · · , n, KY1 , · · · , Yn i.i.d. ∼ N (0, 1). - X(i) − µ Y(i) = , ei = X(i) − E(X(i) ), σ mi = E(Y(i) ), i = 1, 2, · · · , n. 5¿m1 , · · · , mn ´†µ, σ 2 Ã' (½ ê. w,k X(i) = µ + σmi + ei , i = 1, 2, · · · , n, (2.8) Ù¥e = (e1 , · · · , en ) ´þŠ•0, • •V n‘•þ. Š˜† ‹IX, î¶L«X(i) ,p¶L«mi .d(2.8)Œ„,3ù‡‹IX¥(X(1) , m1 ), (X(2) , m2 ), · · · , (X(n) , mn ) ATŒ—¤˜^†‚, ‡ O´d‘ÅØ ei E¤ . N On‡:´ ÄCq3˜^†‚þQ? ·‚Œ±OŽ˜eX = (X1 , · · · , Xn ) Úm = (m1 , · · · , mn ) ƒm ƒ'XêR, n 2 (X(i) − X)(mi − m) i=1 R2 = n n . (X(i) − X)2 (m(i) − m)2 i=1 i=1 7 

8 .w,0 ≤ R2 ≤ 1, R2 C1, X†m ‚5'X ²w. Ïd H0 ¤á, ÃXi ÑlN (µ, σ 2 )ž, R2 C1. Œ„ R2 < c (c• ê, –½)ž–•uĽH0 . duN (0, 1)´é¡©Ù§¤±(Y(1) , · · · , Y(n) )†(−Y(n) , · · · , −Y(1) ) kƒÓ éÜ©Ù§l Y(k) †−Y(n+1−k) Ó©Ù, mk = mn+1−k , k = 1, · · · , n, m = 0,u´ n 2 [n/2] 2 mi X(i) bi (X(n+1−i) − X(i) ) i=1 i=1 R2 = n n = n , (2.9) (X(i) − X)2 m2i (X(i) − X)2 i=1 i=1 i=1 n d?bi = mn+1−i i=1 m2i .ÏdW = R2 ŒŠ•u ÚOþ. g™ì(Shapiro)Ú% Ž(Wilk)é(2.9)Š ? u ÚOþ(•„ë•©z[4] P294 ): [n/2] 2 n 2 W = ai X(n+1−i) − X(i) X(i) − X (2.10) i=1 i=1 3n ≤ 50ž, {ai : i ≤ [n/2]} Š®›¤L, •„NL15. úª(2.10)Œ±^5{zÚOþW OŽ. Œ±y²,u ÚOþW ˜‡-‡5Ÿ: =3 b H0 ¤áž, W ©Ù=† N þnk'(•„[4]¥Ún5.5.4). Ï 3?Øk'ÚOþW ¯KžÃ”b½ 5gN (0, 1)© Ù. Xc¤ã, W ´n‡êéƒm ƒ'Xê ²•, Ïd0 ≤ W ≤ 1.d‚5 .nØŒ• 3 b H0 e, ùn‡êéƒmÄ þ•3‚5'X, W ŠA Cu1. Ïd, ‰½u Y²α , u ¯K(2.7) W u ´ W ≤ Wα ž, ĽH,ÄK ÉH. (2.11) Ù¥W Uúª(2.10)OŽ, .ŠWα ŒdNL16 Ñ. NL16´ŠâW ©Ù=† Nþnk ' ù‡5Ÿ, |^‘Å [{?› ¤ . ~6.6.2 • u ˜1u•<¥ˆ¬< |ØrÝ Cz´ÄÑl ©Ù, l¥‘ Å 10¬ |ØrÝê(d Œü ) •: 57, 66, 74, 77, 81, 87, 91, 95, 97, 109 Áu ù êâ´Ä† ©Ùƒ ?(α = 0.05) ) òêâW\eL i x(i) x(11−i) x(11−i) − x(i) ai 1 57 109 52 0.5739 2 66 97 31 0.3291 3 74 95 21 0.2141 4 77 91 14 0.1224 5 81 87 6 0.0399 8 

9 .Ù¥, ai ù˜ ŠdNL15Šân = 10 . ²OŽ 10 10 1 1 x(i) = 834, x(i) = x = 834 = 83.4 , i=1 10 i=1 10 10 10 (x(i) − x)2 = x2(i) − 10x2 = 71736 − 10 × 6955.56 = 2180.4 , i=1 i=1 5 5 2 ai (x(11−i) − x(i) ) = 46.494, ai (x(11−i) − x(i) ) = 2161.692 i=1 i=1 u´k 5 2 10 2161.7 W = ai (x(11−i) − x(i) ) (x(i) − x)2 = = 0.99 i=1 i=1 2180.4 dα = 0.05, n = 10, NL16 W0.05 = 0.842 < 0.99 = W,¤±ØUáý 5b½. 2. Du u ¯KÚ EXW u ¥¤ã. W u ´k , ŒJ§•·^u Nþ3 ≤ n ≤ 50 . n > 50žéJOŽNL15¥ ƒA Š. •d<‚JÑ Du . ˆ{d. B(Dagostino)ïÆ3n > 50ž^ n n+1 (i − 2 )X(i) i=1 D= (2.12) √ n ( n )3 (X(i) − X)2 i=1 Š•u ÚOþ. dd Ñ u •{, ¡•Du . Œ±y², 3 b H0 ¤áž, D ©Ù=† Nþnk', … √ E(D) ≈ 0.28209479, V ar(D) ≈ 0.02998598/ n òDIOz √ n(D − 0.28209479) Y = 0.02998598 Œ±y²: 5b½H0 ¤á, …n → ∞žk L Y −→ N (0, 1) ´ÚOþY ª•uIO ©Ù „Ýéú, ±—un = 100ž, Y ©Ù†IO ©Ù EkØŒ Ñ . Dagostina ^‘Å [{¼ Y © êŠ(„NL17) . Œþ [L², 3H0 ¤áž, Y Š8¥3"†m, 3 5b½Ø¤áž, Y ŠØ´ Ò´ Œ, Ïdu ¯K(2.7)Y²•α Du ´ Y ≤ Y1−α/2 ½Y ≥ Yα/2 ž, ĽH;ÄKÒ ÉH. (2.13) Ù¥Yα/2 ÚY1−α/2 ©O´Y þýα/2Ú1 − α/2 © ê, ÙŠŒlNL17 Ñ. 9