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I25 I2q hh Vol.25No.22002h4 ACTAMATHEMATICAEAPPLICATAESINICAApr.,2002QqYg WK V i aJCd∗yxz(x i ii x 310012)pk C RM X l M H ky H kE m T E ym Mb o sy e XMb E Xb FM X k ym 1mh nc l r k Ga c A n ZG W Tth GY Q dA n Z B n Z n w t ESj N Z n nG o z F [1−4]. nG Z C mh nG M xQ h P A1 nG Z u mh Z C t s u mh Z CGY U RGG A2 nG Z T mh Z CY G A3! J pS0z J pS1S2···SkQi Z CGY Q! J pSiQJjh Z CGY XijGVSw f(xij)=1θijexp(−xij/θij),xij0,i=0,1,···,k,j=1,2,···,m, θij J pSiQJjh Z CY Gp λij=1/θij.A4J pSiQJjh Z CG # lnθij=aj+bjφ(Si),i=0,1,···,k,j=1,2,···,m, aj,bj B# φ(S) J pSGw#w A5 nG$ v $ G Z z$ GJ p X% Hp &GKo %!zo (MLE),&ED& %!zo G!!c #ZN! J pQp z ZQG %!zo ko uk’ ! Vp z ZQG %!zo Nd# D& %!zo G%q& u’ZNd#$o LG! J’W %!zo G! J’ D&Monte-CarloZ#ZN%$ Wq&v%d#&q Z uq&&L(W(’A#S(e d#GZu(ZR )D2000i3)24(’B*2001i5)8(’Bdcf*∗*)(*)+,+,Vb*2q*++, X l kE, K2552PnlDXfHT-.EIm=1 % - $&G n ![5]# P ’A- nH- a/P ’A. n .P ’A- nv% %P ’A. nU&G u 2.1A0NbGL1j J pSiQ nih n#aP ’A n C rih n Z .2 n q/C Z ti1ti2≤···≤tiri(i=1,2,···,k)./a1,b1 a1,b1GMLE,&ywD a1,b1GFisher/N03 I(a1,b1)=⎛⎜⎜⎜⎜⎝ki=1riki=1riφ(Si)ki=1riφ(Si)ki=1riφ2(Si)⎞⎟⎟⎟⎟⎠. %!zo G!!c#$q&&L1% ! lnθ01=a1+b1φ(S0)∼N(lnθ01,σ2),r σ2=G+Eφ2(S0)−2Iφ(S0)EG−I2,E=ki=1ri,I=ki=1riφ(Si),G=ki=1riφ2(Si).AU1 a1,b1 a1,b1 P ’A. n 2G %!zo θ01=exp a1+b1φ(S0),λ01=θ−101, θ01=θ01e−12σ2, λ01=λ01e−12σ2,4! Eθ01=θ01e12σ2,Eλ01=λ01e12σ2,E θ01=θ01,E λ01=λ01,uMSE( θ01)MSE(θ01),MSE( λ01)MSE(λ01).5 lnθ01!!N(lnθ01,σ2),0x! E(θ01)=θ01e12σ2,D(θ01)=θ201eσ2(eσ2−1).!X! MSE(θ01)=(Eθ01−θ01)2+D(θ01)=θ201(e2σ2−2e12σ2+1).6E θ01=Eθ01e−12σ2=θ01,0x! MSE( θ01)=D( θ01)=e−σ2D(θ01)=θ201(eσ2−1). .! MSE(θ01)−MSE( θ01)=θ201 eσ2(eσ2−1)−2(e12σ2−1)θ201 eσ2−1−2(e12σ2−1)=θ201(e12σ2−1)20.7zClnλ01=−lnθ01!!N(lnλ01,σ2),B! A0 p ZQG u 256 gg125 PC#ZN! J pQp z ZQG %!zo Q ko uk’ ! X θ01z λ01 2 θ01zλ01G!Hko 1U VG %!zo 2.2AZNbGL1j J pSiQ nih n#aP ’A n C 3τi .2 n q/C Z ti1≤ti2≤···≤tiri(≤τi)(i=1,2,···,k)./a1,b1 a1,b1GMLE,Ti=rii=1tij+(ni−ri)τi.D&[6,106s]G u&ywD a1,b1GFisher/N03 I(a1,b1)=⎛⎜⎜⎜⎜⎝ki=1ni(1−pi)ki=1ni(1−pi)φ(Si)ki=1ni(1−pi)φ(Si)ki=1ni(1−pi)φ2(Si)⎞⎟⎟⎟⎟⎠, pi=exp(−τi/θi1) J pSiQ’A3Q /E=ki=1ni(1−pi),I=ki=1ni(1−pi)φ(Si),G=ki=1ni(1−pi)φ2(Si),σ2= G+Eφ2(S0)−2Iφ(S0)/(EG−I2),(1)4! lnθ01=a1+b1φ(S0)∼N(lnθ01,σ2).(2) σ2v B# pi,X θi1G %!zo θi1=Ti/ri,0xpiG %!zo pi=exp(−τiri/Ti),&pi44(1) %Dσ2G %!zo σ2.AU2 a1,b1 a1,b1 P ’A. n 2G %!zo θ01=exp a1+b1φ(S0),λ01=θ−101, θ01=θ01e−12σ2, λ01=λ01e−12σ2,4! Eθ01=θ01e12σ2,Eλ01=λ01e12σ2,E θ01=θ01,E λ01=λ01,uMSE( θ01)MSE(θ01),MSE( λ01)MSE(λ01).5 lnθ01!!N(lnθ01,σ2),0x! Eθ01=θ01e12σ2. %!zo G%q& u#$ri→∞ pia.s.→pi,0xri→∞ σ2/σ2a.s.→1.!X!(2) Dln θ01−lnθ01+12σ2σ=lnθ01−lnθ01σ+σ2−σ22σ2σ∼N(0,1),%! ln θ01∼N(lnθ01−12σ2,σ2),r$’ZB! PC1G’Z 2.3AZNb8L1j nh nJ pS1Q#a n C 3τ1 5J p2eCS2,J pS2Qk τ2 5J p2eCS3,5.9CJ pSkQk τk .2:h n J pSiQq/C Z ti1≤ti2≤···≤tiri(≤τi)(i=1,2···,k),r tij!J p2eCSi 3s 2q*++, X l kE, K257/a1,b1 a1,b1GMLE,Ti=rii=1tij+(ni−r)τi.![7]# a1,b1GFisher/N03 I(a1,b1)=⎛⎜⎜⎜⎜⎝nki=1Ainki=1Aiφ(Si)nki=1Aiφ(Si)nki=1Aiφ2(Si)⎞⎟⎟⎟⎟⎠, Ai=(1−pi)ij=1pj,pi=exp(−τi/θi1)(i=1,2,···,k)./E=nki=1Ai,I=nki=1Aiφ(Si),G=nki=1Aiφ2(Si),σ2= G+Eφ2(S0)−2Iφ(S0)/(EG−I2),(3)4! lnθ01=a1+b1φ(s0)∼N(lnθ01,σ2). σ2v B# pi, &piGMLEpi=exp(−τiri/Ti)44(3) Dσ2Go σ2,B! D AU3 a1,b1 a1,b1 P ’A- n 2G %!zo θ01=exp a1+b1φ(s0),λ01=θ−101, θ01=θ01e−12σ2, λ01=λ01e−12σ2,4! Eθ01=θ01e12σ2,Eλ01=λ01e12σ2,E θ01=θ01,E λ01=λ01,uMSE( θ01)MSE(θ01),MSE( λ01)MSE(λ01).;1m=146f7t5lA6p68K978(7E)n:KTUj8TU;9I8N8n1=n2=20r1=r2=10 θ01θ01−0.029920.100450.258080.34105=n3=n4=20r3=r4=12 λ01λ010.036140.175390.345060.473125ln1=n2=40r1=r2=20 θ01θ01−0.007360.056720.133360.15428n3=n4=40r3=r4=25 λ01λ010.011580.076850.145600.17075N4n1=n2=20p1=p2=0.4 θ01θ010.012830.131240.259110.36270=n3=n4=20p3=p4=0.2 λ01λ01−0.009120.095610.223460.271055ln1=n2=40p1=p2=0.4 θ01θ010.015220.070890.121610.14385n3=n4=40p3=p4=0.2 λ01λ01−0.012350.039260.106420.11671N4n=50p1=p2=0.6 θ01θ010.029000.139180.234110.31344:=p3=p4=0.5 λ01λ01−0.028130.072400.211590.260095ln=100p1=p2=0.6 θ01θ01−0.017290.068820.107960.12495p3=p4=0.5 λ01λ01−0.015930.033020.100260.11051258 gg125 QWD&Monte-CarloZ#Z9: u Wq&r;G θ01/θ01, θ01/θ01,λ01/λ01, λ01/λ01G jP’A ri’A3QpiQ% a1,b1Hp . t=cfx # lnθi1=30−5lnS, J p S1=38.6,S2=44.8,S3=51.2,S4=57.6,! J pS0=30.V&Gq&&L(’A z’A3Q 10000?gq& 3o LGp UVk’zp UV J’ ?g u@1.?g uZ!Hko θ01z λ01Gk’z J’1λ01zθ01 NZRd; q&&L(W(’A#S(ed#Zu(ZR 3RnBDXfHT-.EI![2]# F Wh Z CGP ’A- nH- a/P ’A. n .P ’A- nv% %P ’A. nU&G u xQr /vI=S 3.1A0NbGL1j J pSiQ nih n#aP ’A n C rih n Z .2 n q/C Z 2 (tij,cij),j=1,2,···,r,i=1,2,···,k,r tij J pSiQJjh Z nG Z %ti1≤ti2≤···≤tiri,cij J pSiQJjh Z nG Z C /λij=1/θij,λi=mj=1λij,λ0=mj=1λ0j,θ0=1/λ0,pij=λij/λi,gij J pSiQJjh Z