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(),35,1,19991ActaScientiarumNaturaliumUniversitatisPekinensis,Vol.35,No.1(Jan,1999)1)(19471005)2)..1987:1997209209;:1998207216(MTBF)1)(,,100083)MTBF(),MTBFMTBF;;;;O213120MIL2HDBK2781(1987)2),(Assurancetest),(,,,)(GJB899),MTBFMIL2HDBK2781,(),,MTBF(Meantimebetweenthefailures),MTBF,MTBF()1()X,P(Xx)=1-exp(-xöH),x0;0,x0(111)HMTBF,MTBF:rT,0rT2r,T,r,,r,,r,;r,,T,r,;T,r,:MTBFMTBF,,:1r,MTBF,,,,MTBFx1,x2,,(8,F,PH)(H0),(111)sn=6ni=1xi,n=1,2,,(s0=0)xi=xi(X)(X8),i1A={X:vn,n1,xnr,sn-1T-r}B={X:vn,n1,xir,i=1,2,,n,snT-r}AB,P(A)+P(B)=1(211)wA,S(X)=min{n:n1,xnr,sn-1T-r},wB,S(X)=min{n:n1,x1r,,xnr,snT-r},wöAB,S(X)=F(),:F=sS-1+r,XA;sS,XB;T,XöABN(t)=sup{n:snt,n0},(t0),N(t)[0,t],F=sN(F)+r,XA;sN(F),XB;T,XöAB42()35,(N(t),0tF)(112)(112)H(MTBF)Ft=R(N(u):0ut),F=R(N(u):u0),Ft8(N(u),0ut)R,F(N(t),t0)RH,111111F(N(t),Ft,t0),t0,{Ft}Ft,T-rFT:F=sN(F)+r,FrT-r,F=sN(F),F=sN(T-r)+1T-rFT-r,N(F)=N(T-r),F=sN(F)+r=sN(T-r)+rT-r+r=TN(F)N(T-r),F=sN(T-r)+1=sN(T-r)+xN(T-r)+1T-r+r=TT-rFTtTtT-r,{Ft}FtT-rtT,{F=sN(F),Ft}Ft:{F=sN(F),Ft,N(F)=k}={sk-1T-rsk,xkr,skt}=n=k{sk-1T-rsk,xkr,sntsn+1}:{sk-1T-rsk,xkr,sntsn+1}R(s1,,sn){sntsn+1}=Ft{N(t)=n}Ft,(nk)R(s1,,sn)s1,,snR,{F=sN(F),Ft,N(F)=k}Ftk,{F=sN(F),st}Ft,{FsN(F),Ft,N(F)=k}={sk+rt,xk+1r,skT-rsk+1}={max(sk+r,T-r)t,max(sk+r,T-r)sk+1}=T-rrit{max(sk+r,T-r)t,max(sk+r,T-r)risk+1},ri=T,R(s1,,sk){skrisk+1}=Fri{N(ri)=k}Ft,{FsN(F),Ft,N(F)=k}Ft{Flt}={F=sN(F),Ft}{FsN(F),Ft}Ft,FFFFR,FF={+:+F,Pt0,+{Ft}Ft}(F,N(F))FF[1],FFPHPH0Random2Nikodyn:LF¦dPHdPH0(FF)=H0HN(F)exp-1H-1H0F521:(MTBF)(F,N(F))(N(t),0tF),HH^:H^=FN(F)(113)t0=2Z=(F,N(F)),F,N(F)[1]H,ZE1={(t,i):rtT,i=1,2,},E2={(t,i):T-rtT,i=1,2,}E¦E1E2ZZE1,E;:(t,i)E,(t,i)E,titi,(t,i);(t,i),,;E1HH^=FN(F),(F,N(F));(t,i)H^tiG(u,H)=PH(H^u),H(u,H)=PH(H^u)PH(A)H,AA(0,1),HL(u)=inf{H:G(u,H)A},(2.1)HU(u)=sup{H:H(u,H)1-A}(2.2)[1]HL(H^)HU(H^)H1-A,H0:PH(HHL(H^))1-A,PH(HHU(H^))1-A[1]HL(H^)(HU(H^))H^1-A()()G(u,H)H(u,H),(211)(212)HL(H^)HU(H^),HL(H^),HU(H^),,211PH(A)+PH(B)=1[2],PH=e-rH1+T-rH,PH(B)=PH(vn1,x1r,,xnr,T-rsn)=6k=1PH(x1r,,xkr,sk-1T-rsk)=PH(T-rx1r)+6k=2[PH(x1r,,xkr,sk-1T-r)-PH(x1r,,xkr,sk-1T-r,skT-r)]62()35=exp-T-rH-exp-rH+1-exp-rHT-rH-T-rH-1+exp-T-rH=1-T-rHexp-rH-exp-rHPH(A)+PH(B)=1212x1,x2,,xn(8,F,PH)(H0),(111),x1+x2++xnFH(n,x)=1-6n-1k=0e-xHxHkök!(x0)[3]182111,x1+x2++xn:fn(x)=e-xH1Hnxn-1(n-1)!,x0;0,x0FH(n,x)=x0e-tHH-ntn-1ö(n-1)!dt=-e-xHxHn-1ö(n-1)!+x01He-tHtHn-2ö(n-2)!dt,FH(n,x)=-e-xHxHn-1ö(n-1)!+FH(n-1,x)FH(1,x)=1-e-xH,FH(n,x)=1-6n-1k=0e-xHxHkök!213G1(u,H)=PHFN(F)u,XA=e-rH1+6nröuFH(n,T-r)+6röunTöu[FH(n,T-r)-FH(n,nu-r)]G1(u,H)=PHFN(F)u,XA=e-rH+6n=2Psn-1+rn-1u,snT-r,xnr=e-rH1+6n=1PH(snun-r,snT-r)=e-rH1+6nröuFH(n,T-r)+6röunTöu[FH(n,T-r)-FH(n,nu-r)]214G2(u,H)=PHFN(F)u,XB=PH(x1u,x1r,x1T-r)+6n(T-r)öu-1[(1-e-rH)FH(n,T-r)-FH(n+1,T-r)]+6(T-r)öu-1nröu-1e-(n+1)u-(T-r)H-e-rHFH(n,T-r)721:(MTBF)-e-(n+1)u-(T-r)HFH(n+1,T-r)G2(u,H)=PHFN(F)u,XB=6n=1PHsSSu,B,S=nsST,G2(u,H)=6ntTöuPHsSSu,XB,S=n=6nTöuAnA1=PH(x1u,x1r,x1T-r)3:1)nT-ru-1,(n+1)uT-r:An=PH(sn+xn+1T-r,xn+1r,snT-r)=T-r0PH(rxn+1T-r-x)dFH(n,x)=T-r0e-T-r-xH-e-xHdFH(n,x)=(1-e-rH)FH(n,T-r)-FH(n+1,T-r)2)T-ru-1nru-1,T-r(n+1)ur:An=PH(sn+xn+1(n+1)u,xn+1r,snT-r)=T-r0e-rH-e-(n+1)u-xHdFH(n,x)=FH(n,T-r)e-(n+1)u-(T-r)H-e-(n+1)uH-e-(n+1)u-(T-r)HFH(n+1,T-r)3)ru-1nTu-1,r(n+1)uT:An=PH(sn+xn+1(n+1)u,xn+1r,snT-r)=PH((n+1)u-snxn+1r,snT-r)=T-r(n+1)u-re-(n+1)u-xH-e-rHdFH(n,x)=e-(n+1)u-(T-r)H-e-rHFH(n,T-r)-e-(n+1)u-(T-r)H(FH(n+1,T-r)+e-rHFH(n+1,(n+1)u-r))1),2),3),214211214211211G1(u,H)=PHFN(F)u,XA,G2(u,H)=PHFN(F)u,XB,G(u,H)=G1(u,H)+G2(u,H),u0,G1(u,H)=e-rH1+6nröuFH(n,T-r)+6röunTöu(FH(n,T-r)-FH(n,nu-r)),G2(u,H)=PH(x1u,x1r,x1T-r)+6n(T-r)öu-1[(1-e-rH)FH(n,T-r)-FH(n+1,T-r)]+6(T-r)öu-1nröu-1e-(n+1)u-(T-r)H-e-rHFH(n,T-r)82()35-e-(n+1)u-(T-r)HFH(n+1,T-r)+6röu-1nTöu-1e-(n+1)u-(T-r)H-e-rHFH(n,T-r)-e-(n+1)u-(T-r)HFH(n+1,T-r)+e-rHFH(n+1,(n+1)u-r)FH(n,x)=1-6n-1k=0e-xHxHök!,(x0)212,[4](E,B),(8,F,PH)(H(a,b),-ab+),{PH}RM,Y(8,F,PH)E,M()L(y,H)RL(y,H)0,lnL(y,H)Hy,9lnL(y,H)9H=0,H^(y),H=H^(y)92lnL(y,H)9H20HH^:9lnL(y,H)9H=0,H=H^(y);9lnL(y,H)9H0(H),H^=b;9lnL(y,H)9H0(H),H^=a;:y1,y2E,H(y1)H(y2),y1;y2[4]:215()R,;PH(Y;y)H(a,b),I={H:0PH(Yy)1}212u0,G(u,H)H(0,)Y=(F,N(F)),EY,M=PH01,LF=H0HN(F)e-1H-1H0F,lnLF=N(F)lnH0H-1H-1H0F9lnLF9H=0H=FN(F)92lnLF9H2H=H^=N(F)H2-2FH3H=H0limH0e-xH=0,limHe-xH=1,limH0FH(n,x)=1,limH0FH(n,x)=0,(Px0)limH0G(u,H)=0,limHG(u,H)=1,I(y)=(0,)215,G(u,H)(0,)921:(MTBF)213A(0,1),HL(u)(211),u0,HL(u)G(u,H)=A(2.3)211u0,G(u,H)H,(213)HL(u)018,T=4130,r=2122(:h)u,HHL(H^)1:1Table1NumericalvaluesoflowerconfidencelimitsuHLuHL1.20.7032.61.0811.40.7992.81.1241.60.8603.01.1651.80.9043.21.2032.00.9383.41.2392.20.9863.61.2732.41.0353.81.305,1..,1993,(6):5425522.MTBFMIL2HDBK2781.,1996,2:29313..:,19934SunWanlong.ConfidenceLimitsofParameterinStatisticalModelswithOneParameter.TechReport,1996ConfidenceLimitsaboutTheMeanTimebetweentheFailures(MTBF)inAssuranceTestZHANGXiao(Dept.ofBiomathematics&Biostatistics,BeijingMedicalUniversity,Beijing100083)AbstractOnthebasisofstrictmathematicaldescriptionabouttheassurancetest,theoptimallow2erconfidencelimi