Reliability Analysis of DOOF for Weibull Distribution

http :/www. zju. edu. cn/jzus; E-mail : jzus@ zju. edu. cnISSN 1009 - 3095 Journal of Zhejiang University SCIENCE V.4 ,No.4 ,P .448 - 453 July - Aug. , 2003Reliability analysis of DOOF for Weibull distribution'CHEN Wen-hud陈文华y* ,CUI Ji(崔杰) , FAN Xiao-yau(樊晓燕)LU Xian-biad卢献彪} , XIANG Ping相平尸(' The State Key Laboratory of Fluid Pouver Transnmission and Control , Zhejiang University ，Hangzhou 310027，China ;? Hangzhou Electrical Connector Factory , Hangzhou 310015 , China )↑E-mail : chen-wenhua @ sohu. comReceived Oct.21 2002 ; revision accepted Jan.8 2003Abstract : Hierarchical Bayesian method for estimating the failure probability p; under DOOF by taking thequasi- Beta distribution B( Pi-1 ,1 ,1 ,b )as the prior distribution is proposed in this paper. The weighted LeastSquares Estimate method was used to obtain the formula for computing reliability distribution parameters andestimating the reliability characteristic values under DOOF. Taking one type of aerospace electrical connectoras an example，the corectness of the above method through statistical analysis of electrical connector acceler-ated life test data was verifed.Key words : DOOK Data only one failure ) data , Hierarchical Bayesian estimate , Reliability analysisDocument code : ACLC number : TB114.3INTRODUCTION .Bayesian method for estimating failure probabili-ty by taking the quasi- Beta distribution as theStatistical analysis of reliability test data prior distribution on the basis of zero-failure datashowed that when the failure number exceeds 2，analysis method and validate this method throughthere are many tested methods for processing this statistical analysis of testing data .problem( Zhang et al. 1989 ). However , in thereliability test of product , with the appearances DATA MODEL AND STATISTICAL ANALYSISof high reliability units , even in the accelerated METHODS OF DOOFlife test , it may occur that none of the test unitsfail or only one unit fail before the predetermined1. DOOF modeltest time is reached. Such phenomenon is com-mon especially in the case of small samples .We assume that specimens are put on timeMost papers ( Liu et al. , 2001 ; Anderman et censored life test inspected interval with a prede-al. , 1997 ; Pugh , 1993 ) only mentioned Zero- termined time. The inspected time is noted asFailure problems. Sandoh et al.( 1991 ) and 0< τ1< τ2 < ... τh. Such test can yield DOOFBailey( 1997 ) proposed some theories and meth- defined by( s; r; ,T; ) when only one item failedods using statistical analysis to deal with prob-in the interval of ( Tm-1 ,τm ) and the rest oflems of Zero-Failure. However ,none of the reli-items are in good condition. Here s; is the speci-ability statistical analysis methods of DOOF( Da-men number and r; is the failure number at timeta Only One Failure ) were considered. In order中国煤化工:0 ,or else if i> mto take full advantage of information on the prod-uct failure and increase the precision of reliabili-.MYH.C NMH GS2≥..≥Sh.ty evaluation , it is necessary to study the reli-We can obtain the tollowing information fromability statistical analysis methods of DOOF. In the above DOOF model :this paper , the authors propose the hierarchical( 1 ) Product failure probability Po = 0 whenPrijed?京多携o81 suppoteol by the Naioal Naural Sciene Foundaion of ChinaReliability analysis of DOOF for Weibull distribution449(2 ) The failure probability is defined as BAYESIAN ESTIMATE OF FAILURE POSSIBIL-p;= P( T< τ; ). It is reasonable that we accept TY p; UNDER DOOFP1≤P2≤...≤Pk for 0< T1 < T2 <... Th. Thelarger Sμ is ,the smaller p( i= 1 2.. ,h )are.Among the methods for estimating failure( 3 )The life of all products is not longer than probability p; , the classical estimate method is .Tmbecause one item fails in time period simple in calculation. However , it yields the es-( Tm-1 rtm).timate of p; ，defined by p; = 0.5Ks;+ 1 )( i=1 2.. h ),which is derived from the es-2. Feasible statistical analysis methods of DOOFtimate method under failure data and the result isAt present ，statistical analysis methods inlower precision and cannot truly reflect the reli-the case of DOOF include minimum x2 method, ability level of the produet. So , the Bayesianequivalent number of failures method and weight-method is more attractive. Bayesian analysis ined Least Squares Estimate method. Practical in- volves expression of subjective knowledge or de-formation indicates that the estimates of reliabili- gree-of-belief about model parameter values as aty obtained by the x2 method and the equivalent prior distribution for them. This distribution isnumber of failures method are relatively low then mathematically combined with observed da-( Mao et al. , 1993 ). The reliability estimate by ta to yield the posterior distribution of the param-minimum x method improved by Bayesian x"eter values. The posterior distribution is narrow-method tends to be higher. Compared with mini-er than the prior one , thereby reflecting the add-mum x2 method , equivalent number of failuresed information from the data. The posterior dis-method is better ; but its estimate of reliability istribution yields a Bayesian estimate and proba-prone to be higher. Furthermore , the calculationbility for the true parameter values and theirof this method is laborious and the judging of da-functions. Generally speaking , the estimate per-ta type is necessary. It is difficult to analyze re-formance of the Bayesian method is better. How-liability under DOOF because of the limitation ofever , we prefer to use the classical method in-stead of the Bayesian method , if the prior distri-the equivalent number of failures method.The weighted Least Squares Estimate methodbution is not chosen properly. Therefore , theis a reliability analysis method that reaches thechoice of proper prior distribution of Pi is essen-reliability target by means of fitting the curve ofdistribution. In comparison with the x2 method1. Establishment of prior distributionand equivalent number of failures method ，theThe fact that in the case of DOOF , none ofweighted Least Squares Estimate method , be- the samples fail in the time period (0 r心i )cause of its simplicity and estimate precision ，showed that the reliability in the time period( 0 ,has become the most commonly used method for τ; ) might be very high. So at time τ; , the pos-treating DOOF. Because the weighted Least sibility that failure probability p; is small is verySquares Estimate method is based on the reli- large and that p; is large is very small. In prac-ability analysis method under failure case ,it can tice , quasi-Beta distribution can be taken as thedirectly be used in resolving the problem under prior distribution of p;. Its density function isDOOF. The steps are as follows :( 1 ) Estimateof failure probability p; = P( T< τ; ),fp:8 02 a b;_ (p:-0.)-(02-p:)-1i=12..,h:,attimeτ;.(2)Fitacurveof中国煤化工=b002-01)y+6-T '(1)distribution to the points( τ; p; ),by using theMYHCNMH Gweighted least squares theory. ( 3 ) Estimate the From the above function ,it can be seen that it isreliability from the fitted distribution curve. The a strictly monotonously decreasing function for P;second and third steps are not difficult. The cru- when a and b are constant. It can not only sati-cial point is the first step , how to estimate the sfy the requirement that the possibility that fail-probabil语有效据p( T< τ; ).ure probability p; is small is very high and that450CHEN Wenhua ,CUI Jie et al.p; is large is very small , but also can deal with mate , the thin tail distribution often makes themost testing requirements by changing the inte- Bayesian estimate have poor robustness. Hence ，gral interval of( θ ,θ2 ). Especially when the we choose hyperparameter a as 1.number of sample running in the test does notIt is difficult to confirm the exact value of bvary , Bayesian estimate of p:( i=1 2... ,k: ) further. Though it can be decided on the basis ofcan also meet the requirement that p1 < p2 < ...expert experience ，cases may happen that the< pk. Thus , it is reasonable to take quasi- Betaexpert is not experienced enough to give a valuedistribution B( P;-1 ,1 ,1 ,b )as the prior distri-for b. When a prior distribution contains a hy-perparameter , giving another prior distribution tobution under zero-failure data .In the case of DOOF , though choosing qua-the hyperparameter may lead to a more robust re-si- Beta distribution B( P:-1 ,1 ,1 ,b )as the priorsult by two prior distributions than by one priordistribution. In our problem , the prior distribu-distribution will easily lose the important infor-tion of the hyperparameter b could be chosen asmation that one item has failed when i> m- 1 ;the uniform distribution U( 1 ,C ) , denoted asthe loss can be avoided by modifying the lowerbound value p; . So quasi- Beta distributionπi2(b)=U(1,C)(2)B( Pi-1 ,1 ,1 ,b )can still be chosen as the prior Here C is constant. According to the character-distribution under DOOF .istic of quasi-Beta distribution density function ,the larger b is( under the condition of a = 1 ) ,2. Choosing hyperparameterThe density function of quasi- Beta distribu-the thinner is the right tail of the quasi-Beta dis-tion is an exact monotonously decreasing functiontribution density function ( see Fig. 2 ). Like-of p; when a≤1，b > 1. Such characteristicwise , considering the robustness of the Bayesianestimate，C cannot be taken too large and it iscorresponds to the prior information that failuresuitable to take3-7 as C( Mao etal. , 1996 ). .probability P: is small is very large and that Pi islarge is very small. In this way , we can get afrough scope of a and b. That is ，a≤1 ,b> 1.From Fig.1 ,it can be seen that ,when b is con-stant and a≤1 , the smaller a is , the thinner is2.sFthe right tail of the quasi- Beta distribution dens-25a=1 0;b=4.01.5a=1.0:b=3.0a=1.0;b=2.0作a=1.0a=1.0;b=1.50.5Eb=5.00.20.40.61.0a=0.6b-2.0Fig. 2Density function of quasi-Beta distributiona=0.8when a=1 ,b> 1b=2.0a=0.2t b=2.0pH Aeresent the lower and0.40.60.8upP中国煤化工Pi respectively ,it isreasJIYTHC N M H GThat means the max-Fig. 1 Density function of quasi-Beta distributionimal failure probability is equal to 1. It wouldwhen 0≤a<1 ,b=2naturally be better with an acute upper bound ofity function. According to Berger' s ( 1985 )P: through expert experience. The value of θ1viewpoin万右数据robustness of the Bayesian esti-should accord with the inequality θ≥pi-1，Reliability analysis of DOOF for Weibull distribution451which assures that no backhang phenomenon ex- lihood function isists, so that p1≥p2≥...≥p:. Thus , usingEq.( 1 ) we can obtain the first stage prior dis-[(1 p; )=s;p( 1-p; )-1( 8)tribution at once as follows .Suppose the product fails at time τa ，whereπ;( p:Ip:-1 ,1 ,1 ,b)=( 1-p; )-1m-1 < a < m. In terms of gambling theory ,B( 1 ,bX 1-pi-nYsome failure phenomena can occur to a certain(3 )extent at certain time while products have notfailed at other time. Therefore ，it is reasonableIt is not difficult to generate the prior distributionto suppose the failure probability pa at time τ。of Pi by using the Bayesian formula below :equals mad( pm-1 0.5 ). By using Eq. ( 1 ) weA(p;1p-)=can obtain the prior distribution of Pm1pc( 1- p;)-1C- 1小B(1 ,bX1-p-_yd0(4)(1-pm)-1db(9 )C-1小1 B1 b11-p。YIt is obvious that the above formula is a monoto-It is reasonable to take Eq. ( 9 ) as the prior dis-nously decreasing function of p; .tribution of Pm because it is a monotonously de-3. Bayesian estimate of failure probability p;creasing function of Pm .Suppose we do a life test with s; products ，Under square error loss , the Bayesian esti-and that r; products fail in the time period mate of Pm is its posterior expectation ,i.e.(0 rt; ). As the failure probability of each prod-pm=EpmIsmITm=1)=uct is P; , the likelihood function can be repre-I p2(1-pm)y; +b-2sented by the binomial distributionB161-p。ydpmdb= 1-(1-pa)si,pm(1- Pm):+b-2Ir;p)=|p:( 1-p;)-'(5)J。B1b1-p。ydpndbWhen r; = 0( under the condition of i < m )，+ C+1s;+Cthat means the likelihood function(C-1)pa+(1-pa X1+s; )In-8;+2-s;InS;+1I(0 p; )=(1- p;)(6) ( C-1)p。+( 1-p。)s; lnS;+ C-15+Hi-(s;-1)InsUnder square error loss , combining likelihood( 10)function Eq.( 6 ) with prior distribution Eq.( 4),After acquiring the Bayesian estimate of Pm , wethe Bayesian estimate of P; can be obtained ascan obtain the hierarchical Bayesian estimate offollows .p;( i> m ) under DOOF on the condition ofp; = E(p;|s;)=square error loss .pc plp(1- p:)J1Jp-1 B(1 ,b11-p-Y;dbp;=Ep;Is;r;=1)== pi-1 +p(1- p; ).+b-2pc p1(1-p;Y+9,-TIJo-1区1b美1-P2 ydp,db1 B(1 bX1-pi-1)ydp,db= 1-(1-pi-1)s;+C+1D(1-D)+6-2(1 - ρ-1(1+ s; )ns;+2中国煤化工dbsIrs:+C)(7)(C-1p帕+1-pHy1+5i川DHC NMHGs;+C+1.到+9]/(C-.-silIn5+1)S;+2-s;lnS;+S;+ CS;+C-1If r;= 1( under the condition of i≥m ), it (C-1)2+(1-pu );Ints;+1( s;-1)n-means one product fails with s; products in the(11)test with海有数据ccording to Eq. ( 5 ) , the like-452CHEN Wenhua ,CUI Jie et al.W;X; ,B =w;x? ,C =, W;Yi，RELIABILITY DISTRIBUTION PARAMETERSAND ESTIMATE OF RELIABILITY CHARAC-D=W;x;y; and w; is weight.TERISTIC VALUES UNDER DOOFIn the analysis of the calculating process ofAfter obtaining the Bayesian estimates of allthe weighted least squares method，it can bethe probability P; = P( T< t; ) ,we can estimateseen that it is not T; ,but lnt;，that plays a role .reliability distribution parameters and character-in the calculation. So the weights are chosen byistic values using the weighted I east Squares Es-W; =s;Inτ;timate method.silnτ1 + s2lnτ2 + ... + SμlnτkSuppose the product' s life T follows Weibull(i=12..k)(13)distribution ,whose Cumulative DistributionAgain from the estimates of m and η , weFunction iscan obtain the estimate of reliability at time t .Ft)=1-exp[-(t/η)"](t>0)R( l)=exp[ -( t/介)" ](14 )Herem>1isashapeparameterandη>0isthe characteristic life .Again, suppose that the product' s failureEXAMPLE AND CONCLUSIONprobability is P;，at the time point t = T;，andthat p; is its estimate. Then we haveNow , take the model Y 11X -2221 aerospacep;=1-exp[ -( τ;/η)"]( i=12..h)electrical connector accelerated constant-stresslife test under vibration stress as an example .andFour groups are independently put on time cen-sored life test inspected interval with a predeter-lnτ;= lnη+-l[ -lr( 1-p; )]mined time under different stress levels. Fiven"( i=12..h)samples were tested when the power spectrumdensity was 0.2 g/Hz , five when it was underLet y;=lnτ; ,x; = ln[ - In( 1- p; )], then we0.4 g/Hz , five when it was under 0.6 g/Hzhave yi=-x; + lnηand six when it was under 1.0 g/Hz. The test-Replace x;=lr[ -1n( 1- p; )]in the above for-ing result yielded complete failure data understress levels of 0.4 g/Hz ,0.6 g/Hz and 1.0mula by x;=lr[ -Ir( 1-p: )]to yieldg2/Hz and DOOF under the stress level of 0. 2g/Hz. By the fixed-time of τ( τk = 85h ) , onlyy; =一x; + lnη+ ε;none product failed because its contact resistanceHere ε; is the deviation resulting from the re- exceeded the criterion 3mQ ( GJB101-86 ,1986 ). The data are shown in Table 1placement of Pi by pi.Because the life of the aerospace electricalIn order to estimate the two parameters mand η of Weibull distribution , we can use theconnectorfollowed two- parameter Weibull distri-weighted least squares method. Hence we canbution whose shape parameter was m > 1 ( Chenobtain the weighted Least Squares Estimatorsm，etal. ，1997),we can treat the data with theabove theory and method. In calculation ，we斤of m and η in the following formulaconsider. C=_4.5 _6 The estimates of failureu(y;-上x,- lnn)→min\2prol中国煤化工,h ) and the weightsn~'uncCNMHGarealsoshownifrom which we obtainTable 1Combing the data in Table 1 with Eq. ( 12 )B- A2BC - AD\m=D-AC，力=exp(B-A2,( 12 ) and calculating by computer program , we ob-tained estimates of the distribution parameters .HereWhenC=4，wegotm=3.1609,Reliablity analysis of DOOF for Weibull distribution4539=72.5182. When C=5we gotm=3.2289，3.2891 ,分=75.3594.9=74.8714 and when C=6wegotm=Table1 DOOF and the estimate of failure probability at time r;TestingFailureThe sampleThe estimate of failure probabilityintervalnumberInτ;number byWeight(h) .the time t;C=4.C=5C=60- 383. 637650. 14330.11580.10910. 103238-443.7842，0. 14910.21710.20630. 195744-6104.11090.16200.30730.2929 ,0.278761-674.20470.13250.58310.57760.572567-734.290540.13520.65080.64180.633273- 794.36940.13770.70660.69540.684679- 854.44270. 14000.75290.74050.7283The resuls showed that different valuesof Cmittee of national defense( in Chinese ).had different , but not great influence on the esti- liu, F. H.，Lu, J. C.，Sundaram, V. , Sutter, D. ,mate. It agrees with the fact that the possibilityWhite ,C. and Baldwin ,D.F. ,2001. Reliability as-sessment of microvias in HDI printed circuit boards.of probability P: being large is small , that is ,ρiEletronic Components and Technology Conference ,51 :