Research Article Weibull Model Allowing Nearly Instantaneous Failures

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1 Hindawi Publishing Corporaion Journal of Applied Mahemaics and Decision Sciences Volume 2007, Aricle ID 90842, pages doi:0.55/2007/90842 Research Aricle Weibull Model Allowing Nearly Insananeous Failures C. D. Lai, Michael B. C. Khoo, K. Muralidharan, and M. Xie Received 27 April 2007; Acceped 8 Augus 2007 Recommended by Paul Cowperwai A generalized Weibull model ha allows insananeous or early failures is modified so ha he model can be expressed as a mixure of he uniform disribuion and he Weibull disribuion. Properies of he resuling disribuion are derived; in paricular, he probabiliy densiy funcion, survival funcion, and he hazard rae funcion are obained. Some seleced plos of hese funcions are also presened. An R scrip was wrien o fi he model parameers. An applicaion of he modified model is illusraed. Copyrigh 2007 C. D. Lai e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied.. Inroducion Many generalizaions and exensions of he Weibull disribuion have been proposed in reliabiliy lieraure due o he lack of fis of he radiional Weibull disribuion. A summary of hese generalizaions is given by Pham and Lai []. An exensive reamen on Weibull models is given by Murhy e al. [2]. A modified Weibull disribuion ha allows insananeous or early failures is inroduced by Muralidharan and Lahika [3]. I was poined ou hahe occurrence of insananeous or early (premaure) failures in life esing experimen is a phenomenon observed in elecronic pars as well as in clinical rials. These occurrences may be due o inferior qualiy, fauly consrucion, or nonresponse of he reamens. I has been shown ha he disribuion may be represened as a mixure of a singular disribuion a zero (or 0 ) and a wo-parameer Weibull disribuion. Because of he singular naure of disribuion, we have been unable o define he failure rae (hazard rae) funcion meaningfully. The aim of his paper is o provide a meaningful modificaion so ha he resuling model can be expressed as a mixure of wo coninuous disribuions. This modified form is more realisic as rue insananeous failures rarely occur. The

2 2 Journal of Applied Mahemaics and Decision Sciences modificaion allows us o esablish and sudy he failure rae funcion of his reliabiliy model via mixure disribuions. We also provide some graphical plos o illusrae some possible shapes for he survival funcions as well as he failure rae funcions. The presen paper focuses on he nearly insananeous failure case as i has fewer parameers. This special case is more realisic han he early failure case since many producs exhibi an infan moraliy phenomenon due o iniial defecs. 2. Represenaion of he model Le F()andR() = F() denoe he cumulaive disribuion funcion and he survival funcion of he mixure, respecively. We assume ha F is coninuous and is densiy be given by f () = F (). The componen disribuion funcions and heir survival funcions are F i () andr i () = F i (), respecively, i =,2. The failure rae of a lifeime disribuion is defined as h() = f ()/R() provided he densiy exiss. Insead of assuming an insan or an early failure o occur a a paricular ime poin, as in he original model of Muralidharan and Lahika [3], we now represen his model as a mixure of a generalized Dirac dela funcion and he 2-parameer Weibull as opposed o a mixure of a singular disribuion wih a Weibull. Thus, he resuling modificaion gives rise o a densiy funcion: f () = pδ d ( 0 ) + qαλ α α e (λ)α, p + q =, 0 <p<, (2.) where ( ) δ d 0 = d, d, 0, oherwise, (2.2) for sufficienly small d.herep>0 is he mixing proporion. We noe ha δ ( x x 0 ) = lim d 0 δ d ( x x0 ), (2.3) where δ( ) is he Dirac dela funcion. We may view he Dirac dela funcion as a normal disribuion having a zero mean and sandard deviaion ha ends o 0. For a fixed value of d, (2.2) denoes a uniform disribuion over an inerval [ 0, 0 + d] sohemodified model is now effecivelyamixureofaweibullwihauniformdisribuion.inseadof including a possible insananeous failure in he model, (2.2) allows for a possible near insananeous failure o occur uniformly over a very small ime inerval. Noe ha he case 0 = 0 corresponds o insananeous failures, whereas 0 /= 0(bu small) corresponds o he case wih early failures. Noing from (2.)and(2.2), we see ha he mixure densiyfuncionis no coninuous a 0 and 0 + d. However, boh he disribuion and survival funcions are coninuous. Wriing f () = δ d ( 0 )and f 2 () = αλ α α e (λ)α, α,λ>0; (2.)canbewrienas f () = pf ()+qf 2 (), p + q =, 0 <p<, (2.4)

3 C. D. Lai e al. 3 so F() = pf ()+qf 2 (), (2.5) R() = F() = p + q { pf ()+qf 2 () } = pr ()+qr 2 (). (2.6) Thus, he failure (hazard) rae funcion of he mixure disribuion is h() = pf ()+qf 2 () pr ()+qr 2 (). (2.7) A mixure disribuion involving wo 2-parameer Weibull disribuions has been horoughly sudied in Jiang and Murhy [4]. The mixure considered in his paper is more complex in he sense ha one of he mixing disribuions has a finie range which poses some challenges. Via (2.4) simulaed observaions from his model are made by generaing uniform variaes and Weibull variaes wih proporions p and q = p, respecively. 3. Survival funcion and failure rae funcion of he model Recenly, failure raes of mixures are discussed quie exensively. Lai and Xie [5, Secion 2.8] provide a good summary. As demonsraed by Block e al. [6], various shapes of failure rae funcions can arise wih a mixure of wo increasing failure rae disribuions. Now, a Weibull has an increasing (decreasing) failure rae if is shape parameer α is greaer (smaller) han. The uniform disribuion also has an increasing failure rae if i is uniform over [0, a]. Thus we are ineresed o know wha shapes would resul from our model. The reliabiliy (survival) funcions of he respecive componen disribuions are given by if 0 < 0, R () d + = 0 if d, d 0 if> 0 + d, (3.) R 2 () = e (λ)α, 0, α,λ>0. (3.2) The failure raes are, respecively, 0, 0 < 0, h () = d + 0, d,, > 0 + d, (3.3) h 2 () = αλ(λ) α. (3.4)

4 4 Journal of Applied Mahemaics and Decision Sciences I can be shown from (2.4)and(2.6) ha for any mixure of wo coninuous disribuions, he failure rae funcion can be expressed as h() = f () R() = w()h ()+ [ w() ] h 2 (), (3.5) where w() = pr ()/R() forall 0. In our case, p if 0 < 0, R() w() = pr () if d, R() 0 if> 0 + d (3.6) wih w () = w() [ w() ]{ h 2 () h () }. (3.7) (Noe ha equaion (3.7) is valid for all cases). Also a simple differeniaion shows ha h () = w ()h ()+w()h () w ()h 2 ()+[ w()]h 2(). (3.8) Now, w()h () = pr ()/R() f ()/R () = pf ()/R(), so (3.5) iswelldefinedfor all >0. Summarized expression for R() andh() are, respecively, given as p + qe (λ)α, 0 < 0, R() = pr ()+qr 2 () = p ( d + 0 ) + qe d (λ)α, d, qe (λ)α, > 0 + d, ( qe (λ) α ) αλ p + qe α α, 0 < 0, (λ)α h() = p + dqe (λ)α αλ α α p ( d + 0 ), d, + dqe (λ)α αλ α α, > 0 + d. (3.9) (3.0) Recall ha h() is disconinuous a boh = 0 and = 0 + d. Unlike he model of Muralidharan and Lahika [3], he survival funcion is coninuous hough no differeniable a = 0 and = 0 + d.

5 C. D. Lai e al. 5 f () p = 0.2 p = 0.5 p = 0.8 Figure 4.. Plo of f ():λ =,α = 0.5,d = 0.2, 0 = Nearly insananeous failure case ( 0 = 0) Consider a special case of he model (2.) whereby 0 = 0.Themodelmaybecalledhe Weibull wih nearly insananeous failure model. In his case, (3.3) is simplified giving he failure rae of he uniform disribuion as, 0 d, h () = d, >d, (4.) and (3.) is survival funcion is given as d if 0 d, R () = d 0 if>d. (4.2) The Weibull model wih nearly insananeous failure occurring uniformly over [0,d] has p(d ) + qe R() = d (λ)α, 0 d, (4.3) qe (λ)α, >d, p + dqe (λ)α αλ α α, 0 d, h() = p(d )+dqe (λ)α αλ α α, >d. (4.4) Graphs of Survival, Densiy, and Failure Rae Funcions. Graphical plos are imporan for ageing disribuions. I is no he aim of his paper o presen complee characerizaions

6 6 Journal of Applied Mahemaics and Decision Sciences f () p = 0.2 p = 0.5 p = 0.8 Figure 4.2. Plo of f ():λ =,α = 2,d = 0.2, 0 = 0. R() p = 0.2 p = 0.5 p = 0.8 Figure 4.3. Plo of R():λ =,α = 0.5,d = 0.2, 0 = 0. for he survival, densiy, and he failure rae funcions. Insead, snapshos are aken of some possible shapes from his model, as i is imporan o idenify wheher he model is useful for specific daases for which empirical plos are available. Consider in deail he special case when 0 = 0, ha is, he Weibull wih nearly insananeous failure model. Densiy funcions. In boh Figures 4. and 4.2, hree densiy funcions wih p = 0.2, 0.5, and 0.8 are ploed. In all figures, he smalles mixing proporion p is given by he solid line.

7 C. D. Lai e al. 7 R() p = 0.2 p = 0.5 p = 0.8 Figure 4.4. Plo of R():λ =,α = 2,d = 0.2, 0 = 0. Failure rae funcion h():α = 0.5, d = 0.5, p = Failure rae funcion h():α = 0.5, d = 0.5, p = Failure rae funcion h():α = 2.5, d = 0.5, p = Failure rae funcion h():α = 3, d = 0., p = Figure 4.5. Plos of failure rae funcions. Survival funcions. The survival funcions are given by Figures 4.3 and 4.4 which correspond o he densiy funcions in Figures 4. and 4.2, respecively. Failureraefuncions. The failure rae funcion h()isgivenasin(4.4). Clearly, is shape is he same as he Weibull failure rae afer d. Thus we focus on he segmen from 0 o d.

8 8 Journal of Applied Mahemaics and Decision Sciences Table 5.. Insananeous failures a i = 0, i =,2,...,28. ˆp ˆα / ˆλ Esimaes Sandard error Since he scale parameer λ does no aler he shape, i is se o one. Figure 4.5 shows ha h() can be increasing, decreasing, or bahub shaped for 0 d. From he plos, i can be seen ha he failure rae funcion of he model gives rise o several differen shapes and bumps; his is expeced as mixing disribuions wih a componen disribuion ha has a finie range ofen cause some problems. Alhough he second par can be eiher increasing or decreasing, he firs segmen can achieve various shapes. This finding is consisen wih Block e al. [6]. 5. Daa fiing and an applicaion Mixure disribuions are used widely in he saisics lieraure. Bebbingon e al. [7] have used a mixure of wo generalized Weibull disribuions o fi human moraliy daa. A mixure disribuion may give rise o differen failure rae hus i can provide pseudodemarcaion of various phases of a lifespan. Bebbingon e al. [7] also use heir mixure disribuion o idenify he differences beween (sub)populaions. While sofware for fiing mixure disribuions is available, such as he MIX package for he R environmen (Macdonald [8]), such packages do no handle uniform disribuions, or mixures of unlike disribuions. Tofihemodeloadaase,anRscripwaswrien.OnecanwrieacodeinRofi all 4 parameers (p,λ,α,d) and anoher o fi 3 parameers (p,λ,α) wihd given and held fixed. The second case always works, and works very well, bu he firs never gives good resuls when he edge of he uniform (parameer d) is inside he peak of he Weibull (personal communicaion wih Professor Macdonald). This is because here is no enough informaion in he daa o fi he 4h parameer in his siuaion. In pracice, he value of d can be manually esimaed quie accuraely from he daase. 5.. Applicaion. A sample of 40 boards of woods were checked for heir dryness on a paricular area of a board. The acual observaions were degrees of dryness measured as a percenage. This daase was analysed in Muralidharan and Lahika [3] wih i = 0, i =,2,...,28 and he oher posiive observaions are as follows: , , 0.4, , , 0.649, ,.3585,.772,.86047, 2.225, Treaing he degree of dryness as he failure ime, we apply he Weibull model wih nearly insananeous failures model o his daa (see Table 5.). I is reasonable o spread he zeros uniformly over an inerval [0,d). For illusraion, we selec d = 0.35 so ha = 0, 2 = , 3 = 0.00,..., 28 = Applying he MLE mehod in R, we found he following see Table 5.2. I seems o us he sizes of sandard errors of he hree esimaes are reasonable in view of a small sample size.

9 C. D. Lai e al. 9 Table 5.2. Uniform spread of nearly insananeous failure imes. ˆp ˆα / ˆλ Esimaes Sandard error Table 5.3. Exponenial wih nearly insananeous failures: d = ˆp / ˆλ v esimaes (se) ( ) ( ) v2 esimaes (se) ( ) ( ) Since he shape parameer ˆα =.6 and he fac ha hree parameers are being esimaed from few daa poins, i would be more realisic o specify α = apriori(i.e., an exponenial) as a special case of he Weibull model. This would reduce he number of parameers and herefore he uncerainy in he parameer esimaes. Table 5.3 summarizes he parameer esimaes of he exponenial mixing wih he uniform model. Noe. v consiss of of measuremens from he original daase. The 28 zeros in v are hen calibraed o spread over uniformly over [0,d]. v2 is formed by replacing he firs 28 cells of v by he calibraed values. We noe from he preceding able ha he precision for he second parameer esimae acually deerioraes in comparison wih he Weibull case. Perhaps he Akaike informaion crieria (AIC) or BIC would be a more objecive way o evaluae his comparison Sensiiviy analysis. In he above model fiing, we have chosen d manually and he value d = is chosen because i is sufficienly apar from he firs nonzero observed value We have assessed he sensiiviy of he selecion of he parameer d. For he insananeous failures case, we le d vary beween 0.0 and 0.04; he resuling esimaes and heir sandard errors are virually unchanged. However, i becomes sensiive when d is oo close o = 0 or o he firs Weibull failure ime = If d is se o 0.35, he parameer esimaes are hen given by ˆp = , ˆα = , and / ˆλ = indicaing ha hese esimaes change noiceably as d encroaches on he Weibull erriory. I also shows ha for his value of d, he model wih he uniform mixing wih he exponenial can longer be an alernaive because he esimae for he shape parameer α is now close o Conclusion The Weibull disribuion has been widely used as a life model in reliabiliy applicaions. However, one ofen finds ha i does no fi well in he early par of a lifespan for various reasons. In paricular, in he cases where iniial defecs are presen causing early failures, he Weibull disribuion is found inadequae o model such phenomenon. The proposed model of a modified Weibull mixing wih a uniform disribuion o model he firs phase of a lifespan should provide a useful alernaive.

10 0 Journal of Applied Mahemaics and Decision Sciences Appendix R code for fiing he model v<-rep(0,40) v[29:40]<-c( , ,0.4,0.4257, ,0.649, ,.3585,.772,.86047,2.225,2.2389) v2 <- c(0, , 0.00, 0.005, 0.002, , 0.003, , 0.004, , 0.005, , 0.006, , 0.007, , 0.008, , 0.009, , 0.0, 0.005, 0.0, 0.05, 0.02, 0.025, 0.03, 0.035, , , 0.4, , , 0.649, ,.3585,.772,.86047, 2.225, ) uniweib <-cbind(v, v2) neglluniweib<-funcion(p,x,d){ pp<-p[] shape<-p[2] rae<-p[3] if(pp>0 & shape>0 & rae>0 & pp<) { -sum(log(pp*dunif(x,0,d)+(- pp)*dweibull(x,shape,/rae))) }else{ e+200 } } uniweibmle <- funcion(d,prop,shape,rae,x) { p<-c(prop,shape,rae) fi<-nlm(neglluniweib,p=p,x=x,d=d,hessian=t) fi$se<-sqr(diag(solve(fi$hessian))) fi[c(,2,7,3,4,5,6)] } uniweibmle(0.035,0.,2,,v) uniweibmle(0.035,0.,2,,v2) Acknowledgmens The firs auhor is privileged o have been a former suden and a colleague of Professor Jeff Huner, who has been an inspiraion o many. The auhors wish him well in his reiremen. The auhors are very graeful o Professor Peer Macdonald who helped hem o wrie an R scrip for he model and o he referees for heir helpful commens. The auhors also wish o acknowledge Dr. K. Govindaraju for his help o improve he qualiy of he figures.

11 C. D. Lai e al. References [] H. Pham and C. D. Lai, On recen generalizaions of he Weibull disribuion, IEEE Transacions on Reliabiliy, vol. 54, no. 3, pp , [2] D.N.P.Murhy,M.Xie,andR.Jiang,Weibull Models, Wiley Series in Probabiliy and Saisics, John Wiley & Sons, Hoboken, NJ, USA, [3] K. Muralidharan and P. Lahika, Analysis of insananeous and early failures in Weibull disribuion, Merika, vol. 64, no. 3, pp , [4] R. Jiang and D. N. P. Murhy, Mixure of Weibull disribuions parameric characerizaion of failure rae funcion, Applied Sochasic Models and Daa Analysis, vol. 4, no., pp , 998. [5] C. D. Lai and M. Xie, Sochasic Ageing and Dependence for Reliabiliy, Springer, New York, NY, USA, [6] H. W. Block, Y. Li, and T. H. Savis, Iniial and final behaviour of failure rae funcions for mixures and sysems, Journal of Applied Probabiliy, vol. 40, no. 3, pp , [7] M. Bebbingon, C. D. Lai, and R. Ziikis, Modeling human moraliy using mixures of bahub shaped failure disribuions, Journal of Theoreical Biology, vol. 245, no. 3, pp , [8] P. D. M. Macdonald, MIX for he R Environmen, disribued by ICHTHUS DATA SYSTEMS, hp:// C. D. Lai: Insiue of Informaion Sciences and Technology, Massey Universiy, Palmerson Norh, New Zealand address: c.lai@massey.ac.nz Michael B. C. Khoo: School of Mahemaical Sciences, Universii Sains Malaysia, 800 Pulau Pinang, Malaysia address: mkbc@usm.my K. Muralidharan: Deparmen of Saisics, The Maharaja Sayajirao Universiy of Baroda, Vadodara , India address: lmv murali@yahoo.com M. Xie: Deparmen of Indusrial and Sysems Engineering, Naional Universiy of Singapore, Singapore address: mxie@nus.edu.sg