models with failure-free life—applied review and extensions

IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 10, NO. 2, JUNE 2010 263

Models With Failure-Free Life—AppliedReview and Extensions

Shao-Wei Lam, Member, IEEE, Tony Halim, Member, IEEE, and Kanesan Muthusamy

Abstract—Fundamental engineering considerations may resultin the need for using a statistical distribution with a failure-freelife (FFL) parameter. These include failure-time models as wellas specialized repair-time models (for maintenance planning) andbreakdown-voltage models used for voltage-endurance studiesof electrical insulations. In this paper, some commonly encoun-tered statistical distributions with FFL are described togetherwith a review of their engineering applications. From the liter-ature, the three-parameter Weibull distribution has been foundto be the most common statistical distribution used when FFL isconsidered. Some estimation procedures for the three-parameterWeibull distribution are discussed and a new modified maximum-likelihood-estimation approach which leverages on the empiricalcumulative distribution function is proposed. Two case examplesconsidering complete and censored data are described.

Index Terms—Failure-free life (FFL), modified maximum-likelihood estimation (MMLE), three-parameter Weibull.

I. INTRODUCTION

A CUMULATIVE distribution function (CDF) F (t) thathas support over the nonnegative real line can theoret-

ically be used to model a life distribution. As a corollary,any CDF that has support over a subset of the nonnegativereal line which is greater than a threshold γ can be used tomodel a life distribution with a failure-free life (FFL). Such aconcept can be easily extended to the modeling of repair-timedistributions or any other random variables which possess theproperty of having a zero probability of event occurrence beforethe threshold parameter. As the number of possible life distri-bution is limitless, it is important for us to restrict the scopeof our review for statistical models with FFL to those mostcommonly encountered in reliability-engineering applications.The characteristics of the most popular and reasonable choiceof statistical distributions with FFL will be presented first,followed by a review of some published scientific literatureon their applications. Several common statistical inferentialprocedures would also be discussed for the most commonthree-parameter Weibull distribution. Finally, a new modifiedmaximum-likelihood-estimation (MMLE) approach would bepresented.

Manuscript received March 5, 2010; accepted March 9, 2010. Date ofpublication March 29, 2010; date of current version June 4, 2010.

S.-W. Lam is with the Analytics Consulting International, Singapore 752356(e-mail: [email protected]).

T. Halim is with the Temasek Polytechnic, Singapore 529757 (e-mail:[email protected]).

K. Muthusamy is with the Open University Malaysia, Kuala Lumpur 50480,Malaysia (e-mail: [email protected]).

Digital Object Identifier 10.1109/TDMR.2010.2045758

II. STATISTICAL MODELS WITH FFL

The structure of this review on statistical FFL (s-FFL) dis-tributions can be established along the following two broadcategories: 1) statistical-physics-based FFL (sp-FFL) distribu-tions and 2) s-FFL. In practical engineering applications, it isoften desirable to have some form of physics-of-failure-basedexplanations behind the postulated life distributions. Hence, inadopting an applied perspective, our review shall be cast froman sp-FFL perspective. Prior to describing the various engi-neering applications in the next section, we shall first providesome form of physics-of-failure-based explanation behind thechoice of the FFL distribution. This would enable the reader toappreciate some physical reasoning behind the FFL models.

A. Classification Matrix

Statistical distributions of the sp-FFL form can be classifiedin terms of plausible failure mechanisms, namely, cumulative-damage (CD)- and noncumulative-damage (NCD)-typefailures. This dual CD1 and NCD2 classification is proposedin consideration of the commonly encountered failuremechanisms in engineering reliability associated with modelswhich have their basis in a CD argument and those which donot arise from a CD argument. The descriptions of statisticalfailure models can be further classified using standard statisticalclassification systems of location-scale (LS)-based thresholddistributions and non-LS (NLS)-based threshold distributions[1]. For general LS distributions, the CDF of a random variableT can be expressed as P (T ≤ t) = Φ((g(t) − μ)/σ), wherethe function Φ does not depend on any unknown parameters[1]. The unknown parameters are the location parameter μand scale parameter, σ.When g(t) is a log transformation oft, the distribution is also known as a log-LS distribution. Forsp-FFL distributions, when the threshold term is added, thereparameterized CDF for the LS threshold distributions is givenas P (T ≤ t) = Φ((g(t − γ) − μ)/σ), where γ represents thethreshold parameter.

1CD-based failure distributions are usually employed to model failures whichoccur under a variety of fatigue mechanisms (e.g., chemical, thermal, andphysical fatigue). From a theoretical modeling perspective, fatigue failurestypically result from an accumulation of damage that is due to either the growthand/or coalescence of defects. Under the framework of such fatigue-failuremechanisms, the cumulated damage can be modeled in a linear or nonlinearform, resulting in a wide array of possible CD distributional forms.

2NCD failure distributions have been employed to model failures that couldbe a result of fatigue or otherwise. When NCD distributions are used, theunderlying failure mechanisms may not be due to fatigue, or the actual fatigueprocesses are not predominant at failure.

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264 IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 10, NO. 2, JUNE 2010

TABLE IFAILURE-DISTRIBUTION CLASSIFICATION MATRIX

WITH SOME COMMON EXAMPLES

A matrix which classifies the failure distributions by con-sidering both the statistical and physical failure characteristicsis proposed. This matrix cross classifies failure mechanismsand their distributional forms (see Table I). It has been notedthat existing literature related to the application of models withFFL has extensively utilized only a few forms presented here(notably, exponential and Weibull models), although failurephysics may suggest other suitable models. This matrix offersan alternative perspective on the suitability of other modelforms.

B. Common Statistical Distributions With FFL

In this review, the following sp-FFL distributions commonlyencountered in reliability engineering are considered:

1) two-parameter exponential distribution (NCD-LS);2) three-parameter Weibull distribution (NCD-LS);3) three-parameter inverse Gaussian (CD-NLS);4) three-parameter lognormal distribution (CD-LS).

In general, the statistical models will be presented, andthe possible failure mechanism which motivates these modelswould be discussed.

The CDF for the two-parameter exponential distribution withFFL is P (T ≤ t) = 1 − exp(−(t − γ)/η). The location andscale parameters are μ = log η and σ = 1, respectively. Thisis one of the simplest log-LS members, as the original one-parameter exponential distribution is the simplest and mostcommonly encountered failure distribution in reliability engi-neering. The exponential distribution is essentially a specialcase of the Weibull distribution. A failure-time distribution withFFL which follows an exponential distribution implies constantfailure rate. Such a failure-rate behavior implies that physically,the population under consideration is not wearing out. Thissituation occurs in some types of electronic components that arerelatively robust to wear out such as certain types of capacitorsand high-quality integrated circuits. In general, this distribu-tion is usually suitable for components that exhibit wear-outphenomenon that only manifest itself long after the expectedtechnological (or useful) life of the system. The threshold pa-rameter is a measure of the FFL. If this parameter is sig-nificantly different from zero, the time prior to the thresholdmay have zero failure probability beyond which the failuredistribution assumes the typical exponential failures.

The three-parameter Weibull distribution is a general casefor exponential distribution. In fact, it is also a special caseof the generalized gamma distribution. Many inferential pro-

cedures for the extreme value distribution are available, as thenatural logarithms of the Weibull distribution give a popula-tion which follows a Type I extreme value distribution (alsoknown as a Type I Gumbel distribution). Physically, the Weibulldistribution can be used to model a random variable whichrepresents the minimum of a large number of independent andidentically distributed random variables. Given this underlyingphysical explanation, the Weibull distribution has been used tomodel the time to failure reached as a result of many compet-ing similar and identical defects existing within a component(a weakest link theory). A prominent example in reliability-engineering literature is the failure modeling of dielectricmaterials which contains many flaws competing to lead to aneventual catastrophic failure of the material. Conversely, theweakest link argument might not be readily applicable if thefailure mechanism is typically a degradation type due to cor-rosion or some form of chemical reactions. On the other hand,other arguments for the use of Weibull distribution given em-pirical data have also been put forward in the literature. As anexample, Freudenthal [2] proposed the use of Weibull distribu-tion for the modeling of fatigue failures and suggested the use ofa risk function h(t), which is essentially the intensity or hazardfunction.

The inverse Gaussian distribution (IGD) is an NLS distrib-ution that is commonly used for failure-distribution modelingpossibly due to its theoretical foundations for modeling degra-dation processes that follow an additive CD model. This is incontrast to the lognormal distribution which is suitable for aproportional CD model. The Birnbaum–Saunders distribution,which can be considered from a discrete-time degradationprocess model perspective, is closely associated with the IGDand has been used to model the number of cycles required for afatigue crack to grow to a critical size, triggering fatigue failures[4]. The IGD is suitable for modeling the failure distributionwith basis in CD where accumulated damage is assumed tofollow a stochastic process governed by a deterministic driftterm and a random-noise term with zero mean. By assumingthe random noise as a white Gaussian noise with zero mean,the resulting process is a Wiener process. This random-noiseterm can be assumed to represent a large number of microscopicnoise terms collectively contributing to the damage over time,and the constant λ represents the slope of the growth in CD.By defining failure as the point when such a process crosses aprescribed deterministic threshold, the failure-time distributionof such a process has been found to follow an inverse Gaussianprocess.

The lognormal distribution per se is a common and highlyversatile form of failure distribution. Its mathematical formis essentially a general case of gamma distribution. The haz-ard function of lognormal distribution is characterized by anincrease to a maximum early in life followed by a gradualdecrease from this maximum. Physically, the early hardeningof materials or components may lead to such failure-rate char-acteristics. The lognormal distribution has been proven to betheoretically consistent with a CD form of failure distributionused to describe proportional degradation-based failure mech-anisms [5] and has been widely used to describe the time tofracture due to fatigue crack growth in metals.

LAM et al.: MODELS WITH FAILURE-FREE LIFE—APPLIED REVIEW AND EXTENSIONS 265

III. ENGINEERING APPLICATIONS

In many real-world reliability applications, fundamentalengineering considerations may result in the need for a statis-tical distribution with FFL parameters. These include failure-time models as well as specialized repair-time models (formaintenance planning) and breakdown voltage models usedfor voltage endurance (VE) studies of electrical insulations.In these engineering applications, the three-parameter Weibulldistribution has been the most commonly employed distributiondue to the relevance of its theoretical basis built upon theweakest link concept and highly flexible distributional struc-ture leveraging upon a wide range of possible hazard rateforms. Practical applications of the three-parameter Weibulldistribution on engineering products ranging from small-scaledevices such as electrical components and biomaterials to large-scale engineering infrastructures such as aircrafts and offshoremarine structures would be presented here.

The three-parameter Weibull distributions, which includethe FFL parameter, has been frequently used to describe thereliability of SM assemblies due to wear-out failures of un-derthermal cycling [3], [6]–[12]. This particular distributionwith FFL has been found to effectively model the reliability ofSM assemblies derived from different manufacturing processesand provide valuable information to process engineers in thedesign and evaluation of these manufacturing processes. Thethree-parameter Weibull distribution was found to provide abetter description of failure data for SM devices compared withthe two-parameter Weibull and lognormal distributions [3], [8].FFL scaled for solder crack areas were empirically foundto correlate with the cyclic inelastic strain energy [3]. Thiscorrelation, together with an earlier one with the characteristiclife [8], essentially allows the prediction of failure-free times atdesign stage [3].

A theoretical explanation behind the suitability of theWeibull distribution for SM devices is based on the weakestlink theory. In SM reliability assessments, attachment failuresare usually due to the first interconnection to open amongthe weakest or highest stressed solder joints of a component[3]. The failure-free time can be viewed as the minimumamount of time required for cracks to propagate through theweakest solder joint or for the strain energy of that particularcomponent to reach a threshold value which defines failure.Wear-out failures under thermal cycling essentially do not causeimmediate catastrophic failures but result from an accumulationof damage until a critical level when the weakest joint fails.Empirical observations have consistently shown that the largenumber of electronic packages deployed in products, such astelecommunication equipment, does not experience a signifi-cant number of solder-joint failures [9], [10]. The suitability ofthe three-parameter Weibull distribution is further substantiatedin the characterization of the reliability of SM assemblies underthermal cycling (thermal-fatigue reliability) [6], [7].

The three-parameter Weibull distribution has also been ex-tensively used to model the breakdown-voltage data in VEtests. VE tests are accelerated aging tests typically conductedon electrical insulation systems [13]. Extensive empirical andtheoretical results on the use of three-parameter Weibull distri-

bution to derive the threshold breakdown voltage of dielectricinsulation have been described [14]–[17]. The primary concernis to derive the FFL and the threshold voltage below whichno breakdown of insulation will occur. The three-parameterWeibull distribution has also been used to study the mechanical-fatigue reliability in the field of biomaterials by modelingthe mechanical-fatigue reliability of antibiotic powder loadedacrylic bone cement [18].

The concept of FFL has been introduced for the reliabilityprediction and maintenance of large-scale systems such asaircrafts, offshore marine structures, and ships. In such mainte-nance planning activities, distributions with FFL parameters arenot only useful in modeling the failure distribution but also inthe repair-time distribution (see, for example, Rao and Badhury[19]). For combat aircraft, a new concept encompassing the ideaof FFL in reliability management of air fleets has been pro-posed [20]. Further developments in lieu of these failure-freeconcepts and requirements were enhanced by the developmentof a comprehensive customer-focused paradigm in achievingfailure- and maintenance-free guaranteed periods in order todrive down the costs of logistics and maintenance [21]–[25].

Reliability studies utilizing the concept of FFL also includethe study of fatigue failures in the design of large-scale float-ing structures such as liquefied natural gas tankers [26] andother marine infrastructures [27]. A three-parameter Weibullmodel was used to analyze and predict the reliability of fatiguelife and to determine the allowable stress range for structuralcomponents of large-scale floating structures [26]. The physicalbasis of fatigue failures in these marine structures are known toreside in the reliability of welded joints. The stress number ofcycle relationship is typically constructed based on the weld-material or weld-affected zones. Under such an assumption,fatigue failures under cyclic loading occur only after a nonzerofinite amount of time.

IV. ESTIMATION PROCEDURES

From the review on existing literature, the three-parameter Weibull distribution with reliability functionR(t) = exp(−((t − γ)/η)β) can be considered as the mostimportant distribution for describing models with FFL inexisting engineering applications. Inferential procedures forthree-parameter Weibull based on the maximum likelihoodestimation (MLE) concept, however, present unique theoreticalproblems. One of the reasons for this is that the likelihoodtypically tends to infinity when the failure-free parameter tendstoward the lowest failure time when the shape parameter is lessthen one. Essentially, the MLE estimators of a three-parameterWeibull distribution may not exist and numerical problemsare known to occur even when the likelihood functionis finite. Fortunately, apart from MLE, many inferentialprocedures have been developed based upon existing resultsfor the Type I extreme value distribution that have beenproven useful for estimating the three-parameter Weibulldistribution. Briefly, these include method of moments;maximum likelihood methods; best linear unbiased estimates(BLUEs), empirical CDF (ECDF)-based estimation, andgraphical-based estimation. These methods are briefly


reviewed here. In addition, a new likelihood-based approachdefined as a constrained MLE (McLE) approach, whichleverages on the information generated from an empiricaldistribution function, is proposed.

A. Method of Moments

There are several moment-based estimators for estimatingthe threshold parameter γ and the shape and scale parametersβ and η, respectively. These approaches are typically availableonly for complete data (see, for example, Weibull [28]). Apractical implementation is described by Cran [29] for completedata. A modified moment-estimation procedure was proposedby Cohen et al. [30] with motivation from the traditionalmethod-of-moments approaches utilizing the expectation andvariance of the failure-time distribution together with a first-order statistic of the failure time. This method is also onlyapplicable for complete data, and Cohen et al. [30] have pointedout some computational difficulties may be encountered forthis modified-moment estimator when β < 0.5. Although theasymptotic variances and covariances of maximum-likelihoodestimates may not be strictly applicable for modified momentestimates, some close agreement were found through simula-tion studies conducted by Cohen and Whitten [31]. For thecase when all three parameters are estimated, the asymptoticvariances are generally applicable when β > 2 (conservatively,β > 2.2) [31]. Instead of the simple modified-moment equationfor the first-order statistic, Cohen and Whitten [32] have alsoconsidered other possible estimates.

B. MLE

Similar to the method of moments, the MLE approachis also dependent on the nature of the data (complete andcensored data). In order to derive the maximum-likelihoodestimator for a sample of n complete data, the likelihoodfunction is

∏ni=1 f(DATA;β, η, γ) where f(DATA;β, η, γ) =

d(1 − R(t))/dt. For right Type I censored data (with cen-soring time τ ), the likelihood function is L(β, η, γ; DATA) =∏n

i=1(f(ti))δi(R(τ))1−δi , where δi = 0 if observation is un-censored (ti ≤ τ), and δi = 1 if the observation is censored.The full likelihood function for complete data has been givenin literature (see, for example, Murthy et al. [33]). Given thelikelihood functions, the MLE estimates of the three-parameterWeibull distribution can be found by taking the first partialderivatives of the log likelihood and solving the resulting equa-tions. However, the maximum-likelihood estimators for a three-parameter Weibull distribution resulting from such a procedurehave been known to be problematic for β ≤ 2 [36]. VariousMMLE methods were proposed by Cohen and Whitten [32]to circumvent the difficulty of estimating all three parameterssimultaneously for nonregular cases when β ≤ 2.

C. Best Linear Unbiased Estimation

In order to describe the BLUEs for a three-parameter Weibulldistribution, we need to make use of the standard two-parameterWeibull distribution. The three-parameter Weibull distribution

can essentially be written as a simple linear model with thecovariate being the expectation of a two-parameter Weibullrandom variable [35]–[37].

D. ECDF-Based Estimates

ECDF-based approaches are essentially estimation ap-proaches based on the ECDF that is typically defined asfollows:

H(t) =

⎧⎨⎩

0, t < t(1)rn , t(r) ≤ t ≤ t(r+1)

1, t(n) ≤ t

where t(i) represents the ith ranked failure observations. Othercommon definitions of the ECDF include replacing r/nwith (r − 0.3)/(n + 0.4) or with the median rank estimators.ECDF-based estimators are typically derived from the mini-mization of a discrepancy measure which reflect the differencebetween the postulated three-parameter Weibull function andan ECDF. Such discrepancy measures can be broadly classifiedinto the supremum and quadratic classes [38]. The supremumclass of statistics essentially computes the maximum deviationsbetween the ECDF and the hypothesized CDF (in this case,the three-parameter Weibull distribution). This class includesthe well-known Kolmogorov–Smirnov distance measure. Thequadratic class of ECDF-based statistics computes a quadraticmeasure of deviation between the ECDF and hypothesizedCDF. This class includes the Cramer von Mises family whichencompasses statistics such as the Anderson–Darling measures[39]. The essential difference between statistics in this class isthat the weighting function is defined differently so as to weightdeviations with respect to the distribution function.

E. Graphical Approaches

Many different graphical-based-estimation approaches forthe estimation of three-parameter Weibull distribution havebeen proposed. These approaches essentially leverage on theexpected linearity of the Weibull probability plot. Examplesof such approaches include simple graphical estimation proce-dures or the geometric approach proposed by Drapella [40]. Asthese numerical methods suffer from the drawback of havinglocal minima (e.g., for the SSE minimization approach) or max-ima (e.g., for the R2 maximization approach), a good startingvalue for the numerical search is usually required. Typically,this value has been set at the minimum failure time.

V. PROPOSED MMLE APPROACH

The McLE approach, which leverages on the expected lin-earity of the Weibull probability plot, is proposed here for esti-mating the three-parameter Weibull distribution. It makes use ofthe ECDF information obtained through the Weibull probabilityplot for MLE instead of estimates based on the first-orderstatistics or mean time to failure in existing MMLE methods.Compared with existing MMLE approaches based on simplesummary statistics, it utilizes the entire ECDF information. It


Fig. 1. Flowchart for McLE approach for complete data (ε: Arbitrary smallpositive value).

can be considered a hybrid of ECDF-, graphical-, and MLE-based approaches (see Fig. 1). The variance–covariance matrixof this estimator can be derived based on simulations similar tothose given by existing MMLE approaches. Here, we describethe approach based on the log-failure times.

Given complete failure time data in a sample of sizen, the log-likelihood function of a three-parameter Weibulldistribution is

n lnβ

η−

(1β− 1

) n∑i=1

zi −n∑

i=1

ezi

where

zi = β ln(

t(i) − γ

η

).

The partial derivatives for this log likelihood can be found inMurthy et al. [33]. In order to derive the MLE, the partialderivatives can be equated to zero and solved. However, it iswell known that the likelihood function of a three-parameterWeibull distribution is unbounded for β < 1. In addition, thefirst-order derivative of the likelihood function may not have asolution for β ≥ 1, and the global maximum for the likelihoodfunction may occur at boundary, β = 1. Given these limitations

for the case of typical MLE estimates, a new hybrid McLEestimator based on a combination of ECDF, graphical, andMLE approaches is developed. The proposed McLE approachis based on the least square estimates of β̂γ and η̂γ from thefollowing linearized model:

yi = βγxi − βγ ln(ηγ)

where

xi = ln(t(i) − γ

)yi = ln

(− ln(1 − H

(t(i)

)))

H(t) =1

n + 0.4

⎛⎝ n∑

j=1

Ij(t) + 0.3

⎞⎠ .

Here, Ij(t) = 1 for t(j) ≤ t, and 0 otherwise.In addition

β̂γ =

n∑i=1

(xi − x)(yi − y)

n∑i=1

(xi − x)2η̂γ = exp(β̂γx − y).

x and y are the sample means of xi and yi, respectively.Subsequently, the constrained log likelihood evaluated at leastsquare estimates β̂γ and η̂γ is maximized. This constrained loglikelihood is

n lnβγ

ηγ−

(1βγ

− 1

) n∑i=1

zi −n∑

i=1

ezi .

This method is defined as the McLE. A schematic showingthe procedure for obtaining parameter estimates from completefailure data is shown in Fig. 1. Since γ gives the FFL and η + γgives the characteristic life for a reliability of 0.632, these quan-tities can sometimes be estimated via engineering techniquessuch as finite-element analysis to provide appropriate startingvalues of γ. Alternatively, the minimum failure time can be used(see Fig. 1).

It is frequently desirable to have a consistent estimate for β,such as in the analysis of accelerated life test data. Additionalconstraints may be set for β being similar at different accelera-tion levels. In the case when β is to be estimated using McLE,least square estimates of γβ and ηβ can be obtained through thelogarithm-transformed linear functions

yIIi = ηβxII

i + γβ xIIi =

{− ln(1 − H

(t(i)

))}1/β

where yIIi = t(i).

Here, the parameter estimates are given as

η̂β =

n∑i=1

(xII

i − x II) (

yIIi − y II

)n∑

i=1

(xII

i − x II)2

γ̂β = y II − η̂βx II.

x II and y II are sample means of xIIi and yII

i , respectively. Inthis case, good starting values of β can be obtained using either


TABLE IICOMPLETE VEHICLE FAILURE DATA

the minimize SSE or maximize R2 approaches. The maximumconstrained likelihood estimator here is

McLE(β | γβ , ηβ)

= sup{β:β≥0}

{n ln

β

ηβ−

(1β− 1

) n∑i=1

zi −n∑

i=1

ezi

}.

VI. CASE STUDY

The complete vehicle failure data shown in Table II [41]were used for showing the proposed MLE procedure. Usually,although a three-parameter Weibull distribution may be fitted,the two-parameter Weibull distribution is preferred unless thereare a priori reasons for suspecting the presence of an FFL. Inthis case, we made use of the proposed MLE procedure to solvefor the γ, β, and η parameters in a three-parameter Weibulldistribution.

The maximization of the log likelihood can be performedusing Microsoft Excel Solver. The results are shown in Table IIIand Fig. 2. The McLE estimate for γ is found to be 123.46at least square estimate, β̂γ = 1.15, and η̂γ = 1081.7. Giventhese least square estimates, the likelihood profile for the FFLestimate γ̂ is shown in Fig. 2. The maximum log likelihoodevaluated was −156.85. The R2-based estimate of γ̂ is foundto be at 125.23 with maximum R2 of 0.963.

Another set of right Type I censored data shown in Table IVwas used for testing the proposed McLE procedure. Type Icensoring is usually preferred in reliability tests since the lengthof the test time is finite. For such data, the analysis procedureis identical to that of complete data. For this case study, theECDFs are shown in Table IV. For multicensored data, theECDF can be estimated with the Kaplan–Meier product limitestimator [42].

For the data shown in Table IV, the least square estimatesof γβ and ηβ are evaluated before finding the estimate of theshape parameter using the maximum-likelihood approach. Thetotal log likelihood for right Type I censored data with n − sfailures can be evaluated [33]. Given the range of possible γ lessthan the minimum observed failure time, the McLE estimate forβ appears to be at β̂ = 3.73. The least square estimates γ̂β andη̂β were found to be γ̂β = 2713 and η̂β = 4254, respectively,using the McLE approach. Using the maximize R2 approach,the least square estimates of the parameters were found to beβ̂ = 4.622, γ̂β = 1977, and η̂β = 4982. The likelihood profileand R2 values over the range of possible FFL are shown inFig. 3.

TABLE IIICOMPUTED ECDF AND LOG LIKELIHOOD VEHICLE FAILURE DATA

Fig. 2. Constrained log likelihood (log L(γ|βγ , ηγ)) for vehicle data.

VII. CONCLUSION

A review of existing statistical distributions with FFL, to-gether with their practical applications, has been presented inthis paper. A classification structure is proposed for definingthe type of distributions based on considerations of the failuremechanism and distributional structure. From the literaturereview, the three-parameter Weibull distribution has been foundto be the most frequently used statistical distribution with FFL.Some common estimation approaches for the three-parameterWeibull distribution has been discussed, and a new hybridestimation approach has been proposed. Such an estimatoris consistent with an existing graphical-based approach and


TABLE IVPBGA PACKAGING FAILURE DATA AND COMPUTED ECDF

Fig. 3. Constrained log likelihood (log L(β|γβ , ηβ)) for PBGA failure data.

easily implementable in Microsoft Excel. The assessment of thequality of this proposed approach, together with the estimationof inferential errors, is a subject of future investigation.

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Shao-Wei Lam (M’07) received the B.Eng. degreein mechanical engineering and the M.Eng. degree inindustrial and systems engineering from the NationalUniversity of Singapore, Singapore, in 1999 and2001, respectively.

Apart from scientific and technical research ex-perience as a Research Engineer with the DesignTechnology Institute and Research Fellow with theNational University of Singapore, he was a SeniorPolicy Analyst with the National Science andTechnology Agency, Singapore. He is currently a

Principal Consultant with Analytics Consulting International, Singapore.Mr. Lam is a Certified Reliability Engineer as certified by the American

Society of Quality.

Tony Halim (M’10) received the B.Eng. degree inelectrical engineering and the M.Eng. degree in in-dustrial and systems engineering from the NationalUniversity of Singapore, Singapore, in 2000 and2004, respectively.

He was previously a Research Fellow with theNational University of Singapore. Along with hisscientific research experience, he has held variousprofessional positions with the National InfocommAuthority and STMicroelectronics. He is currentlya Lecturer with the Department of Business Process

and Systems Engineering, Temasek Engineering School, Temasek Polytechnic,Singapore, Singapore. His main research area includes systems engineering,reliability, and preventive maintenance applied in facilities management, andhe has published widely in the mentioned knowledge domain.

Kanesan Muthusamy received the Ph.D. degree ininformation engineering (operational research) fromthe Graduate School of Engineering, Osaka Univer-sity, Osaka, Japan, in 2002.

He has held various positions with Robert Bosch,Motorola, and Panasonic. He is currently an Asso-ciate Professor and Deputy Dean with the Faculty ofScience and Technology, Open University Malaysia,Kuala Lumpur, Malaysia. His research interest in-cludes six-sigma methodology and optimal schedul-ing methods.

Dr. Muthusamy is a Professional Engineer registered with the Board ofEngineers, Malaysia.

models with failure-free life—applied review and extensions

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