accelerated degradation analysis for the quality of a system based on the gamma process

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IEEE TRANSACTIONS ON RELIABILITY 1

Accelerated Degradation Analysis for the Qualityof a System Based on the Gamma Process

Man Ho Ling, Kwok Leung Tsui, and Narayanaswamy Balakrishnan

Abstract—As most systems these days are highly reliable withlong lifetimes, failures of systems become rare; consequently, tra-ditional failure time analysis may not be able to provide a preciseassessment of the system reliability. In this regard, a degradationmeasure, as a percentage of the initial value, is an alternate wayof describing the system health. This paper presents accelerateddegradation analysis that characterizes the health and quality ofsystems with monotonic and bounded degradation. The maximumlikelihood estimates (MLEs) of the model parameters are derived,based on a gamma process, time-scale transformation, and a powerlink function for associating the covariates. Then, methods of es-timating the reliability, the mean and median lifetime, the condi-tional reliability, and the remaining useful life of systems undernormal use conditions are all described. Moreover, approximateconfidence intervals for the parameters of interest are developedbased on the observed Fisher information matrix. A model valida-tion metric with exact power is introduced. A Monte Carlo simu-lation study is carried out for evaluating the performance of theproposed methods. For an illustration of the proposed model, andthe methods of inference developed here, a numerical example in-volving light intensity of light emitting diodes (LED) is analyzed.

Index Terms—Accelerated degradation analysis, asymptoticconfidence interval, gamma process, maximum likelihood esti-mate, remaining useful life, system health.

ACRONYMS AND ABBREVIATIONS

MLE maximum likelihood estimate

LED light emitting diodes

RMSE root mean square error

CP coverage probability

AW average width

CI confidence interval

AIC Akaike information criterion

Manuscript received March 18, 2013; revised February 02, 2014; acceptedMarch 25, 2014. This work was supported by the Collaborative Research Fund(Ref. CityU8/CRF/12G), granted by the Research Grants Council and CityUStrategic Research Grant (SRG-Fd) (7004085), supported by City Universityof Hong Kong. Associate Editor: L. Walls.M. H. Ling is with the Department of Mathematics and Information Tech-

nology, The Hong Kong Institute of Education, Tai Po, Hong Kong, China.K. L. Tsui is with the Department of Systems Engineering and Engineering

Management, The City University of Hong Kong, Kowloon Tong, Hong Kong,China.N. Balakrishnan is with the Department of Mathematics and Statistics,

McMaster University, Hamilton, ON L8S 4K1, Canada.Digital Object Identifier 10.1109/TR.2014.2337071

NOTATION

number of devices tested in each test group

number of test groups

number of types of stress factors

stress level in the -th test group

-th inspection time

degradation of the -th tested item in the-th test group at inspection time

Gamma distribution

Birnbaum-Saunders distribution

shape parameter in a gamma distribution inthe -th test group at inspection timescale parameter in a gamma distribution inthe -th test groupparameter in

probability density function of

cumulative distribution function of

cumulative distribution function of failuretimeparameters in the link function for

time-scale parameter

parameters in the link function for

vector of parameters

likelihood function based on observed data

mean time-to-failure

median time-to-failure

reliability at time

conditional reliability at time , givendegradation at timeremaining useful life, given degradation

at timegamma function

upper incomplete gamma function

lower incomplete gamma function

digamma function

trigamma function

hypergeometric function

0018-9529 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


2 IEEE TRANSACTIONS ON RELIABILITY

first-order derivative of lower incompletegamma functionpre-specified threshold

observed Fisher information matrix

mean of

variance of

parameter of interest

standard error

probability-based test statistics

I. INTRODUCTION

R ELIABILITY assessment plays an important role inengineering design and maintenance management, and

greatly assists in the development of products. Traditionalfailure time analysis has been well developed over the years[1]–[4]. However, due to a strong demand from customerson highly reliable, safe systems, many systems are designedto function for a long period of time. Because there will bevery few or even no failures observed within a limited timeunder usual life testing experiments, traditional reliabilityprediction methods may result in inaccurate prediction of thesystem reliability, which would then lead to wrong decisions inmaintenance scheduling. Consequently, the impact on safety,availability, and cost becomes very high.Lu et al. [5] compared degradation analysis with traditional

failure time analyses, and mentioned that degradation analysisprovides more accurate estimation of the mean lifetime thandoes traditional failure time analysis. Singpurwalla [6] useddegradation models with covariates for modeling lifetime in adynamic environment. Ebrahimi [7] also proposed degradationmodels for fatigue cracks in a system, and assessed the dis-tribution of the failure time at which the crack size of at leastone of the cracks exceeds a given threshold value. Further, toobtain degradation information quickly and efficiently withina reasonable period of time, accelerated life-tests, whereinspecimens are subjected to higher stress levels, are commonlyused. Many papers have appeared on accelerated degradationmodels for analyzing the reliability of integrated circuit devicesand newly developed light emitting diode (LED) lamps in theliterature. Interested readers may refer to articles [8]–[10].Due to many attractive properties of the normal distribution,

the Wiener process is commonly used to model the degradationof products; see articles [11]–[14] for degradation models usingWiener processes. A Wiener process is useful for some specificdatasets, but is not appropriate for modeling monotonic degra-dation, because of a characteristic feature of Wiener processesthat the performance of a product can increase and decrease overtime. In such cases, it is natural to consider a gamma processfor modeling the degradation. Lawless and Crowder [15] ana-lyzed fatigue crack growth rates in metals of steel and aluminumalloy with the use of gamma processes. Park and Padgett [16]presented accelerated degradation models for failure based ongamma processes. They subsequently modeled the increase inresistance in carbon-film resistors through accelerating degra-dation models based on a gamma process, incorporated with

temperature and voltage as accelerating factors; see [17]. Panand Balakrishnan [18] modeled degradation of products withtwo performance characteristics based on a bivariate gammaprocess.In degradation analysis, a failure time can be viewed as a

first-passage time past a specified threshold for a degradationprocess; in the case of a Wiener degradation process, the first-passage time is known to follow an inverse Gaussian distribu-tion; see Chhikara and Folks [19]. Park and Padgett [16] alsodiscussed the mean-time-to-failure of products when the degra-dation is a gamma process. In many applications, the health orquality of a system is usually quantified in terms of percentageto the initial value. For example, Chaluvadi [20] presented adataset of LEDs with light intensities by percent under accel-erated degradation life-tests, and the failure is declared whenthe light intensity falls below 50%. This paper presents acceler-ated degradation analysis for the light intensity of the LEDs, anddevelops inferences on different lifetime characteristics of theLEDs. However, due to the degradation being bounded in thisway between 0 and 1, the development of inferential methodsfor the lifetime characteristics of LEDs becomes a challengingtask.The rest of this paper proceeds as follows. In Section II,

an accelerated degradation model for the considered problemis formulated, and the corresponding likelihood function ispresented. In Section III, inferences on lifetime characteristicsof systems such as the reliability, the mean, and the medianlifetime at normal use conditions are all described. For purposesof prognostics and system health management, the conditionalreliability and the remaining useful life of systems are alsostudied. Methods for determining the maximum likelihoodestimates (MLEs) and the construction of confidence intervalsfor the parameters of interest are discussed in Section IV. Then,a test statistic with an exact power is introduced for the purposeof model validation in Section V. In Section VI, a simulationstudy is carried out at different levels of reliability and samplesizes to evaluate the performance of the proposed methods. InSection VII, for illustrative purpose, an application of the pro-posed methods to a real dataset on LEDs is elaborated. Finally,in Section VIII, some concluding remarks are made whereinsome further issues of interest in this direction are pointed out.

II. MODEL DESCRIPTION, AND LIKELIHOOD FUNCTION

Suppose that items are to be placed at a constant stresslevel of for each elevated condition , and in-spected at times . The items’degradations are measured as . As discussed above, thequality of a system is quantified in terms of the percentage tothe initial value, and so is taken to be 1 for all and. When the degradation is bounded between 0 and 1, neitherthe Wiener nor the gamma process is suitable as they are notbounded by 1. So, we consider the logarithmic (log-) trans-formation to the degradation. In addition, a time-scale trans-formation introduced by Whitmore and Schenkelberg [13] isconsidered. Under this transformation, we assume that

has a gamma distribution with shape, and scale . In this case, we

have the corresponding probability density function (pdf), andthe cumulative distribution function (cdf) as


LING et al.: ACCELERATED DEGRADATION ANALYSIS FOR THE QUALITY OF A SYSTEM BASED ON THE GAMMA PROCESS 3

(1)

and

(2)

respectively. Note thatcan be viewed as an increment

in the logarithm of the degradation.We now assume the inverse power law as an acceleration

model to relate both model parameters and to the stresslevel . The inverse power law is commonly used as a linkfunction when dealing with electrical current and stress (see[8]). We thus have

(3)

and

(4)

Under this setting, we have the likelihood function as

(5)

where .

III. INFERENCES ON LIFETIME CHARACTERISTICS OF PRODUCT

Suppose is the time-to-failure of a system under normal usecondition at which the degradation level crosses a pre-spec-ified threshold, . By using the property that the sum of gammarandom variables with the same scale parameters is again dis-tributed as gamma, we readily get , where

, and. This result enables us to develop in-

ferences on the lifetime characteristics of products such as themean and the median time-to-failure, and , as wellas the reliability at a mission time .

A. Reliability at a Mission Time

When the failure of an item is defined by , we readilysee that the probability of the failure of the item by time is

(6)

where denotes the lower incomplete gamma function.Then, the reliability of the system at mission time is given by

(7)

B. Median-Time-to-Failure

The median-time-to-failure is the time at which 50% ofthe systems would have their degradation level crossing thethreshold . Banneheka and Ekanayake [21] presented a goodapproximation for the median of any gamma distribution withshape parameter being greater than 1. In the present situation, itis equivalent to seeking a time such that

(8)

We thus get

(9)

C. Mean-Time-to-Failure

Park and Padgett [16] derived the exact distribution ofbased on the gamma process, and also provided an approxima-tion to by using a two-parameter Birnbaum-Saunders dis-tribution; see Birnbaum and Saunders [22]. In our case, we have

(10)where and . See that

then follows the Birnbaum-Saunders distribution,. It then follows that

(11)

where is a hyperge-ometric function that can be readily computed using Matlab orMaple. The derivation of the above expression of the mean-time-to-failure is presented in the Appendix.

D. Conditional Reliability

In addition to the mean and the median time-to-failure,the conditional reliability of a system is also of great interestin prognostics and system health management. Banjevic andJardine [23] discussed the importance of current diagnosticinformation in maintenance decisions as it can greatly im-prove the maintenance policy. In our case, due to the fact that

, the conditionalreliability is the probability that the degradation level stillmaintains above the threshold at time , given the degradation

at the current time , given by

(12)

where . It is seen that the conditional reli-ability reduces to the reliability as , and .



E. Remaining Useful Life

The remaining useful life is one of the critical assessments incondition-based maintenance, as well as prognostics and systemhealth management. It is defined as the length from the cur-rent time, , to the time-to-failure based on the current degra-dation . Si et al. [24] provided a comprehensive reviewon remaining useful life estimation based on different statis-tical models. Gebraeel and Pan [25] discussed a Bayesian ap-proach for computing and updating the remaining useful life ofbearings. Recently, Fan et al. [26] developed physics-of-failurebased damage models to predict the remaining useful life ofLEDs. However, unit-to-unit variation is not taken into account.Banjevic and Jardine [23] derived the remaining useful life interms of conditional reliability, and showed it to be

(13)

(14)

For a fixed , , and , it is found thatis decreasing in , and

when ,when ,when .

It is also expected that

when ,when ,when .

In other words, the information on determines the behavior ofthe remaining useful life of a system. More specifically, whentwo systems have the same level of degradation but with dif-ferent operating times, the remaining useful lives of the two sys-tems are the same when . However, the older system willhave a longer remaining useful life when , and a shorterremaining useful life when .

IV. METHODS OF POINT AND INTERVAL ESTIMATION

A. Maximum Likelihood Estimates

Here, the Newton-Raphson method is utilized to determinethe MLEs of the model parameters. The first-order and second-order derivatives of with respect to the modelparameters, , , that are required for theimplementation of the Newton-Raphson method, are as follows.

(15)

(16)

(17)

(18)

(19)

where , and and are thedigamma and trigamma functions.Let

and be the corresponding observed Fisher informationmatrix. The MLEs of the model parameters, , can then beobtained when convergence occurs to a desiredlevel of accuracy as

(20)

The MLEs of all the parameters of interest can then bereadily determined by substituting into the correspondingexpressions.

B. Initial Values of Model Parameters

In the implementation of the iterative method to ob-tain the MLEs while solving (15) and (16), we requiregood initial values for the model parameters. Given that

for each , by using theknown properties of the gamma distribution, for each pair of

, we have a pair of as follows.

(21)

(22)

With these expressions, the initial values of the model param-eters can be determined through the least-squares method. In-terested readers may refer to Balakrishnan and Ling [28] fordetails.

C. Approximate Confidence Intervals

Once the observed Fisher information matrix for the modelparameters is obtained as above, the asymptotic confi-dence interval can be obtained by using the delta method asan approximate confidence interval for any parameter of in-terest, . It simply requires the first-order derivatives of the pa-rameter of interest with respect to the model parameters, i.e.,

, and . For example, the



first-order derivatives of in (7) with respect to the modelparameters are

(23)

(24)

(25)

(26)

(27)

whereis the first-order derivative of the lower in-

complete gamma function, and its expression can be found inBalakrishnan and Ling [27], along with further details. Let

The standard error of the estimate of can then be obtainedas

(28)

Due to the skewness in the sampling distribution of the esti-mated reliability when the sample size is not large, the logit-transformation is commonly employed to improve the confi-dence intervals for reliability (see Meeker et al. [9], and Balakr-ishnan and Ling [28]). With the use of the logit-transformation,the approximate confidence interval for canbe obtained as

(29)

where .Similarly, for estimating the mean-time-to-failure, and to de-

termine its standard error, the required first-order derivativeswith respect to the model parameters are

(30)

(31)

(32)

(33)

(34)

Because the first-order partial derivatives of the expected meanlifetime with respect to do not have a closed-form, we use anapproximation of the rate of change around with a very small. For the mean-time-to-failure, the log-transformation is usu-ally employed for constructing confidence intervals (see Balakr-ishnan and Ling [28]). With the use of the log-transformation,the approximate confidence interval for isgiven by

(35)

Let be the -percentile of time-to-failure, that is,, for . Then, we readily have

(36)

where is the inverse distribution function. To derive thestandard error of the -percentile of the time-to-failure,with

(37)

we have the required first-order derivatives with respect to themodel parameters , given by

(38)

where , which is as given above. We

can then obtain , where .

Alternatively, becauseas in (10), it is natural to consider as an estimator of themedian-time-to-failure, where . Based onthis estimator, are required, and then the com-putation of the standard error of the median-time-to-failure,

, becomes straightforward. Thus, with the use of thelog-transformation once again, the approximateconfidence interval for is given by

(39)

Similarly, the conditional reliability and the remaining usefullife also have no explicit expressions. So, their first-orderderivatives can be approximated by the rate of change aroundthe estimates with small values of . Then these approximationsmay be used to obtain standard errors, as well as to constructapproximate confidence intervals. Similar to the intervals in



(29) and (35), the approximate confidence inter-vals for , and are, respectively,given by

(40)and

(41)

where ,, and

.

V. MODEL VALIDATION METRIC

Model validation metrics with exact powers would help en-gineers and scientists to select a suitable model for prediction,as well as for making maintenance decisions. For this reason, aprobability-based test statistic of the form

(42)

is proposed as a measure for evaluating the fit of the assumedmodel to the observed data. Also, under the assumed model

, the corresponding p-value canbe approximated as

(43)

Pertinent details of the derivation of the p-value in (43) arepresented in the Appendix. The probability-based test statisticsimply quantifies the probability for each observed data. Wewould expect to observe a small p-value ( 0.05) when the as-sumed model does not fit the data, because the test statisticwould tend to 0.5.

VI. SIMULATION STUDY

To evaluate the performance of the proposed methods, aMonte Carlo simulation study was carried out to estimate thebias, root mean square error (RMSE), coverage probability(CP), and average width (AW) of 95% confidence interval(CI) for all the lifetime characteristics of interest based on

(small sample size), (medium sample size),and (large sample size). The corresponding averagevalue of the tolerance, and the number of cases of divergenceare presented as well. Suppose there were 2 elevated levels( , ) in the accelerated degradation life-test, ineach of which the degradation of each specimen was measuredat inspection times , where . Let

; and (high level of reli-ability), (moderate level of reliability), and(low level of reliability). The data generating scheme for eachdegradation at the -th stress level at the inspection timeis as follows.

TABLE IVALUES OF BIAS AND ROOT MEAN SQUARE ERROR OF THE MLESOF THE MODEL PARAMETERS AND ASSOCIATED INFERENCE, ALONGWITH AVERAGE TOLERANCE AND NUMBER OF CASES OF DIVERGENCE,

FOR DIFFERENT SAMPLE SIZES (HIGH LEVEL OF RELIABILITY)

1) Generate from a gamma distribution with scale andshape from (3) and (4), for

.2) Compute .To find the true parameter of interest at typical use condition

, for example, and can be obtained by sub-stituting the true parameter values into (7) and (9), whilecan be obtained from a simulation of degradation with 10,000samples, wherein is the

first passage time crossing the given threshold . Theiteration for the estimates of the model parameters was termi-

nated when . For the iterative processof the least-squares method, the initial value of was set to 0.1,and then an incremental step of 0.1 was used for subsequentcomputations. The results obtained from this simulation study,based on 10,000 Monte Carlo simulations, are summarized inTables Ithrough VIII.It is clear from Tables I and II that, as the sample size gets

larger, the accuracy of the estimates of the model parametersand the corresponding inference improve, and the number ofcases of divergence in the proposed methods of finding the esti-mates decrease. However, more samples are required to reducethe RMSE of in the shape parameter for the influence of thecovariate on the shape parameter to be identified well. More-over, the RMSE of the model parameters are all quite stable forthe considered levels of reliability. Tables III and IV reveal thatthe accuracy of the approximate confidence intervals becomesbetter as the sample size gets larger. Except for the reliability,



TABLE IIVALUES OF BIAS AND ROOT MEAN SQUARE ERROR OF THE MLESOF THE MODEL PARAMETERS AND ASSOCIATED INFERENCE, ALONGWITH AVERAGE TOLERANCE AND NUMBER OF CASES OF DIVERGENCE,FOR DIFFERENT SAMPLE SIZES (MODERATE LEVEL OF RELIABILITY)

TABLE IIIVALUES OF COVERAGE PROBABILITY AND AVERAGE WIDTH OF THE 95%CONFIDENCE INTERVALS FOR THE MODEL PARAMETERS AND ASSOCIATEDINFERENCE FOR DIFFERENT SAMPLE SIZES (HIGH LEVEL OF RELIABILITY)

and the conditional reliability, the coverage probabilities of theproposed confidence intervals are in good agreement with the

TABLE IVVALUES OF COVERAGE PROBABILITY AND AVERAGE WIDTH OFTHE 95% CONFIDENCE INTERVALS FOR THE MODEL PARAMETERSAND ASSOCIATED INFERENCE FOR DIFFERENT SAMPLE SIZES

(MODERATE LEVEL OF RELIABILITY)

TABLE VCOMPARISON OF VALUES OF REMAINING USEFUL LIFE

FOR DIFFERENT VALUES OF MODEL PARAMETERS, AND

nominal level for all the considered cases. When the number ofspecimens gets larger, the coverage probabilities of the confi-dence intervals for the reliability and the conditional reliabilityalso become quite satisfactory.Moreover, Table III shows that is generally better thanas determination of the standard error of the median-time-



TABLE VIMLES OF THE MODEL PARAMETERS ALONG WITH THE CORRESPONDING

95% APPROXIMATE CONFIDENCE INTERVALS

TABLE VIIREMAINING USEFUL LIFE OF LED UNDER NORMAL USE ELECTRICAL

CURRENT OF 30 MA, ALONG WITH THE CORRESPONDING95% APPROXIMATE CONFIDENCE INTERVALS

Fig. 1. Degradation of light intensity of 24 LEDs at different levels of electricalcurrent.

to-failure in the sense of coverage probability. However, the av-erage width of the confidence intervals based on is very largewhen the sample size is small, because, in few cases, the stan-dard errors of the median-time-to-failure are very large. In thiscase, the estimator is an alternate approximation for thestandard error, as well as for the construction of the confidenceinterval.In addition, Table V shows the influence of on the estimated

remaining useful life. As expected, the remaining useful life be-comes larger when the degradation stays at a higher level for allthe considered cases. Also, as mentioned earlier, with differentvalues of , the remaining useful life at different time pointsmay not be the same, even though the degradation stays at the

TABLE VIIIDEGRADATION DATA OF LIGHT INTENSITY OF LEDSAT DIFFERENT LEVELS OF ELECTRICAL CURRENT

same level. Again, when two systems are with the same level ofdegradation but different operating times, the remaining usefullives of the two systems are the same when . However,the older system will have a longer remaining useful life when

, and a shorter remaining useful life when .

VII. ILLUSTRATIVE EXAMPLE USING REAL DATA FROM LEDS

As an illustration, we perform accelerated degradation anal-ysis on the light intensity of 24 LEDs, presented in Chaluvadi[20], by using the methods proposed in the preceding sections.The data consist of two accelerating stress levels of electricalcurrent, and at each level, outputs of the light intensity of 12LEDs were measured at 5 inspection times. Fig. 1 displays thedegradation of light intensity of the LEDs. The light intensitydegrades slowly at a low level of electrical current. The data arepresented in Table VIII in the Appendix.Table VI presents the MLEs of the model parameters,

along with the corresponding confidence intervals for differentmodels. Model 1 involves the accelerating factor, electricalcurrent, on both shape and scale parameters; while Models 2,and 3 involve the accelerating factor on either the shape, orscale parameter, respectively. In Model 1, multi-colinearityoccurs due to the electrical current being accounted for in bothshape and scale parameters, and for this reason, this model isnot recommended for analysis. Models 2 and 3 show that theelectrical current has a significant influence on the degradation.Moreover, the value of implies that when two LEDs are



with the same level of degradation, the older LED will havea longer remaining useful life than the newer one. Table VIIdisplays the remaining useful life of LEDs under a typicaluse electrical current of 30 mA, with different levels of lightintensity, at different mission times for Models 2 and 3, andwhen the light intensity degrades below a threshold of 50% ofthe initial value, it is regarded as a failure. Finally, the maxi-mized log-likelihood value under Model 2 of 1198 is betterthan the corresponding maximized log-likelihood value underModel 3 of 1246; but, Model 3 is preferred for modeling thedegradation data based on the test statistic .

VIII. CONCLUDING REMARKS

We have studied here the degradation of light intensity ofLEDs based on accelerated degradation analysis assuming agamma degradation process and time-scale transformation, andalso proposed methods for estimating the remaining useful life,given the information on current time and the degradation level.Moreover, there are several useful acceleration laws for accel-erated degradation analysis. One may consider AIC criterion orMSE or both for selecting a proper acceleration law. For ex-ample, Park and Padgett [16] selected an appropriate law basedon AIC criterion. In addition, the proposed model also helps indetermining the behavior of the remaining useful life, and thatin turn would assist in making good maintenance decisions.In the study of light intensity of LEDs, given information on

age, and degradation level, we can compute the conditional reli-ability, and the remaining useful life of LED.With due consider-ations to costs of maintenance and failure, based on the currenthealth information, optimal inspection and maintenance deci-sions can also be determined to lower the costs. The proposedmethods can be used for other applications such as maintenanceof roads, railways, bridges, buildings, and industrial plants. Re-cently, Tsai et al. [29] discussed the optimal design for degrada-tion tests under a pre-specified budget, and determined the op-timal termination time for burn-in tests based on gamma degra-dation processes; see [30], [31]. From a practical viewpoint,Nicolai et al. [32] considered a gamma deterioration process formodeling the deterioration process of the coating, and then usedit to find a set of maintenance actions so as to minimize the ex-pected maintenance cost of coatings on steel structures.Many interesting aspects of the accelerated degradation anal-

ysis require further study. When the remaining useful life islarge, the conditional reliability decreases very slowly. In thiscase, the recursive formula in (14) would require an intensivecomputation for determining the remaining useful life. It willbe therefore of interest to propose a more efficient way of com-puting the remaining useful life. Our discussion of the analysisof LEDs shows the difference in the prediction on the remaininguseful life based onModels 2 and 3, which demonstrates that theway the covariate is incorporated becomes critical in the predic-tion. Even though the maximized log-likelihood values providesupport to Model 2 in the analysis, but the test statistic pro-vides support to Model 3. In this regard, Balakrishnan and Ling[27] gave an example to show that the maximized log-likeli-hood value may not be a good criterion for model validation,and so it would be natural to study the effect of misspecificationin the prediction of the remaining useful life when covariates

are incorporated in different ways. Finally, accelerated degra-dation analysis with three or more stress levels are preferable,from which the relationship between covariates and stress couldbe identified better, which may then result in better prediction.

APPENDIX

Suppose . Then,, and

(44)

Now setting , we have

(45)

The p-value for the test statistic can be then determined as

(46)

ACKNOWLEDGMENT

We express our sincere thanks to the Associate Editors, Dr.Lesley Walls, and the anonymous reviewers for their commentsand useful suggestions on the original manuscript which led tothis improved version.

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Man Ho Ling received his B.Sc. and M.Phil. degrees from Hong Kong BaptistUniversity, Hong Kong, in 2005, and 2008, respectively. He finished his Ph.D.in Statistics from McMaster University, Hamilton, ON, Canada, in 2012.He is a lecturer in the Department of Mathematics and Information Tech-

nology at the Hong Kong Institute of Education. His research interests includereliability and survival analyses, statistical inference, censoring methodology,and statistical computing methods.

Kwok Leung Tsui is Head and Chair Professor of Industrial Engineering in theDepartment of Systems Engineering and Engineering Management at the CityUniversity of Hong Kong, and the founder and Director of Center for SystemsInformatics Engineering. Prior to his current position, he was Associate Pro-fessor and later Professor at the School of Industrial and Systems Engineering atthe Georgia Institute of Technology between 1990 to 2011. His current researchinterests include data mining, surveillance in healthcare and public health, cali-bration and validation of computer models, bioinformatics, process control andmonitoring, and robust design and Taguchi methods.Prof. Tsui was a recipient of the National Science Foundation Young Inves-

tigator Award. He is Fellow of the American Statistical Association, AmericanSociety for Quality, and International Society of Engineering Asset Manage-ment; he is also a U.S. representative to the ISO Technical Committee on Sta-tistical Methods. He was Chair of the INFORMS Section on Quality, Statis-tics, and Reliability; and the Founding Chair of the INFORMS Section on DataMining.

Narayanaswamy Balakrishnan received his B.Sc. and M.Sc. degrees in Sta-tistics from the University of Madras, India, in 1976 and 1978, respectively. Hefinished his Ph.D. in Statistics from the Indian Institute of Technology, Kanpur,India, in 1981.He is a Professor of Statistics at McMaster University, Hamilton, ON,

Canada. He is a Fellow of the American Statistical Association, and a Fellow ofthe Institute of Mathematical Statistics. He is currently the Editor-in-Chief ofCommunications in Statistics. His research interests include distribution theory,ordered data analysis, censoring methodology, reliability, survival analysis,nonparametric inference, and statistical quality control.

accelerated degradation analysis for the quality of a system based on the gamma process

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