a tool for evaluating time-varying-stress accelerated life test plans with log-location-scale...

620 IEEE TRANSACTIONS ON RELIABILITY, VOL. 59, NO. 4, DECEMBER 2010

A Tool for Evaluating Time-Varying-StressAccelerated Life Test Plans With Log-Location-Scale

DistributionsYili Hong, Haiming Ma, and William Q. Meeker

Abstract—Accelerated life tests (ALTs) are often used to maketimely assessments of the lifetime distribution of materials andcomponents. The goal of many ALTs is the estimation of a quan-tile of a log-location-scale failure time distribution. Much of theprevious work on planning accelerated life tests has focused onderiving test-planning methods under a specific log-location-scaledistribution. This paper presents a new approach for computingapproximate large-sample variances of maximum likelihood es-timators of a quantile of a general log-location-scale distributionwith censoring, and time-varying stress. The approach is based ona cumulative exposure model. Using sample data from a publishedpaper describing optimum ramp-stress test plans, we show thatour approach and the one used in the previous work give the samevariance-covariance matrix of the quantile estimator from the twodifferent approaches. Then, as an application of this approach,we extend the previous work to a new optimum ramp-stress testplan obtained by simultaneously adjusting the ramp rate with thelower start level of stress. We find that the new optimum test plancan have a smaller variance than that of the optimum ramp-stresstest plan previously obtained by adjusting only the ramp rate. Wecompare optimum ramp-stress test plans with the more commonlyused constant-stress accelerated life test plans. We also conductsimulations to provide insight, and to check the adequacy of thelarge-sample approximate results obtained by the approach.

Index Terms—Cumulative exposure model, large-sample ap-proximate variance, lognormal, maximum likelihood, ramp-stress,step-stress, Weibull.

ACRONYMS

ALT Accelerated life tests

df cumulative distribution function

pdf probability density function

ML maximum likelihood

NOR -normal distribution

SEV smallest extreme value distribution

LOGIS logistic distribution

LEV largest extreme value distribution

Manuscript received April 19, 2009; revised June 25, 2010; accepted June 27,2010. Date of publication October 25, 2010; date of current version November30, 2010. Associate Editor: J. C. Lu.

Y. Hong is with the Department of Statistics, Virginia Tech, Blacksburg, VA24061 USA (e-mail: [email protected]).

H. Ma is with the AT&T Modeling Group, Milwaukee, WI 53000 USA(e-mail: [email protected]).

W. Q. Meeker is with the Department of Statistics, Iowa State University,Ames, IA 50011 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TR.2010.2083252

NOTATION

, , Initial low level, pre-specified use level, andhighest possible level of the original stress

Transformed stress; when voltage is theramp-stress,

Standardized stress

Ramp rate

Time to reach the highest possible level ofstress

, Failure time, and censoring time

, Location, and scale parameters of alocation-scale distribution

, Parameters of the log linear regression modelwhen

, Parameters of the log linear regression modelwhen

Cumulative exposure at time .

, pdf, and cdf of a location-scale distribution

, Probabilities that a unit will fail by time atuse, and the highest stress levels, respectively

quantile of a standard location-scaledistribution

quantile of a location-scale distribution atstress level

Total number of test units

NOR -normal distribution with mean , andstandard deviation

Transpose of vector

I. INTRODUCTION

A. Accelerated Testing Background

A CCELERATED LIFE TESTS (ALT) are commonly usedin product design processes. Because there is limited time

to launch new products, engineers use accelerated tests to obtain

0018-9529/$26.00 © 2010 IEEE

HONG et al.: EVALUATING TIME-VARYING-STRESS ALT PLAN WITH LOG-LOCATION-SCALE DISTRIBUTION 621

needed information on the reliability by raising the levels of cer-tain acceleration variables like temperature, voltage, humidity,stress, and pressure. Techniques for performing an ALT includeconstant stress, step stress, and ramp stress, among others. Eval-uation of the variance of an estimator of a log-location-scaledistribution quantile (e.g., Weibull or lognormal) with varyingstress has many practical applications. Statisticians help engi-neers design -efficient ALT plans, and assess estimation preci-sion as a function of sample size. Reviews of work in this areacan be found in, for example, Nelson [12]–[15], Meeker andHahn [7], and Meeker and Escobar [8].

Most previously developed methods to calculate the large-sample approximate variance of Maximum Likelihood (ML) es-timators of distribution quantiles treated only a particular log-location-scale distribution, and a particular type of test plan(e.g., constant-stress, monotone step-stress, or monotone ramp-stress). For example, Miller and Nelson [10] presented a theoryfor optimum step-stress test plans for an exponential distribu-tion. Bai and Kim [4] developed a step-stress theory of the large-sample approximate variance for the Weibull distribution. Al-hadeed and Yang [1] give an optimum simple step-stress planfor the lognormal distribution.

To consolidate and extend this previous work, it is useful todevelop an approach for computing the large-sample approx-imate variance for a general log-location-scale distribution,and a given general form for stress variation. Besides the ele-gance of having just one algorithm, the generalization allowsevaluations for log-location-scale distributions beyond themore commonly used exponential, Weibull, and lognormaldistributions, including the loglogistic and Fréchet distributions(e.g., Meeker and Escobar [8, Chapter 4]). Recently, Ma andMeeker [6] provided an approach to calculate the large-sampleapproximate variance in step-stress test plans for a generallog-location-scale distribution. In this paper, we adapt theresults of Ma and Meeker [6] to the case of a test plans with thestress varying continuously over time, and show how to use theresults to find ramp-stress test plans that have good statisticalproperties.

B. Ramp-Stress Accelerated Test Background

Ramp-stress ALTs have been used in practice (e.g., Nelson[13, Chapter 10]). Yin and Sheng [16] studied the propertiesof the ML estimator of the exponential life distribution param-eter from a ramp-stress test plan. Nelson [10, Chapter 10] pre-sented theory to calculate the ML estimator of a quantile associ-ated with a ramp-stress ALT using a Weibull distribution, and aparticular cumulative exposure model. Bai et al. [2] consideredramp-stress ALTs with two ramp rates for the Weibull distribu-tion under Type-I censoring. Bai et al. [3] developed a methodfor finding an optimum ramp-stress ALT test plan for a Weibulldistribution by choosing a ramp rate to minimize a large-sampleapproximate variance.

In a ramp-stress ALT, units begin the test at a low level ofstress, and the level of stress increases at a constant rate. Thetest plan can be adjusted by the starting stress level, and the rate,given the use and highest levels of stress, and a censoring time.There are two ways to specify test termination. One is that the

level of stress increases linearly until a specific censoring time.The other is that initially the level of stress increases linearly.After a time fraction of the censoring time, the level of stressreaches the pre-specified highest level. Then the highest level ofstress will be applied to the surviving test units until the specificcensoring time.

A ramp-stress ALT can be viewed as a limit of a multiplestep-up stress ALT when the number of steps approaches in-finity, while the change in stress level at each step approacheszero, and the sum of the step jumps is held constant. Like step-upstress ALTs, ramp-stress ALTs can result in failures happeningmore quickly by increasing the ramp rate, and such tests onlyrequire a single temperature-controlled chamber for testing.

Different choices of the stress start level and the ramp ratecan affect estimation precision of a quantity of interest, such asa quantile of the life distribution at use conditions. Our goal isto find test plans with the optimum (i.e., the smallest) varianceof ML estimators of such quantiles.

C. Overview

The rest of this paper is organized as follows. Section II de-scribes the ramp-stress test plans used in this paper. Section IIIpresents the general time-varying stress model, and the likeli-hood for a general log-location-scale life distribution. This sec-tion also shows how to compute the large-sample approximatevariance of the ML estimator of a specified quantile of the lifedistribution under the general time-varying model, and log-lo-cation-scale distributions. Section IV obtains optimum ramp-stress plans that minimize the large-sample approximate vari-ance of the ML estimator of a specified quantile, and com-pares these optimum plans with previously-suggested optimumramp-stress plans and optimum constant-stress plans in termsof the approximate scaled large-sample variance. This sectionalso gives the results of a simulation to check the adequacy ofthe large-sample approximate variances. Section V states someconclusions, and outlines areas for future research.

II. TEST PLANS

In most ALT models there is an original stress scale (e.g.,voltage), and an implied one-to-one transformation of stress(e.g., log of voltage). We use , and to denote the original,and transformed stresses, respectively. The stress applied to testspecimens in a ramp-stress test will always be between an initiallower level of stress , and the highest possible level of stress

. can be either higher or lower than the use level of stress. As in Bai et al. [3], in the ramp stress test plans we consideras a voltage, and use a transformed stress, .

Sometimes, for convenience, we use a standardized stress. This standardized stress has the nice prop-

erties that , and . The maximum test length istime units (e.g., hours). If we define the time for the stress levelto reach to be , then , and ,where is the constant rate of change in(e.g., volts per hour).

Fig. 1 shows the two possible test schemes: (a) , i.e.,; and (b) , i.e., . By simultaneously opti-

mizing and , one can usually obtain a smaller variance of


Fig. 1. Ramp-stress test schemes considered in this paper.

the quantile estimator by using scheme (a), relative to scheme(b).

III. THE MODEL AND LOG LIKELIHOOD

A. Model

This section shows how to compute the large-sample approx-imate variance of the ML estimator of a quantile of a generallog-location-scale distribution at use conditions from continu-ously time-varying stress accelerated life tests. We begin with amultiple step-stress test. Then we let the number of steps ap-proach infinity as the change of stress level at each step ap-proaches zero, holding the sum of the step jumps constant.

We assume that at any level of stress the failure time followsa log-location-scale distribution with cdf

where the location parameter is , and the scaleparameter is constant. We also assume that Nelson’s [13] cu-mulative exposure model holds. This model implies that the dis-tribution of the remaining life of a test unit depends only on thecumulative exposure it has received no matter how it was ex-posed.

In a multiple step-stress test, we define the standardized logtime at stress level to be ,

, and , where, , is the time at which the

stress level change from to , , and is the totalnumber of stress levels in the experiment. The time shift inducedby the change in stress levels from the beginning of the test tothe beginning of step is

Under the cumulative exposure model, we have, . When , under the limiting

process described at the beginning of this section, we have, where

is the cumulative exposure at time . Note that must beintegrable. This is a weak condition that is met in both step-stress, and ramp-stress test plans, whether the changes in stressare monotone or not. The cdf, and pdf of when are

and

respectively. The failure probabilities at the highest level ofstress , and the use level of stress canbe expressed as , and

, respectively.

B. Log Likelihood

The log likelihood for a single test unit having a log-location-scale failure time distribution under stress is

(1)

where if , and otherwise. Here, and . The total

log likelihood is obtained by summing (1) over all test units. TheML estimators , , and are those values that maximize thetotal log likelihood. The first and the second partial derivativesof (1) with respect to the model parameters are given in theAppendix.

C. The Large-Sample Approximate Variance

Under the standard regularity conditions (which hold for log-location-scale distributions), the large-sample approximate vari-ance-covariance matrix of , and , denoted by , isthe inverse of the Fisher information matrix (FIM). The FIM isthe expectation, with respect to the data, of the negative Hessianmatrix (the second derivatives of log likelihood in (1) with re-spect to the model parameters), evaluated at the model parame-ters. The Appendix provides expressions for the elements of theFIM.

The ML estimator of the quantile at standardized stress is. The large-sample approximate variance

of is . Our goal isto minimize at (i.e., at the use conditions). Thetest plan properties to be optimized are , and . We used thefunction in R [11] to optimize.

To compare the relative efficiency of test plans with differentsample sizes, we use the scaled large-sample approximate vari-ance defined as . The large-sample approxi-mate variance-covariance matrix and properties of optimum testplans depend on the planning values of the parameters , ,and . As usual, when dealing with locally optimum designs


Fig. 2. Simulated scaled variance as a function of the expected total number of failures under the two-dimensional optimum ramp-stress test plan for the Weibull,and lognormal distributions as compared to the corresponding large-sample approximate variance (horizontal dashed line). The number of simulations at each pointis 5,000. The optimum two-dimensional test plan has the following conditions for the Weibull: � � ��, � � ��, � � ��, � � �� , � � � �,� � �� , � � �� , � � �� and � � ��, and for lognormal � � ��, � � ��, � � ��, � � �� , � � � �, � � �� ,� � �� , � � �� , and � � ��.

(i.e., when the optimum depends on the model parameters), suchplanning values are obtained from some combination of pre-vious experience and engineering judgment.

In the case of the ramp-stress test plans we consider,when , and otherwise.

Note that ramp stress refers to a test in which is linearin . will not, in general, be linear in . Simple deductionshows that defines the shape of time dependenceof , and thus . Therefore, in addition to , , and ,the large-sample approximate variance-covariance matrix andproperties of the optimum test plans will also depend on .

IV. OPTIMUM RAMP-STRESS PLANS

A. Comparison With Existing Optimum Ramp Stress Plans

In Bai et al. [3], a one-dimensional optimum ramp-stress testplan is proposed. This plan is similar to that given in Fig. 1 ex-cept that , and the ramp rate is optimized. The locationparameter is expressed as , which is a little dif-ferent from our notation described in Section III. It is easy toconvert from one parametrization to the other. Bai et al. [3] con-sidered a case where , , , ,

, , and . The quan-tile of interest is . Bai et al. [3] also provided simu-lated sample data yielding estimates , , and

. Our approach gives the same variance-covariancewith respect for as that given in [3], where the au-thors used a different theoretical approach that is only valid forthe Weibull distribution.

Using , , and as input to the algo-rithm proposed in this paper, we find that the one-dimensionaloptimum is 1635.1 at when

is fixed at 0 kV. We extend the one-dimensional optimizationof the ramp-stress test plan in [3] into a two-dimensional opti-mization. By selecting , and ,we obtain the optimum , a nearly

8.6% reduction in variance with respect to that of the one-di-mensional optimization. We also obtained the results for the log-normal case. To make the results comparable, we use ,

, , , , and obtained, for the lognormal case. For the one dimen-

sional case, we have , and the scaled optimum vari-ance is 509.6. For the two dimensional case, we have ,

, and the scaled optimum variance is 475.5. There isalso a nearly 6.7% improvement. In the rest of this section, wewill only work with the optimum test plan from our two-dimen-sional optimization.

B. Comparison of the Large-Sample Approximate Variance toa Variance Obtained With Simulations

To assess the adequacy of the large-sample approximate vari-ances used in this paper, we conducted simulations to com-pare with the actual variance as a function of expected totalnumbers of failures during the test. Fig. 2 shows the simulatedscaled variance as a function of the expected total number offailures under the two-dimensional optimum ramp-stress testplan. As expected, when the expected total number of failuresgets large, the simulated variance approaches the large-sampleapproximate variance. In this test plan, the probability that a testunit will fail during the test is 99.99%. Thus the expected totalnumber of failures is almost exactly the same as the sample size.

C. Comparison of Optimum Ramp Stress Plans With OptimumConstant-Stress Plans

Ma and Meeker [6] compared the optimum step-stressand constant-stress test plans in terms of the large-sampleapproximate variance of the ML estimators of a quantile ofinterest. In this paper, we extend the comparison to optimumramp-stress tests. For the constant-stress tests, the two-dimen-sional optimization is done by adjusting the lower level ofstress as well as the allocations of test units to the constant


Fig. 3. Scaled optimum variance as a function of � for the Weibull distribution at three possible quantiles of interest for � � �� and � � �� with � � �

(left), and � � �� (right).

Fig. 4. Scaled optimum variance as a function of � for lognormal distribution at three possible quantiles of interest for � � ��, and � � �� with � � �

(left), and � � �� (right).

stress levels. For step-stress tests, the optimization is done byadjusting the lower level of stress as well as the fraction oftest time spent at each level. Note that a ramp-stress test planis a limit of a multiple step-stress test plan under the limitingprocess described in Section III-A. Note also that the optimumtwo-level step-stress has the smallest variance for reasonableplanning values without any constraint on the fraction of timeat each of the stress levels. Thus, the variance of a ramp-stresstest plan is not expected to be smaller than the variance of theoptimum two-level step-stress test plan given the same planningvalues. Therefore, we compare the ramp-stress test plans withthe commonly used constant-stress test plans using the sameplanning values used in [6].

Fig. 3 shows as a function of under theoptimum ramp-stress test plans with (left), and(right) for , 0.10, and 0.50, respectively, at ,and for the Weibull distribution. Fig. 4 shows sim-ilar results for the lognormal distribution. The scaled approxi-mate variance decreases as increases. Thescaled variance at is the largest, followed by the vari-ance for , and then . These results are similarto those that were observed in [6] for step-up-stress test plans.

Figs. 3 and 4 also show that the optimum variance with a largeris slightly larger than that with a smaller under the same

planning values.Our major interest is variance comparison. In [6], the op-

timum two-level constant-stress are 95, 120,and 149 for , 0.10, and 0.50, respectively with

, and under the Weibull distribution. Comparedwith the values of on the left of Fig. 3, wefind an optimum two-level constant-stress test plan has a smaller

than that of an optimum ramp stress test planin this particular situation. The conclusion is similar for the log-normal distribution. Our other results (not shown here) indicatethat this conclusion also holds for the three other pairs of plan-ning values ( , and ), ( , and

), and ( , and ) when isbetween 0.5 and 1.5.

V. CONCLUDING REMARKS

In this paper, we propose an approach for computingthe large-sample approximate variance of ML estimators ofquantiles of the widely-used log-location-scale family of dis-tributions with continuous time-varying stress accelerated life


tests and censoring. We applied this approach to a situation thatcorresponds to the ramp-stress sample data described in [3].The results obtained by our more general method of computinglarge-sample approximate variances agree with the special-caseresults reported in [3]. We also extend previously-developedone-dimensional optimum ramp-stress test plans to the two-di-mensional optimum ramp-stress test plans. The variance of theML estimator of a quantile at the planning values was investi-gated for the Weibull and lognormal distribution, and was foundto be larger for the two-dimensional optimum ramp-stress testplans than for the simple optimum two-level constant-stresstest plans.

One may raise the point that it is difficult to distinguish be-tween a Weibull and a lognormal distribution when sample sizesare small. In real applications of accelerated testing, engineersconducting the test often have a good idea of the distributionfrom previous experience with the failure mechanism understudy. When such information is not available, analyses areoften done with both the lognormal and Weibull distributions,and perhaps other distributions that are consistent with the data.Then it is often deemed prudent to use the more conservativeanswer (generally provided by the Weibull distribution if thereis extrapolation in time). The practical purpose of optimumreliability tests is to provide insight into how to plan actual teststhat have good statistical properties, but also meet practicalconstraints (such as robustness to departures from inputs). Thisinsight is described further in Nelson [13, Chapter 6], andMeeker and Escobar [8, Chapter 20].

There are a number of possible extensions of the work pre-sented here.

• The formulas in the Appendix could be extended to mul-tiple-stress situations (e.g., temperature and voltage) whereeither or both could be increasing in the test.

• We have considered only log-location-scale distributions.It would be possible to derive similar results for other non-log-location-scale distributions (e.g., the gamma distribu-tion).

• The response in an accelerated life test is time to failure.In some accelerated tests, the response is degradation. Insome applications, degradation can be monitored or mea-sured periodically (see, for example, Meeker et al. [9]). Inother applications, degradation is a destructive measure-ment (see, for example, Nelson [12], and Escobar et al.[5]). It would be interesting to develop methods parallelto ours where the degradation measure follows a log-loca-tion-scale distribution.

APPENDIX A

This Appendix provides the derivatives of the log likelihoodgiven in Section III, and the details of the approach that we usedto calculate needed expectations, and the large-sample approx-imate variance-covariance matrix of the ML estimators of ,

, and .The log likelihood for a single test unit having a log-location-

scale failure time distribution under stress is given in (1).

The first partial derivatives of the log likelihood with respect toparameters for a single observation can be expressed as

Here is the failure indicator, andNote that

here denotes the derivative with respect to instead of .The second derivatives are

Here,


whereNote that here denotes the second derivative with respectto instead of . Also,

and

Table I gives the detailed formulae for commonly-used loca-tion-scale distributions (normal, smallest extreme value, largestextreme value, and logistic, corresponding to the lognormal,Weibull, Fréchet, and loglogistic log-location-scale distribu-tions). The scaled expectations of the second derivatives are

(2)

(3)

(4)

(5)

(6)

(7)

TABLE IDETAILED FORMULAE FOR COMMONLY-USED LOCATION-SCALE

DISTRIBUTIONS

Let . The Fisher information matrix of atime-varying stress ALT can be computed as

After factoring out the common factor of , the Fisher infor-mation matrix still depends on through , ,and . Under the standard regularity conditions, thelarge-sample approximate variance-covariance matrix of theML estimators of , , and is (See equation at bottomof page) To compute the , one needs to compute thescaled expectations of the second derivatives in (2)–(7). Withthe specified distribution, and the stress function , one needsto substitute the corresponding formulae in Table I into (2)–(7),and do the integrations. Numerical integration is needed for thecomputations.

ACKNOWLEDGMENT

The authors would like to thank an anonymous referee andthe Associate Editor for providing helpful suggestions that im-proved this paper.


REFERENCES

[1] A. A. Alhadeed and S. S. Yang, “Optimum Simple Step-Stress Plan forCumulative Exposure Model Using Log-Normal Distribution,” IEEETrans. Reliability, vol. 50, pp. 64–68, 2005.

[2] D. S. Bai, M. S. Cha, and S. W. Chung, “Optimum simple ramp-testsfor the Weibull distribution and type-I censoring,” IEEE Trans. Relia-bility, vol. 41, pp. 407–413, 1992.

[3] D. S. Bai, Y. R. Chun, and M. S. Cha, “Time-Censored Ramp Tests withStress Bound for Weibull Life Distribution,” IEEE Trans. Reliability,vol. 46, pp. 99–107, 1997.

[4] D. S. Bai and M. S. Kim, “Optimum Simple Step-Stress AcceleratedLife Tests for Weibull Distribution and Type I Censoring,” Naval Re-search Logistics, vol. 40, pp. 193–210, 1993.

[5] L. A. Escobar, W. Q. Meeker, D. L. Kugler, and L. L. Kramer, “Accel-erated Destructive Degradation Tests: Data, Models, and Analysis,” inMathematical and Statistical Methods in Reliability, B. H. Lindqvistand K. A. Doksum, Eds. : World Scientific Publishing Company,2003.

[6] H. Ma and W. Q. Meeker, “Optimum Step-Stress Accelerated Life TestPlans for Log-Location-Scale Distributions,” Naval Research Logis-tics, vol. 55, pp. 551–562, 2008.

[7] W. Q. Meeker and G. J. Hahn, How To Plan An Accelerated LifeTest-Some Practical Guideline. Milwaukee, Wisconsin: AmericanSociety for Quality Control, 1985, Volume 10 in the American Societyfor Quality Control Basic References in Quality Control: StatisticalTechniques..

[8] W. Q. Meeker and L. A. Escobar, Statistical Methods for ReliabilityData. New York: Wiley, 1998.

[9] W. Q. Meeker, L. A. Escobar, and C. J. Lu, “Accelerated Degrada-tion Tests: Modeling and Analysis,” Technometrics, vol. 40, pp. 89–99,1998.

[10] R. Miller and W. Nelson, “Optimum Simple Step-Stress Plans for Ac-celerated Life Testing,” IEEE Trans. Reliability, vol. 32, pp. 59–65,1983.

[11] W. N. Venables and D. M. Smith, An Introduction to R. Bristol, U.K.:Network Theory Ltd., 2004.

[12] W. Nelson and T. J. Kielpinski, “Theory for Optimum Censored Ac-celerated Tests for Normal and Lognormal Life Distributions,” Tech-nometrics, vol. 18, pp. 105–114, 1976.

[13] W. Nelson, Accelerated Testing Statistical Models, Test Plans and DataAnalysis. New York: Wiley, 1990.

[14] W. Nelson, “A Bibliography of Accelerated Test Plans, PartI–Overview,” IEEE Trans. Reliability, vol. 54, pp. 194–197, 2005.

[15] W. Nelson, “A Bibliography of Accelerated Test Plans, Part II–Refer-ences,” IEEE Trans. Reliability, vol. 54, pp. 370–373, 2005.

[16] X. K. Yin and B. Z. Sheng, “Some Aspects of Accelerated Life Testingby Stress,” IEEE Trans. Reliability, vol. 36, pp. 150–155, 1987.

Yili Hong received a BS in statistics (2004) from University of Science andTechnology of China and MS and PhD degrees in statistics (2005, 2009) fromIowa State University. He is currently an Assistant Professor in the Departmentof Statistics at Virginia Tech. His research interests include reliability data anal-ysis and engineering statistics.

Haming Ma received MS and PhD degrees in statistics (2003, 2009) from IowaState University. He is currently working as a modeler in the AT&T modelinggroup in Milwaukee, WI.

William Q. Meeker is a Professor of Statistics and Distinguished Professor ofLiberal Arts and Sciences at Iowa State University. He is a Fellow of the Amer-ican Statistical Association, and of the American Society for Quality. He is anelected member of the International Statistical Institute, and a past Editor ofTechnometrics. He is co-author of the books Statistical Methods for ReliabilityData with Luis Escobar (1998), Statistical Intervals: A Guide for Practitionerswith Gerald Hahn (1991), nine book chapters, and numerous publications inthe engineering and statistical literature. He has won the American Society forQuality (ASQ) Youden prize five times, the ASQ Wilcoxon Prize three times,and the American Statistical Association Best Practical Application Award. Heis also an ASQ Shewhart Medalist. He has consulted extensively on problemsin reliability data analysis, reliability test planning, accelerated testing, nonde-structive evaluation, and statistical computing.

a tool for evaluating time-varying-stress accelerated life test plans with log-location-scale...

Documents