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Equivalent step-stress accelerated life tests with log-location-scale lifetime distributions under Type-IcensoringCheng-Hung Hua, Robert D. Planteb & Jen Tangb

a Department of Industrial and Information Management, National Cheng Kung University,Tainan City, Taiwan, 70101 E-mail:b Krannert Graduate School of Management, Purdue University, West Lafayette, IN47907-2056, USA E-mail:Accepted author version posted online: 06 Jun 2014.Published online: 21 Nov 2014.

To cite this article: Cheng-Hung Hu, Robert D. Plante & Jen Tang (2014): Equivalent step-stress accelerated life tests withlog-location-scale lifetime distributions under Type-I censoring, IIE Transactions, DOI: 10.1080/0740817X.2014.928960

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IIE Transactions (2015) 47, 1–13Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/0740817X.2014.928960

Equivalent step-stress accelerated life tests withlog-location-scale lifetime distributions under Type-Icensoring

CHENG-HUNG HU1, ROBERT D. PLANTE2 and JEN TANG2,*

1Department of Industrial and Information Management, National Cheng Kung University, Tainan City, Taiwan, 70101E-mail: chhu@mail.ncku.edu.tw2Krannert Graduate School of Management, Purdue University, West Lafayette, IN 47907-2056, USAE-mail: jtang@purdue.edu

Received December 2011 and accepted March 2014

Accelerated Life Testing (ALT) is used to provide timely estimates of a product’s lifetime distribution. Step-Stress ALT (SSALT) isone of the most widely adopted stress loadings and the optimum design of a SSALT plan has been extensively studied. However,few research efforts have been devoted to establishing the theoretical rationale for using SSALT in lieu of other types of stressloadings. This article proves the existence of statistically equivalent SSALT plans that can provide equally precise estimates to thosederived from any continuous stress loading for the log-location-scale lifetime distributions with Type-I censoring. That is, for anyoptimization criterion based on the Fisher information matrix, SSALT is identical in comparison to other continuous stress loadings.The Weibull and lognormal distributions are introduced as special cases. For these two distributions, the relationship among statisticalequivalencies is investigated and it is shown that two equivalent ALT plans must be equivalent in terms of the strongest version ofequivalency for many objective functions. A numerical example for a ramp-stress ALT, using data from an existing study on miniaturelamps, is used to illustrate equivalent SSALT plans. Results show that SSALT is not only equivalent to the existing ramp-stress testplans but also more cost-effective in terms of the total test cost.

Keywords: Accelerated life test, optimum ALT plan, step-stress loading, Fisher information matrix

1. Introduction

1.1. Accelerated life test background

Continued advances in design and manufacturing technol-ogy and intense global competition, as well as increas-ing customer requirements for high-quality and reliableproducts, have led to a generation of products that operatewithout failure for relatively longer periods of time. This in-creasingly longer life of products makes testing under nor-mal operating conditions more costly and time-consuming.The need to obtain reliability information in a timely man-ner has led to the development of Accelerated Life Testing(ALT). In an ALT, test units are tested at higher levels ofstress (e.g., pressure, temperature, humidity, etc.) to induceearly failures. The observed failure data are then analyzedand extrapolated to the designed use stress level to esti-mate the lifetime distribution of a product under normaloperating conditions.

∗Corresponding author

1.2. Literature review for different stress loadings

Prior to conducting an ALT, practitioners should carefullychoose the type of stress factors, the stress loading, the lev-els of stress, the allocation of sample units to each stresslevel, the test termination time, etc. Several types of stressloadings have been proposed and studied in the literature,including parallel constant-stress, step-stress, ramp-stress,cyclic-stress, and combinations thereof. For graphical rep-resentations of these stress loadings, see Elsayed et al.(2009). For different types of stress loading, the problem ofdesigning optimum ALT plans has been studied by manyscholars. A general criterion for designing an ALT planis to minimize the asymptotic variance of the MaximumLikelihood Estimate (MLE) of a function of lifetime pa-rameters; for example, the pth percentile and Mean Time toFailure (MTTF). In most models, the asymptotic varianceof the MLE is related to the Fisher information matrix,which provides sample information for the model and theassociated parameters. Hence, to design an optimum ALTplan, one typically begins by finding the Fisher informationmatrix for a specific type of stress loading. The followingare commonly used ALT plans.

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Parallel constant-stress ALT plan: Generally, in a ParallelConstant-stress ALT (PCSALT), k constant levels (k > 1)of stress are simultaneously used. Ng et al. (2007) discussedthe optimal allocation of test units to different stress lev-els under the assumption of a Weibull distribution. Maand Meeker (2010) proposed a strategy for planning a PC-SALT when the sample size is small. Zhu and Elsayed(2013) investigated the design of ALT plans under pro-gressive censoring under competing failure modes for theminimization of the asymptotic variance of the mean timeof first failure. For other papers that discuss PCSALT, seeZhang and Meeker (2006), Yang (2010), and Liu and Tang(2013). In practice, a PCSALT is easy to implement; how-ever, it may suffer from long test durations and the need formore test units under lower stress levels to induce enoughfailures.

Step-stress ALT plan: A Step-Stress ALT (SSALT) usesa step function as the time-varying stress loading and atest plan is determined by its stress levels and stress changetimes. When there are only two stress levels (i.e., one changetime), it is referred to as a simple SSALT. Miller and Nel-son (1983), Bai and Kim (1993), and Alhadeed and Yang(2005) developed optimum SSALT plans under differentlifetime distribution assumptions. Yuan et al. (2012) pro-posed Bayesian methods for planning an optimal simpleSSALT when the model parameters are uncertain. Hanand Ng (2013) compared the optimal k-level PCSALT toSSALT for exponential failure data in order to quantify theadvantages of using the SSALT. Wang et al. (2012) com-pared the efficiency of step-up and step-down ALT for fourcriteria by using Monte Carlo simulation. For more recentresearch efforts regarding SSALT, see, for example, Fardand Li (2009), Liu and Qiu (2011), Fan et al. (2013), andLee et al. (2013).

Ramp-stress ALT plan: A ramp-stress assumes that thestress path is a linear function of time. By changing theintercept and slope of the linear function, one obtains aramp-stress test plan. References for ramp-stress test plansinclude Bai et al. (1992), Srivastava and Mittal (2012), andZhang and Liao (2014). For an overview of statistical ALTmodels, methods, optimum plans, and applications, referto Nelson (1990, 2005a, 2005b) and Elsayed (2012).

1.3. Motivation and overview

Traditionally, the type of a stress loading is pre-selected andthe optimum test plan is obtained accordingly. However, aspointed out by Elsayed et al. (2009, p. 152):

Each stress loading has some advantages and drawbacks.This has raised many practical questions such as: Can ac-celerating test plans involving different stress loadings bedesigned such that they are equivalent? What are the mea-sures of equivalency? Can such test plans and their equiv-alency be developed for multiple stresses especially in thesetting of step-stress tests and other profiled stress tests?

Liao and Elsayed (2010) also mentioned that the lackof research conducted to demonstrate the equivalency be-tween SSALT and PCSALT has limited the use of such testsin industry. To address this issue, they proposed several def-initions of Statistical Equivalency (SE) and provided analgorithm to search for equivalent ALT plans. How-ever, conditions that guarantee the existence of equivalentSSALT plans were not discussed in these papers.

Our goal in this article is to show that, under certainmodel assumptions, there exist SSALT plans that can pro-vide identical information to that from any continuousstress loading design. Furthermore, we investigate the rela-tionship among the equivalency definitions and show that,for many common objectives, two ALT plans can only beequivalent in the sense of the strongest version of SE givenin Liao and Elsayed (2010). Moreover, a cost model forALT plans is used as an optimization criterion to minimizethe total test costs without sacrificing equivalency when astatistically equivalent plan is not unique. The remainder ofthis article is organized as follows. In Section 2, we presentthe model assumptions used in this article and review thedefinitions of equivalent ALT plans. In Section 3, we de-rive the log-likelihood function and the Fisher informationmatrix for a time-varying stress loading for distributionsin the log-location-scale family. Using the derived matrix,we show how a continuous time-varying stress loading canbe equivalenced by a finite steps step-stress loading. TheWeibull and lognormal models are illustrated as special ex-amples in Section 4. For these two distributions, we showthat equivalent ALT plans only exist at the strongest ver-sion of SE for many objective functions. In Section 5, weuse a numerical example from Liao and Elsayed (2010)to illustrate equivalent SSALT plans. Additionally, whileretaining equivalency, a cost model is used to obtain aminimum-cost SSALT plan for practical usage. Section 6draws conclusions and outlines promising future researchtopics.

2. Model assumptions

In this section, we describe the parametric model assump-tions used throughout this article. First, let x(t) > 0 be atime-varying stress loading (or function), with step- andramp-stress loadings as examples. We assume an Acceler-ated Failure Time (AFT) model (also referred to as the Ad-ditive Accumulative Damage model; see Gerville-Reacheand Nikulin (2007)) for the lifetime Cumulative Distribu-tion Function (CDF) under x(t). Under this assumption,the effect of stress is multiplicative on the random lifetimeor a function of the lifetime. The AFT model is a verygeneral model and has extensive applications. Many well-known models used in ALT are special cases; for example,the basic (linear) Cumulative Exposure (CE) model (Nel-son, 1990; Bai et al., 1992), the model used in Srivastava andMittal (2010) for partial SSALT, and the general model for

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Step-stress accelerated life tests 3

SSALT in Zhao and Elsayed (2005). Specifically, our modelassumptions are as follows.

1. The life testing model is an AFT model, with the lifetimeCDF under the stress loading x(·):

Fx(·)(t) = F0

⎛⎜⎜⎜⎜⎝

ln

(t∫

0a(x(z))dz

)− μ0

σ0

⎞⎟⎟⎟⎟⎠ , (1)

where F0(·) is a baseline standard CDF, independentof a(·). In this article, we assume that F0(·) comes fromthe log-location-scale family with unknown location andscale parameters (μ0, σ0).

2. The acceleration function or life-stress relationship, a(·),is a positive continuous, non-decreasing function and itsparameters are assumed to be estimated and then fixedin advance based on an engineer’s experience or pilotstudies of similar products.

3. The test is terminated at a pre-specified censoring timeη (Type-I or time censoring).

Assumption 1 is general, given that many commonlyused distributions in reliability engineering are from thelog-location-scale family. Note that Assumption 1 impliesthat the form of the lifetime CDF under a time-varyingstress, x(t), is the same as that under normal use condi-tions, x0. Obviously, Fx(·)(t) at t depends on the stress func-tion x(·) but only up to t—i.e., {x(z); 0 ≤ z ≤ t}—throughthe acceleration function a(·). If a Weibull or lognormaldistribution is assumed, F0(u) = F0,SEV(u) = 1 − e−eu

or =�(u) with SEV representing the Smallest Extreme Valuedistribution and �(·) being the CDF of the standard nor-mal distribution.

The acceleration function a(·) determines the relation-ship between the parameters of a lifetime distribution un-der accelerated and normal use stress. Assumption 2 impliesthat, following the life testing, the only parameters neededto be estimated are those in F0(u). Several well-known rela-tionships, such as the Arrhenius, inverse power, and Eyringrule, have their own special acceleration factors (see Zhaoand Elsayed (2005)). Elsayed (2003) presented several ex-amples of real ALT data and demonstrated methods toestimate the acceleration factors under different life-time distributions and life/stress relationships using leastsquares regression.

Remark 1: It is important to notice that although themodel assumed here is general, it remains necessary tocheck whether the data collected from an ALT supports themodel assumptions. Procedures for testing the Goodness-of-Fit (GOF) of a specific distribution to data are available.For example, graphical tools such as probability plot (or Q-Q plot) can be used to assess the adequacy of the fitteddistribution. Probability plots of standardized residu-als versus fitted values are useful for checking the lin-

ear life/stress model and constant scale parameter as-sumptions (Meeker and Escobar (1998, pp. 122–152,445–447). For formal GOF tests, one could use theKolmogorov–Smirnov and Anderson–Darling tests (Gib-bons and Chakraborti (2011, ch. 4). Moreover, if there arecompeting lifetime models, some criteria such as Akaike’sInformation Criterion and Schwarz’s Bayesian Criterioncan be used to choose the best-fitted model for the col-lected data.

Under three assumptions above, we will prove the ex-istence of an equivalent step-stress loading to any givencontinuous stress loading. Before proceeding further, wefirst describe the definitions of SE of ALT plans providedby Liao and Elsayed (2010) as follows.

Let θ be the vector of unknown model parameters, G(θ)be a scalar function of the parameters, a(θ) = ∂G(θ )

/∂θ be

the gradient vector of G(θ ), and θ̂ denote the observed MLEof θ from a given ALT plan with observed asymptotic co-variance matrix X(θ̂). Another ALT plan with asymptoticcovariance matrix Y(θ) is said to have one of the following.

1. General SE to the given ALT plan if a(θ̂ )′X(θ̂)a(θ̂ ) =a(θ̂)′Y(θ̂ )a(θ̂ ) where a(θ̂ )′stands for the transpose of thevector a(θ̂ ).

2. Type-I SE to the given ALT plan if a(θ̂)′X(θ̂) =a(θ̂)′Y(θ̂ ).

3. Type-II SE to the given ALT plan if it has Type-I SEand also the norm of every column of Y(θ̂) is the sameas the norm of the associated column in X(θ̂).

4. Type-III SE to the given ALT plan if X(θ̂) = Y(θ̂ ).

The definition of Type-III SE above provides thestrongest version of equivalency among the four types ofequivalency between two ALT plans. Liao and Elsayed(2010) also provided geometric interpretations of these SEdefinitions. In this article, we investigate Type-III statisti-cally equivalent SSALT plans to a continuous stress loadingin a more general sense by directly comparing two covari-ance matrices without fixing the value of θ at θ̂ from thegiven plan. One reason for focusing on Type-III statisti-cally equivalent SSALT plans is that they are automaticallyequivalent to the given plan with respect to any equiva-lent criterion based on the covariance matrix. Also, as wewill show later, for many commonly used objectives, twoALT plans are equivalent to any type if and only if they areType-III statistically equivalent for the Weibull and lognor-mal distribution models.

It is also possible to define other types of equivalencyand optimality if one is interested only in estimating theparameter vector θ instead of G(θ ). One may focus on theasymptotic covariance matrix of θ̂ and use criteria such asD-criterion (based on the determinant of the asymptoticcovariance matrix) or A-criterion (based on the trace ofthe asymptotic covariance matrix; see Ng et al. (2007)), fordefining SE between two ALT plans.

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3. Equivalent Type-III SSALT plans

In this section we show that, under Assumptions 1 to 3 inSection 2 and for any objective function that completelydepends on the asymptotic covariance matrix, there ex-ist SSALTs with a finite number of steps that are Type-IIIstatistically equivalent to any ALT with continuous x(·), in-cluding ramp-stress, modified step-stress (Park and Yum,1998), and cyclic stress functions. To show this main result,we first derive the Fisher information matrix of parame-ters under an arbitrary continuous stress loading x(t) inSection 3.1.

3.1. The Fisher information matrix under x(t)

Recall that (μ0, σ0) are respectively the unknown locationand scale parameters of F0(·), which are assumed to beindependent of the acceleration factor. Under model (1)and a continuous stress loading x(·), the likelihood andlog-likelihood function of parameters from a single testunit are, respectively,⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

L(μ0, σ0) = Iyt ×{

1σ0

f0

(yt − μ0

σ0

)}

+ (1 − Iyt ) ×{

1 − F0

(T − μ0

σ0

)},

l(μ0, σ0) = Iyt ×{−ln(σ0) + ln

(f0

(yt − μ0

σ0

))}

+ (1 − Iyt ) ×{

ln(

1 − F0

(T − μ0

σ0

))},

,(2)

where

yt ≡ ln

⎛⎝ t∫

0

a(x(z))dz

⎞⎠ , T ≡ ln

⎛⎝ η∫

0

a(x(z))dz

⎞⎠ ,

Iyt ≡{

0, if yt > T1, if yt ≤ T

,

and f0(·) is the Probability Density Function (PDF) forF0(·). To ease notation, we did not explicitly express x(t)but note that all of the quantities (yt, T, Iyt ) would dependon it.

The symmetric Fisher information matrix can be ob-tained by taking the expected value of the negative secondderivative of the log-likelihood function in Equation (2)with respect to (μ0, σ0):

Ix(·)(μ0, σ0) ≡[

I11 I12I21 I22

]

=

⎡⎢⎢⎣

E[−∂2l(μ0, σ0)

∂μ20

]E[−∂2l(μ0, σ0)

∂μ0∂σ0

]

E[−∂2l(μ0, σ0)

∂σ0∂μ0

]E[−∂2l(μ0, σ0)

∂σ 20

]⎤⎥⎥⎦ , (3)

where (derivations in Appendix A)

I11 =α∫

−∞

{−(∂2 f0(z)

/∂μ2

0

)f0(z) − (

∂ f0(z)/∂μ0

)2

f0(z)

}dz

+{

−(1/σ0) (

∂ f0(α)/∂μ0

)(1 − F0(α)) − ((

1/σ0)

f0(α))2

1 − F0(α)

},

I12 =α∫

−∞

{−(∂2 f0(z)

/∂μ0∂σ0

)f0(z) − (

∂ f0(z)/∂μ0

) (∂ f0(z)

/∂σ0

)f0(z)

}dz

+⎧⎨⎩−

(− (1/σ 2

0

)f0(α) + (

1/σ0) (

∂ f0(α)/∂σ0

))(1 − F0(α)) − α

σ 20

( f0(α))2

1 − F0(α)

⎫⎬⎭ ,

I22 =α∫

−∞−{

1

σ 20

+(∂2 f0(z)

/∂σ 2

0

)f0(z) − (

∂ f0(z)/∂σ0

)2

f0(z)

}dz

+{

−(− (

2α/σ 2

0

)f0(α) + (

α/σ0) (

∂ f0(α)/∂σ0

))(1 − F0(α)) − ((

α/σ0)

f0(α))2

1 − F0(α)

},

and α ≡ (T − μ0)/σ0. The α, which is a function of

x(t), can be considered as the standardized censoring time(Meeker and Nelson, 1977). We note that the matrix (andthus the covariance matrix of MLE) depends only on(μ0, σ0, α) (or (μ0, σ0, T)).

3.2. Equivalent SSALT plans

Since an estimated covariance matrix of the MLE of θ isproportional to the inverse of the estimated Fisher informa-tion matrix, two plans are Type-III statistically equivalentif they have identical Fisher information matrices The fol-lowing proposition summarizes our first main result.

Proposition 1. For an ALT plan with a continuous stressloading x(t), if Assumptions 1 to 3 in Section 2 are satisfied,there exist SSALTs with a finite number of steps that areType-III statistically equivalent to the given ALT.

Proof. From the information matrix in Equation (3), wenotice that the entire matrix depends on x(t) but onlythrough T ≡ ln

(∫ η

0 a(x(z))dz). Consequently, if any stress

loading possesses the same T-value as x(t), its Fisher infor-mation matrix is identical to that from x(t) and hence thetwo plans have Type-III statistically equivalent. �

Since both x(·) and a(·) are assumed to be continuous,a(x(·)) is also continuous. Then, by the first mean-valuetheorem for integrals (Salas et al., 1995), for any positiveinteger m and any finite partition {0 ≡ t0, t1, . . . , tm ≡ η}of [0, η], there exists a point, say t′

i , within each subinterval[ti , ti+1] such that

ti+1∫ti

a(x(z))dz = a(x(t′i ))(ti+1 − ti ), i = 0, 1, . . . , m − 1.

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Step-stress accelerated life tests 5

Consequently, we have

T ≡ ln

⎛⎝ η∫

0

a(x(z))dz

⎞⎠ = ln

⎛⎝m−1∑

i=0

⎛⎝ ti+1∫

ti

a(x(z))dz

⎞⎠⎞⎠

= ln

(m−1∑i=0

(a(x(t′i ))(ti+1 − ti ))

).

Thus, the step function s(t) with s(t) = x(t′i ), for ti < t ≤

ti+1 (i = 0, 1, ... , m − 1), gives the identical T-value asx(t). Since x(t) and s(t) have the same Fisher informationmatrix, they are Type-III statistically equivalent.

According to our proof of Proposition 1, for any given fi-nite partition of [0, η] (and thus the number of steps), thereexists a statistically equivalent SSALT. Hence, a statisticallyequivalent SSALT is likely to be non-unique. In this case, anexperimenter can consider a second (or multiple) criterion,such as the test cost criterion discussed in Section 5.3, forselecting appropriate statistically equivalent SSALT plans.On the other hand, if one considers a more complicatedstatistical model (e.g., more parameters to be simultane-ously estimated, resulting in more constraints to satisfyin Type-III SE), it is possible that a Type-III statisticallyequivalent SSALT plan does not exist. In this case, one canconsider other types of lifetime models or a less stringentequivalency criterion (e.g., Type-I or Type-II SE to reducethe number of constraints) so that the feasible region forthe design variables of the proposed statistically equivalentplan is not empty. In addition, if the initial stress loading,x(t), is not continuous, statistically equivalent SSALT plansmay not exist. For example, Ma and Meeker (2008) com-pared a PCSALT with a SSALT and demonstrated that, forcertain model parameter values, the optimal PCSALT canresult in a smaller asymptotic variance for the estimated pthpercentile of a lifetime distribution than the optimal simpleSSALT. This result suggests that one stress loading maybe preferable than the other. Therefore, the tradeoffs (e.g.,in terms of variance of MLE of quantiles) between usingdifferent stress loadings should be evaluated first before anALT is conducted.

As an application of the proposition, if x(t) is an optimalcontinuous stress loading according to an optimization cri-terion that depends completely on the Fisher informationmatrix, there always exist statistically equivalent SSALTplans with the same criterion value (and hence are opti-mal). Consequently, one only needs to consider the classof SSALT plans when studying the optimal design of ALT.This is practically useful because an optimal continuousstress plan is in general difficult to obtain and implement.The result in Proposition 1 is also true for AcceleratedDegradation Tests under certain model assumptions (seeLee and Tang (2013)).

4. SE for the Weibull and lognormal distributions

In Section 3, we proved Proposition 1 for any distributionin the log-location-scale family. The two most frequentlyused lifetime distributions from this family are the Weibulland lognormal distributions. Application of Proposition1 to these two distributions is illustrated in this section.Furthermore, for these two distributions and for many ob-jective functions, we study the possibility of the existenceof statistically equivalent plans that do not have Type-IIISE.

4.1. The Weibull and lognormal examples

A lognormal lifetime distribution is often used for metalfatigue, solid-state components, and electrical insulation.On the other hand, a Weibull model is used to describethe life of roller bearings, electronic components, ceramics,and capacitors and dielectrics (Pascula and Montepiedra,2003). For these two distributions, we obtain the Fisherinformation matrices by substituting appropriate functionsinto Equation (3).

4.1.1. The Weibull distributionFor the Weibull distribution with (μ0, σ0) ≡ (μw, σw), theCDF of the lifetime under x(·) is

Fx(·)(t) = 1 − exp

⎛⎜⎜⎜⎜⎜⎜⎜⎝

− exp

⎛⎜⎜⎜⎜⎜⎜⎜⎝

ln

⎛⎝ t∫

0

a(x(z))dz

⎞⎠− μw

σw

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎠

.

It can be shown that the Fisher information matrix is

Ix(·)(μw, σw) =

⎡⎢⎢⎣

E[−∂2l(μw, σw)

∂μ2w

]E[−∂2l(μw, σw)

∂μw∂σw

]

E[−∂2l(μw, σw)

∂σw∂μw

]E[−∂2l(μw, σw)

∂σ 2w

]⎤⎥⎥⎦ ,

(4)where

E[−∂2l(μw, σw)

∂μ2w

]= 1

σ 2w

⎧⎨⎩

α∫−∞

ez−ezdz + eα−eα

⎫⎬⎭ ,

E[−∂2l(μw, σw)

∂μw∂σw

]= E

[−∂2l(μw, σw)

∂σw∂μw

]

= 1σ 2

w

⎧⎨⎩

α∫−∞

(ez + zez − 1) e−ezdz

+eα−eα + αeα−eα

⎫⎬⎭ ,

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6 Hu et al.

E[−∂2l(μw, σw)

∂σ 2w

]= 1

σ 2w

⎧⎨⎩

α∫−∞

(2zez + z2ez − 2z − 1

)e−ez

dz

+ (2αeα + α2eα

)e−eα

⎫⎬⎭ .

Each entry in the Fisher information matrix is a functionof variables (μw, σw, T), as described in Section 3.1.

4.1.2. The lognormal distributionSuppose the lifetime distribution is a lognormal distribu-tion with (μ0, σ0) ≡ (μL, σL), the CDF is then

Fx(·)(t) = �

⎛⎜⎜⎜⎜⎜⎜⎜⎝

ln

⎛⎝ t∫

0

a(x(z))dz

⎞⎠− μL

σL

⎞⎟⎟⎟⎟⎟⎟⎟⎠

.

Similarly, it can be shown that the Fisher informationmatrix under stress loading x(t) is

Ix(·)(μL, σL) =

⎡⎢⎢⎣

E[−∂2l(μL, σL)

∂μ2L

]E[−∂2l(μL, σL)

∂μL∂σL

]

E[−∂2l(μL, σL)

∂σL∂μL

]E[−∂2l(μL, σL)

∂σ 2L

]⎤⎥⎥⎦,(5)

where

E[−∂2l(μL, σL)

∂μ2L

]

= 1

σ 2L

{�(α) −

(α (1 − �(α)) − φ(α)

1 − �(α)

)φ(α)

},

E[−∂2l(μL, σL)

∂μL∂σL

]= E

[−∂2l(μL, σL)

∂σL∂μL

]

= 1

σ 2L

⎧⎨⎩

α∫−∞

2zφ(z)dz

+(

αφ(α) − (α2 − 1)(1 − �(α))1 − �(α)

)φ(α)

⎫⎬⎭ ,

E[−∂2l(μL, σL)

∂σ 2L

]= 1

σ 2L

⎧⎨⎩

α∫−∞

(3z2 − 1)φ(z)dz

+(

αφ(α) − (α2 − 2) (1 − �(α))1 − �(α)

)αφ(α)

⎫⎬⎭ ,

where φ(z) is the standard normal PDF. The entire Fisherinformation matrix for the lognormal distribution again de-pends on variables (μL, σL, T) only. The details for obtain-ing the matrices in Equations (4) and (5) are also available

in Hong et al. (2010), where Fisher information matrices forboth distributions are derived under the acceleration factora(x(z)) = exp[−γ1x(z)]. In the next subsection, we investi-gate the relation among the definitions for SE of these twodistributions.

4.2. The relationship among the definitions of SE

Given the Fisher information matrix, the asymptotic co-variance matrix of parameter MLEs can be obtained byinverting the information matrix. The existence of Type-III SE plans is established by Proposition 1. However, inmany applications, the purpose of an experiment is sim-ply to estimate some functions of model parameters (e.g.,the MTTF, lifetime distribution percentiles, reliability func-tion, etc.). In these cases, the objective function may be afunction of the covariance matrix and hence searching anoptimal plan among the Type-III SE plans may be toorestrictive. That is, one might only require two plans gen-erating the same variance of estimated functions insteadof having identical covariance matrices (i.e., finding gen-eral SE plans for some G(θ) is sufficient). This immediatelyraises a question regarding the existence of other types ofstatistically equivalent plans. In other words, can one findgeneral statistically equivalent ALT plans that are not Type-III SE plans? The next proposition shows that actually twoequivalent ALT plans must also have Type-III SE for manycommonly used objective functions. However, before pre-senting the proposition, we first provide a result regardingthe covariance matrix that will be useful in proving theproposition. That is, for both distributions and for fixedvalues of location and scale parameters, each element in thecovariance matrix is a decreasing function of α (and hencedecreasing functions of T). For example, for the lognor-mal distribution, the variances and covariance of (μ̂L, σ̂L)are Var(μ̂L) = (σ 2

L

/N)A(α), Var(σ̂L) = (σ 2

L

/N)B(α), and

Cov(μ̂L, σ̂L) = (σ 2L

/N)C(α) for some functions A(α), B(α),

C(α), and sample size N. We tabulate the values of A(α),B(α), and C(α), representing the elements of the covariancematrix, for various values of α in Table 1.

A similar table for the Weibull distribution can be foundin Meeker and Nelson (1977). Results in both tables nu-merically demonstrate that the variance and covariance ofMLEs are decreasing functions of α. Thus, for any fixedparameters, the variance and covariance of MLEs are alsodecreasing functions of variable T . We will use this empir-ical result in the following proposition.

Proposition 2. For both Weibull and lognormal distributions,let a(θ ) = [

a1(θ ) a2(θ)]′ be the gradient vector of a function

of model parameters θ . If a(θ ) �= 0 with a1(θ ) × a2(θ ) ≥ 0and if the variances and covariance of MLE of parameters aredecreasing function of T, then two ALT plans have general,Type-I, and Type-II SE if and only if they have Type-III SE.

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Step-stress accelerated life tests 7

Table 1. The variances and covariances of MLEs of parametersunder various standardized censoring times for the lognormaldistribution.

α �(α) (%) A(α) B(α) C(α)

−3.0 0.13 6001.3120 647.9308 1960.6767−2.8 0.26 2751.2315 338.3129 957.6665−2.6 0.47 1297.2683 183.0520 482.6072−2.4 0.82 628.5797 102.5904 250.6861−2.2 1.39 312.7282 59.5263 134.0784−2.0 2.28 159.6612 35.7402 73.7498−1.8 3.59 83.6383 22.1926 41.6638−1.6 5.48 44.9858 14.2432 24.1386−1.4 8.08 24.8920 9.4423 14.3192−1.2 11.51 14.2242 6.4616 8.6818−1.0 15.87 8.4477 4.5614 5.3696−0.8 21.19 5.2612 3.3192 3.3807−0.6 27.43 3.4729 2.4879 2.1619−0.4 34.46 2.4532 1.9194 1.4007−0.2 42.07 1.8631 1.5229 0.9172

0.0 50.00 1.5171 1.2415 0.60520.2 57.93 1.3118 1.0388 0.40130.4 65.54 1.1888 0.8911 0.26660.6 72.57 1.1144 0.7826 0.17680.8 78.81 1.0692 0.7025 0.11671.0 84.13 1.0417 0.6434 0.07631.2 88.49 1.0249 0.6000 0.04941.4 91.92 1.0147 0.5685 0.03141.6 94.52 1.0085 0.5458 0.01961.8 96.41 1.0049 0.5299 0.01202.0 97.72 1.0027 0.5190 0.00712.2 98.61 1.0015 0.5117 0.00412.4 99.18 1.0008 0.5069 0.00232.6 99.53 1.0004 0.5040 0.00132.8 99.74 1.0002 0.5022 0.00073.0 99.87 1.0001 0.5012 0.0003

Proof. The “if” part follows immediately from the defini-tions of SE. To prove the “only if” part, first denote thecovariance matrix of MLEs as[

v1(T) v2(T)v2(T) v3(T)

],

and each element is assumed to be a decreasing function ofT. Suppose there exist two ALT plans (say A and B) that aregeneral statistically equivalent but not Type-III statisticallyequivalent. Let TA and TB be the respective values of T .Then, from the assumptions above, TA �= TB but

[ a1 a2 ][

v1(TA) v2(TA)v2(TA) v3(TA)

] [a1a2

]= [ a1 a2 ]

[v1(TB) v2(TB)v2(TB) v3(TB)

]

×[

a1a2

]

for some[

a1 a2] = [

a1(θ ) a2(θ )].

Without loss of generality, assume TA > TB. We begin byassuming that both ai parameters are positive. The equality

above implies

[a1 a2

] [v1(TA) − v1(TB) v2(TA) − v2(TB)v2(TA) − v2(TB) v3(TA) − v3(TB)

] [a1a2

]= 0.

This equation and assumptions imply[v1(TA) − v1(TB) v2(TA) − v2(TB)v2(TA) − v2(TB) v3(TA) − v3(TB)

] [a1a2

]∈ Null Space([ a1 a2 ]′),

but [v1(TA) − v1(TB) v2(TA) − v2(TB)v2(TA) − v2(TB) v3(TA) − v3(TB)

] [a1a2

]�= 0.

The null space of[

a1 a2]′

has dimension one and a basis[−a2 a1]′

, which suggests that[v1(TA) − v1(TB) v2(TA) − v2(TB)v2(TA) − v2(TB) v3(TA) − v3(TB)

] [a1a2

]= λ

[−a2a1

]or, equivalently,

a1

[v1(TA) − v1(TB)v2(TA) − v2(TB)

]+ a2

[v2(TA) − v2(TB)v3(TA) − v3(TB)

]= λ

[−a2a1

]for some λ ∈ R.

However, this is not possible because the signs of matriceson both sides do not agree:

+[+

+]

+ +[+

+]

�= λ

[−+]

for any λ ∈ R.

A similar result holds when only one of the ai param-eters is zero or both ai parameters are negative. Thus, Aand B show general SE only if they are Type-III SE plans.Furthermore, since general SE is the least restrictive typeof equivalency (i.e., the feasible region of general SE planscontains feasible regions of all other types of SE plans), wecan conclude that two ALT plans are equivalent in termsof any SE definition given in Section 2 if and only if theyare Type-III statistically equivalent. �

For many practically used functions of parameters, therequired condition on the gradient vector, a(θ ), in Propo-sition 2 is satisfied. Some examples of these functions aregiven in Appendix B. Based on Proposition 2 we now fo-cus on investigating Type-III statistically equivalent SSALTplans in the following numerical example.

5. An example on miniature lamps

Liao and Elsayed (2010) proposed an experiment in whicha PCSALT is conducted to estimate the MTTF of a type ofminiature lamp under use conditions: 2 V DC and ambi-ent temperature 20◦C. The highest operating voltage of thelamp is known to be 5 V DC. A pilot experiment was con-ducted (with different combinations of voltage and temper-ature) to obtain information regarding the validity of the

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8 Hu et al.

model assumptions. Five combinations of constant-stresslevels were used with 100 test units allocated at each level.Each CSALT was censored at different termination timesand the largest censored time was 941.0 hours. One canrefer to Liao and Elsayed (2010) for a detailed experimen-tal setup and the observed failure time data from this pilotexperiment.

A preliminary analysis showed that temperature has nosignificant impact on the lifetimes and the data can bedescribed by a lognormal distribution. Thus, the follow-ing lognormal distribution PDF with a single stress fac-tor (the voltage) was used to model the random lifetimeof the lamp under a constant standardized test voltage,xi = (Vi − 2)

/(5 − 2) where Vi = 2, 3.5, and 5 V DC,

respectively:

f (t; xi ) = 1√2π σLt

exp

[−1

2

(ln(t) − (θ0 + θ1xi )

σL

)2]

,

i = 1, 2, and 3.

The location parameter of the lognormal PDF above wasassumed to be linearly related to the standardized stresslevel and the scale parameter (σL) was assumed to be in-dependent of the applied stress. The MLEs of the modelparameters were obtained by maximizing the log-likelihoodfunction, resulting in θ̂0 = 7.0285, θ̂1 = −2.5291, and σ̂L =1.0346. The estimates of the asymptotic variance of eachparameter were Var(θ̂0) = 0.0259, Var(θ̂1) = 0.0470, andVar(σ̂L) = 0.0907, respectively. Moreover, based on thesebaseline estimates of parameters, the estimated MTTFand its asymptotic variance under use conditions were1927.1147 hours and 3.7259 × 105. The MLE values abovewere used in this example to obtain statistically equivalentALT plans, which are called ML SE plans.

In practice, MLE values of unknown parameters areoften used to obtain optimal plans when the objectivefunction depends on unknown parameters. Such plans arereferred to as “locally optimum” (Chernoff, 1953; Meekerand Escobar, 1993). Alternatively, Zhang and Meeker(2006) and Yuan et al. (2012) considered plans obtainedby using Bayesian approaches. However, when one prefixesthe values of the hyper-parameters for the joint prior dis-tribution of parameters, the resulting Bayesian plan is alsolocally optimal. Zhang and Meeker (2006) demonstratedthat, when a very informative prior and a large sample sizeare used, the optimal Bayesian plan is very close (or equal)to the locally optimal ML plan. Furthermore, Chalonerand Larntz (1992) used some criteria to compare severalBayesian plans with locally optimal ML plans. They foundthe local optimum ML plans to be highly efficient with re-spect to the Bayesian plans; also see Meeker and Escobar(1993, Section 4.7). In other words, ALT plans obtained bysubstituting unknown parameters with pre-estimated MLEvalues are practically useful.

5.1. General statistically equivalent ramp-stress plans

In order to shorten the test time or to reduce the samplesize, Liao and Elsayed (2010) proposed general statisticallyequivalent ramp-stress test plans that retained the sameprecision of the estimated MTTF as achieved in the pilotexperiment. To model the lifetime CDF for a ramp-stresstest with a standardized linearly increasing stress loadingx(t) ∈ [0, 1], for all t ≥ 0, the following lognormal distri-bution and the AFT model in (1) was used:

Fx(·)(t) = �

⎛⎜⎜⎜⎜⎜⎜⎜⎝

ln

⎛⎝ t∫

0

a(x(z))dz

⎞⎠− θ0

σL

⎞⎟⎟⎟⎟⎟⎟⎟⎠

= �

⎛⎜⎜⎜⎜⎜⎜⎜⎝

ln

⎛⎝ t∫

0

e−θ1x(z)dz

⎞⎠− θ0

σL

⎞⎟⎟⎟⎟⎟⎟⎟⎠

.

They searched for general statistically equivalent ramp-stress test plans with a specific x(t) of the form

x(t) = t/η, 0 < t ≤ η.

Nonlinear programming models were used to search fortwo general statistically equivalent ramp-stress plans: onewith minimizing the sample size N and the other with mini-mizing the censoring time η as the objective function. Theyrequired that the resulting ramp-stress plan should providethe same variance of the estimated MTTF and that thetest was not longer than the longest censoring time in thegiven PCSALT. One of the two resulting ramp-stress testplans that were generally statistically equivalent to the orig-inal PCSALT used in the pilot experiments is N =100 andη = 146.73 hours.

5.2. Type-III statistically equivalent SSALT plansto the ramp-stress plan

Since the proposed ramp-stress loading is a continuousfunction of time, there exist SSALT plans that have exactlythe same Fisher information matrix. In addition, becausethe distribution and objective function used in this exam-ple satisfy the conditions required in Proposition 2, anyALT plan equivalent to the original ramp-stress test in Sec-tion 5.1 must be Type-III statistically equivalent. Therefore,we focus on searching for Type-III statistically equivalentSSALT plans to the ramp-stress test plan with one, two,three, and four levels of stress by applying Proposition 1.Since a multi-level plan has more decision variables (i.e.,stress levels and stress change times) than constraints, we

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Step-stress accelerated life tests 9

Table 2. Step-stress plans with Type-III SE to the ramp-stressplan with η = 146.73 hours.

Step-stress plan Stress levels (Vi )Stress change times

(τi )

Constant ALT 3.8 NoneSimple SSALT (2.83, 4.33) 73.37Three-level SSALT (2.53, 3.53, 4.53) (48.91, 97.82)Four-level SSALT (2.39, 3.14, 3.89, 4.64) (36.68, 73.37, 110.05)

focus only on equally spaced transition times. Results areshown in Table 2.

We notice that the highest allowable operating voltagein this experiment was 5 V, but none of the SSALT plansin Table 2 use this highest voltage. From a statistical opti-mization point of view, it has been suggested in the liter-ature (e.g., Miller and Nelson (1983) and Ma and Meeker(2008)) that for many objectives based on the Fisher infor-mation matrix, an optimal SSALT plan would include anacceleration stress level as high as possible. This suggeststhat the SSALT plans in Table 2, although equivalent tothe original ramp-stress plans, may not be globally opti-mal SSALT plans for the given objective. In other words,we may be able to find SSALTs better than those alreadyobtained. For example, for the same objective (minimizingthe asymptotic variance of the estimated MTTF under 2 VDC), we modified the simple SSALT plan in Table 2 by let-ting the highest level be as high as possible (i.e., the simpleSSALT plan becomes (V1, V2, τ1) = (2.83, 5.0, 73.37)) andrecalculated the asymptotic variance. The resulting vari-ance was around 1.7448 × 105, which is smaller than theoriginal variance of 3.7259 × 105. Similar results were ob-served across all other SSALT plans in Table 2. These re-sults combined with Proposition 1 suggest that, for a givencontinuous stress loading, x(t), using SSALT is not only

equivalent to x(t) but may provide even better estimationprecision.

5.2.1. Sensitivity analysisTo obtain SE SSALT plans in Table 2, we fixed θ1 of theacceleration factor a(·) at the MLE value, θ̂1 = −2.5291.These plans are called locally optimal plans (Chernoff,1953). Since we do not know the true value of θ1, the besthope is that the planning value, θ̂1 = −2.5291, is close tothe true value of θ1 so that a locally optimal plan is closeor equal to the optimal plan based on the true parame-ter value. In general, the ML method provides reasonableplanning values because, under some regularity conditions,MLEs are asymptotically unbiased and consistent. For theexample provided in Liao and Elsayed (2010), the estimatedasymptotic standard deviation/error of θ̂1 = −2.5291 wassd(θ̂1) = 0.3012 (sample size was 100). Hence, it is reason-able to assume that a range of θ1 is provided by the 95%approximate confidence interval (i.e., θ̂1 ± 1.96 × sd(θ̂1))around θ̂1 = −2.5291. For some possible values of θ1 withinthis interval, we followed the same procedure as in Section5.2 to obtain the corresponding SE SSALT plan. Resultsare presented in Table 3. (The stress change times of the cor-responding SSALT plans in Table 3 are the same as thosein Table 2.)

From Table 3, for each number of stress levels (eachcolumn in Table 3), the SE SSALT plans obtained fromvarious θ1-values are almost identical, which implies thatthe proposed statistically equivalent SSALT plan based onθ̂1 = −2.5291 (the center of the interval) is near-optimalor optimal (the confidence level is 95%). In particular, themulti-level SSALT plans, which are commonly used in prac-tice, are more robust (insensitive to θ1) than the single-levelconstant stress plans because there was little change in thesolutions for all cases considered in this analysis. Another

Table 3. Stress levels of SSALT plans with Type-III SE to the ramp-stress test under various values of θ1.

Constant ALT Simple SSALT Three-level SSALT Four-level SSALT

θ1 = θ̂1 − 1.96 × sd(θ̂1)3.86 (2.83, 4.35) (2.53, 3.53, 4.55) (2.39, 3.14, 3.89, 4.66)

θ1 = θ̂1 − sd(θ̂1)3.83 (2.83, 4.34) (2.53, 3.53, 4.54) (2.39, 3.14, 3.89, 4.65)

θ1 = θ̂1 (Current)3.80 (2.83, 4.33) (2.53, 3.53, 4.54) (2.39, 3.14, 3.89, 4.65)

θ1 = θ̂1 + sd(θ̂1)3.77 (2.83, 4.32) (2.53, 3.53, 4.53) (2.39, 3.14, 3.89, 4.65)

θ1 = θ̂1 + 1.96 × sd(θ̂1)3.74 (2.82, 4.31) (2.53, 3.52, 4.53) (2.39, 3.14, 3.89, 4.64)

Note: θ̂1 = −2.5291 and sd(θ̂1) = 0.3012. The respective stress change times of each plan are given in the last column ofTable 2.

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10 Hu et al.

reason for using multi-level SSALT plans is that they mayincur lower expected test cost, as shown in the next section.

5.3. Minimum-cost SSALT plans

When testing cost is a concern, minimum-cost test plansshould be considered. Test cost can be used as a secondarycriterion to select an ALT plan when a statistically equiva-lent plan is not unique. We searched for the minimum-costSSALT plans by using cost functions (functions of stress)as our optimization objective while maintaining SE. Weconsidered two sources of test cost as in Takemoto andArizono (2003).

1. Sampling cost (denoted as CU per test unit, in $): thiscost is related to the number of test units in a sample anddoes not depend on how an ALT would be implemented(i.e., independent of x(t)). Thus, the sampling cost isN × CU, where N is the number of test units in a sample.

2. Testing cost (denoted as CT(x) per unit time at stresslevel x, in $): the testing cost includes the cost of runninga test device and using measuring equipment to detectand record failures. We assumed that this cost is relatedto the stress level used, so a relatively higher stress levelincurs higher testing cost. This assumption is reasonablesince deterioration of measuring equipment tends to bemore severe at higher stress levels (Tang et al., 2004).

Based on the two costs defined above, the total test costusing stress loading x(t) is

Total Test Cost = N × CU +η∫

0

CT(x(t))dt. (6)

Suppose one would like to obtain a k-level SSALT planwhere the design parameters of this SSALT are as follows:the stress levels xi , i = 1, 2, . . . , k, and the stress changetimes τ j , j = 1, 2, . . . , k − 1. The following nonlinear

programming model is used to search for the minimumcost equivalent SSALT plan:

Minimize the total test cost in (6)subject to:

1. The resulting SSALT plan is Type-III SE to the givenramp-stress test plan in Section 5.1,

2. 0 ≤ xi ≤ 1, for all i = 1, 2, . . . , k,

3. 0 ≤ τ1 < τ2 < . . . < τk−1 ≤ η.

It is guaranteed by Proposition 1 that the feasible regionof this nonlinear programming problem is not empty. Also,note that the optimum solution from this nonlinear pro-gramming model is at most a k-level SSALT plan becausethe stress levels of two adjacent intervals might be the same.As an example, we consider the following cost functions:

CU = 1 dollar per test unit andCT(x) = α + βxp dollars per unit time for all x ∈ [0, 1],

where α, β ∈ {10, 100} and p ∈ {0.5, 1, 2}. For each com-bination of cost functions, we obtained the minimum-costsimple SSALT plan (i.e., k = 2). The reason we only consid-ered a simple SSALT plan is because it can be considered asa special case of a multi-level SSALT plan. Thus, allowing aplan to have more steps will not increase the total test cost.In Table 4, we compare the costs of resulting simple SSALTplans, the original ramp-stress plan, and single-level Type-III statistically equivalent CSALT plans.

From the results in Table 4, all tests using a simple step-stress loading have a lower total test cost than the corre-sponding statistically equivalent single-level CSALT plan.Furthermore, the Type-III statistically equivalent simpleSSALT plans are also more cost-effective than the originalramp-test plan. The most extreme case in Table 4 can save46% of test cost of the ramp-stress plan by using a simpleSSALT.

Table 4. Test costs for different Type-III SE stress loadings when N = 100, CU = 1, and η = 146.3.

Minimum cost simple SSALT

CT(x) $ Ramp-test $ CSALT $ Cost Plan (x1, x2, τ )

10 + 10√

x 2546 2704 2020 (2.0, 5.0, 101.4) (5.0, 2.0, 45.3)100 + 10

√x 15 751 15 910 15 226 (2.0, 5.0, 101.4) (5.0, 2.0, 45.3)

10 + 100√

x 11 349 12 933 6098 (2.0, 5.0, 101.4) (5.0, 2.0, 45.3)100 + 100

√x 24 555 26 139 19 303 (2.0, 5.0, 101.4) (5.0, 2.0, 45.3)

10 + 10x 2301 2448 2020 (2.0, 5.0, 101.4) (5.0, 2.0, 45.3)100 + 10x 15 507 15 653 15 226 (2.0, 5.0, 101.4) (5.0, 2.0, 45.3)10 + 100x 8904 10 371 6098 (2.0, 5.0, 101.4) (5.0, 2.0, 45.3)100 + 100x 22 110 23 577 19 303 (2.0, 5.0, 101.4) (5.0, 2.0, 45.3)10 + 10x2 2056 2095 2002 (2.0, 5.0, 106.5) (5.0, 2.5, 40.3)100 + 10x2 15 262 15301 15 207 (2.0, 5.0, 106.5) (5.0, 2.5, 40.3)10 + 100x2 6458 6850 5910 (2.0, 5.0, 106.5) (5.0, 2.5, 40.3)100 + 100x2 19 664 20 055 19 116 (2.0, 5.0, 106.5) (5.0, 2.5, 40.3)

Note: The Type-III SE 1-level CSALT is V(t) = 3.8V , for all t ∈ [0, η].

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Step-stress accelerated life tests 11

6. Summary

ALT is increasingly important since modern products arebecoming more and more reliable. Several types of stressloadings have been proposed in the literature; however,practical questions have been raised about finding equiva-lent test plans to shorten the test duration and to accom-modate a limited number of test items.

Step-stress ALT is one of the most widely used stressloadings in practice; it can shorten the test duration whilereducing the required number of test items and test de-vices. However, the rationale for using SSALT instead ofother types of continuous stress loadings has been rarelyinvestigated. In this article, we provide a rationale for usingstep-stress loading by showing that any continuous stressloading has Type-III statistically equivalent SSALT plans.Focusing on Type-III statistically equivalent SSALT plansis reasonable because two ALT plans are equivalent if andonly if they are Type-III statistically equivalent in manyapplications.

In addition, while retaining equivalency, minimum-costsSSALT plans are obtained. Results show that a SSALTcan be more cost-effective without sacrificing estimationprecision. Promising areas for future research include (i)extension of the results to other ALT models and (ii) ap-plication of time-varying ALT methods to accelerated teststhat provide degradation data.

Funding

The research of Cheng-Hung Hu was partially supportedby a grant from the National Science Council of the Repub-lic of China, Taiwan (grant NSC-101-2218-E-155-002-).

References

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Appendices

Appendix A: Derivation of matrix (3)

We take the first and second derivatives of the log-likelihoodfunction in Equation (2) as follows:

∂l(u0, σ0)∂u0

= Iyt

{∂ f0(zt)

/∂u0

f0(zt)

}+ (1 − Iyt )

{(1/σ0)

f0(α)

1 − F0(α)

},

∂l(u0, σ0)∂σ0

= Iyt

{− 1

σ0+(∂ f0(zt)

/∂σ0

)f0(zt)

}

+ (1 − Iyt )

{(α/σ0)

f0(α)

1 − F0(α)

},

∂2l(u0, σ0)

∂u20

= Iyt

{(∂2 f0(zt)

/∂u2

0

)f0(zt) − (

∂ f0(zt)/∂u0

)2

( f0(zt))2

}

+ (1 − Iyt )

×{(

1/σ0) (

∂ f0(α)/∂u0

)(1 − F0(α)) − ((

1/σ0)

f0(α))2

(1 − F0(α))2

},

∂2l(u0, σ0)∂u0∂σ0

= Iyt

{(∂2 f0(zt)

/∂u0∂σ0

)f0(zt) − (

∂ f0(zt)/∂u0

) (∂ f0(zt)

/∂σ0

)( f0(zt))

2

}

+ (1 − Iyt ){(− (1/σ 2

0

)f0(α) + (

1/σ0) (

∂ f0(α)/∂σ0

))(1 − F0(α)) − α

(f0(α)

/σ0)2

(1 − F0(α))2

},

∂2l(u0, σ0)

∂σ 20

= Iyt

{1

σ 20

+(∂2 f0(zt)

/∂σ 2

0

)f0(zt) − (

∂ f0(zt)/∂σ0

)2f 20 (zt)

},

+ (1 − Iyt )

×{

(−(2α/σ 20 ) f0(α) + (α/σ0)(∂ f0(α)/∂σ0))(1 − F0(α)) − ((α/σ0) f0(α))2

(1 − F0(α))2

}.

where zt = (yt − u0)/σ0 and α = (T − u0)

/σ0. Then, each

entry in Equation (3) is obtained by taking the expectedvalue of the negative second derivatives presented above.

Appendix B: Functions satisfying the requirementin proposition 2

Some commonly used functions of parameters of lifetimedistributions, as well as their corresponding gradient vec-tors, are given. All gradient vectors presented here satisfythe assumption in Proposition 2 in Section 4.2.

For the Weibull distribution.

• The location parameter of the SEV distribution (μw):a(μw, σw) = [ 1 0 ]′.

• The scale parameter of the SEV distribution (σw):a(μw, σw) = [ 0 1 ]′.

• The scale parameter of the Weibull distribution (eμw ):a(μw, σw) = [ eμw 0 ]′.

• The shape parameter of the Weibull distribution (1/σw):a(μw, σw) = [ 0 −1/σ 2

w ]′.• The MTTF of the Weibull distribution

(eμw × ∫∞0 t1+σw e−tdt) : a(μw, σw) = MTTF ×

[ 1 ln(1 + σw) ]′.• The p-th percentile of the Weibull distribution

(eμw+F−10,SEV(p)σw ) : a(μw, σw) = (eμw+F−1

0,SEV(p)σw ) ×[ 1 F−1

0,SEV(p) ]′ for all p ≥ 1 − e−1.• The reliability function at a specific time t0 of

the Weibull distribution (e−e(ln(t0)−μw/σw)) : a(μw, σw) =

(e−e((ln(t0)−μw)/σw)) × [ e((ln(t0)−μw)/σw)

σw

e((ln(t0)−μw)/σw)

σ 2w

]′.

For the lognormal distribution.

• The location parameter of the normal distribution (μL):a(μL, σL) = [ 1 0 ]′.

• The shape parameter of the lognormal distribution (σL):a(μL, σL) = [ 0 1 ]′.

• The MTTF of the lognormal distribution (eμL+0.5σ 2L ) :

a(μL, σL) = MTTF × [ 1 σL ]′.• The pth percentile of the lognormal distribution

(eμL+�−1(p)σL ) : a(μL, σL) = (eμL+�−1(p)σL ) × [ 1 �−1(p) ]′for all p ≥ 0.5.

• The reliability function at a specific time t0 of the log-normal distribution (1 − �((ln(t0) − μL)/σL)) :a(μL, σL) = [ φ((ln(t0)−μL)/σL)

σL

φ((ln(t0)−μL)/σL)σ 2

L]′.

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Step-stress accelerated life tests 13

Biographies

Cheng-Hung Hu is an Assistant Professor in the Department of Indus-trial and Information Management, National Cheng Kung University,Tainan City, Taiwan. He received his B.S. and M.S. degrees in AppliedMathematics and Statistics, respectively, from the National Tsing-HuaUniversity, Taiwan. In 2012, he received his Ph.D. degree from KrannertSchool of Management, Purdue University. His current research interestis in accelerated life/degradation testing, particularly on planning a testusing step-stress lodgings.

Robert D. Plante is the James Brooke Henderson University Profes-sor of Management at the Krannert School of Management, where hisresearch interests include the development of state-of-the-art statisti-cal quality control and improvement models. His efforts have focusedon the following classes of problems: (i) robust product/process design;(ii) screening procedures for process control and improvement; (iii) sta-tistical/process/dynamic process control models; and (iv) specializedprocess improvement problems. He has more than 50 research publi-cations in these areas, which have appeared in numerous journals, in-cluding Operations Research, Management Science, Decision Sciences,Journal of the American Statistical Association, International Journal

of Production Research, The Accounting Review, Auditing: Journalof Practice and Theory, Naval Research Logistics, IIE Transactions,Technometrics, Production and Operations Management, Computers &Operations Research, Journal of Quality Technology, European Jour-nal of Operational Research, Manufacturing and Service OperationsManagement, Journal of Operational Research, Information Systemsand Operational Research, and Journal of Business and EconomicStatistics.

Jen Tang is a Professor of Management at the Krannert School of Man-agement, where his current research interests include the areas of appliedstatistics, statistical quality control, and reliability analysis. He publishedvarious internal technical reports/references for Bell CommunicationsResearch. His journal publications have appeared in Biometrika, Statis-tics and Probability Letters, The Australian Journal of Statistics, Inter-national Journal of Production Research, IIE Transactions, Journal ofStatistical Computation and Simulation, Management Sciences, NavalResearch Logistics, IEEE Transactions on Reliability, Journal of QualityTechnology, Journal of American Statistical Association, Manufacturingand Service Operations Management, Production and Operations Manage-ment, European Journal of Operational Research, Statistica Sinica, andTechnometrics.

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