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Computational Statistics and Data Analysis 53 (2009) 2702–2709

Contents lists available at ScienceDirect

Computational Statistics and Data Analysis

journal homepage: www.elsevier.com/locate/csda

Testing for the validity of the assumptions in the exponential step-stress

accelerated life-testing model

Bing Xing Wang

Zhejiang Gongshang University, Hangzhou, PR China

a r t i c l ei n f o

Article history:

Received 27 March 2007

Received in revised form 25 December 2008

Accepted 19 January 2009

Available online 24 January 2009

a b s t r a c t

In the application of the exponential step-stress accelerated life-testing model, there are

usually three assumptions required: (1) for any stress level, the lifetime distribution of a

test unit is exponential; (2) for any stress level, the mean life of a test unit is a log-linear

function of stress; (3) a cumulative exposure model holds. This paper explores the validity

of assumptions 1 and 3. It is proved that assumption 3 is unnecessary to the exponential

step-stress accelerated life-testing model. A test statistic is proposed to test the validity

of the assumptions 1. The null distribution of the test statistic is derived. A Monte Carlo

simulation is given to study the power of the proposed test procedure. Finally, an example

is given to illustrate the proposed test procedure.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

An accelerated life test quickly yields information on the lifetime distribution of materials and products by testing them

at higher than usual levels of stress to induce early failure. The results obtained at the accelerated conditions are analyzed

in terms of a model to relate life length to stress; they are extrapolated to the design stress to estimate the life distribution.

A step-stress accelerated life test was developed to reduce operation time and costs further. One way of applying stress to

the test units is the step-stress scheme which allows the stress setting of a unit to be changed at pre-specified times or upon

the occurrence of a fixed number of failures. The former is called step-stress accelerated life test with type I censoring and

the latter is step-stress accelerated life test with type II censoring.

The problem of modeling data from the step-stress accelerated life test and making inferences from such data have been

studied by many authors. DeGroot and Goel (1979) proposed the tampered random variable model. Nelson (1980) proposed

the cumulative exposure model. Bhattacharyya and Soejoeti (1989) proposed the tampered failure rate model. Wang and

Fei (2004) discussed conditions for the coincidence of these models. On the basis of their results, these models coincide if

the lifetime has an exponential distribution. Khamis and Higgins (1998), and Bagdonavičius et al. (2002) proposed some

new models for the step-stress accelerated life test. Xu and Tang (2003) pointed out that the Khamis/Higgins model is a

special case of the multiple step-stress tampered failure rate model under Weibull distribution. Tang et al. (1996) obtained

the maximum likelihood estimations for parameters in multi-censored accelerated life tests. Xiong (1998) discussed the

maximum likelihood estimation for the exponential step-stress accelerated life test with type II censoring. Teng and Yeo

(2002) used the method of least squares to estimate the life-stress relationship in a step-stress accelerated life test. Dorp

et al. (1996) gave a Bayes approach to a step-stress accelerated life test. Dorp and Mazzuchi (2004) developed a general

Bayesexponentialinferencemodelforacceleratedlifetests.Wang(2006)obtainedunbiasedestimationsfortheexponential

distribution based on step-stress accelerated life test data. Balakrishnan et al. (2007) obtained point and interval estimation

for the exponential simple step-stress model. Bai et al. (1989) obtained the optimum simple step-stress accelerated life test

plans.KhamisandHiggins(1996)obtainedtheoptimum3-stepstep-stressacceleratedlifetestplans.Cheng(1994)extended

E-mail address: wangbxsn@yahoo.com.cn.

0167-9473/$ – see front matter © 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.csda.2009.01.008

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B.X. Wang / Computational Statistics and Data Analysis 53 (2009) 2702–2709

2703

theresultsofBaietal.(1989)tothecaseofthek-stepstep-stressacceleratedlifetest.Tangetal.(1999)discussedanoptimum

plan for two-parameter exponential distribution. Gouno et al. (2004) discussed the optimal step-stress accelerated life test

plans under progressive type-I censoring. Many examples of accelerated life test, as well as an excellent introduction to the

methodology, are given in Nelson (1990) and Mao and Wang (1997).

Statistical inference for the exponential step-stress accelerated life test usually depends on the following assumptions:

1. For any stress level xi, the lifetime distribution of a test unit is exponential with cumulative distribution function (cdf):

Fi(t) = 1 − exp(−t/θi),

where θi> 0 is the mean life of a test unit at stress level xi.

2. The log mean life of a test unit log(θi) is a linear function of stress level xi:

t > 0,

(1)

log(θi) = α + βxi

where α and β are unknown parameters.

3. A cumulative exposure model holds: the remaining life depends only on the current cumulative failure probability and

current stress level regardless of how the probability is accumulated.

The cumulative exposure model may be replaced by the tampered random variable model or the tampered failure rate

model (see DeGroot and Goel (1979) and Bhattacharyya and Soejoeti (1989)).

The literature on the step-stress accelerated life test mostly focuses on a few areas such as: (1) derivation of optimum

test plans; (2) developing inference procedures with various data types; (3) constructing new models. But there is little

investigation for validity of the assumptions in the step-stress accelerated life-testing models. In this paper, we consider

this problem.

The paper is organized as follows. In Section 2, it is proved that assumption 3 is unnecessary to analyze the exponential

step-stress accelerated life-testing data. A test statistic is also proposed to test the validity of assumption 1 based on a type

II censored step-stress accelerated life-testing sample. The null distribution of the test statistic is derived. In Section 3, a

Monte Carlo simulation is given to study the power of the proposed test procedure under several different alternatives. In

Section 4, an example is given to illustrate the proposed test procedure.

(2)

2. Main results

We first prove that assumption 3 is unnecessary to analyze the exponential step-stress accelerated life-testing data.

Consider the k-step step-stress accelerated life test with type I censoring. Suppose that all n test units are initially placed

at the lowest stress level x1and run until time τ1. At time τ1, the stress level is changed to x2. The test is continued,

triggering stress level changes at times τi. At the stress level xk, the test is continued until time τk. At the stress level

xi(i = 1,2,...,k), rifailure times ti,j(j = 1,2,...,ri) of test units are observed. The set of observed failure times

(t1,1,...,t1,r1(<τ1),...,tk,1,...,tk,rk(<τk)) is called a type I censored step-stress accelerated life-testing sample.

Theorem 1. Suppose that (t1,1,...,t1,r1(<τ1),...,tk,1,...,tk,rk(<τk)) is a type I censored step-stress accelerated life-testing

sample. Then, under assumption 1, the likelihood function is given by

?

i=1

L(θ1,...,θk) =

n!

k ?

?

?

n −

i=1

ri

?

!

k ?

i=1

θ−ri

i

exp

−

k ?

Si

θi

?

,

(3)

where

Si=

ri ?

j=1

(ti,j− τi−1) +

n −

i ?

j=1

rj

?

(τi− τi−1)

is the total time on test at stress xi(i = 1,2,...,k), τ0= 0.

Proof. The joint probability density function (pdf) of (t1,1,...,t1,r1,...,tk,1,...,tk,rk) can be written as

f1(t1,1,...,t1,r1)

k ?

i=2

fi(ti,1,...,ti,ri|ti−1,1,...,ti−1,ri−1,τi−1),

(4)

where f1,f2,...,fkrepresent the pdf of the variables indicated.

Specifically, the joint pdf f1(t1,1,t1,2,...,t1,r1) is given by

f1(t1,1,t1,2,...,t1,r1) =

n!

(n − r1)!θ−r1

1

exp

?

−S1

θ1

?

.

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B.X. Wang / Computational Statistics and Data Analysis 53 (2009) 2702–2709

Given t1,1,t1,2,...,t1,r1,τ1, the survivor function of the lifetimes T of units left in the experiment is given by

S(t|t1,1,t1,2,...,t1,r1,τ1) = Pr(T > t|t1,1,t1,2,...,t1,r1,τ1)

= Pr(T − τ1> t − τ1|t1,1,t1,2,...,t1,r1,τ1)

= exp

θ2

The last equation in Eq. (5) is due to the ‘‘lack-of-memory’’ property of the exponential distribution. Thus, given

(t1,1,t1,2,...,t1,r1,τ1), t2,1,t2,2,...,t2,r2is a type I censored sample with censored time τ2in a random sample of size

n − r1from distribution (5). That is, the pdf f2(t2,1,...,t2,r2|t1,1,...,t1,r1,τ1) in Eq. (4) is given by

(n − r1)!

(n − r1− r2)!

Similarly, we obtain

?

?

Finally, we obtain

?

?

n!

?

The proof is completed.

?

Now,considerthek-stepstep-stressacceleratedlifetestwithtypeIIcensoringasfollows.Supposethatallntestunitsare

initiallyplacedattheloweststresslevelx1andrununtilr1unitshavefailed.Atfailuretimet1,r1,thestresslevelischangedto

x2. The test is continued, triggering stress level changes at failure times ti,ri. At the stress level xk, the test is continued until rk

unitshavefailed.Atthestresslevelxi(i = 1,2,...,k),rifailuretimesti,j(j = 1,2,...,ri)ofthetestunitsareobserved.The

set of observed failure times(t1,1,...,t1,r1,...,tk,1,...,tk,rk) is called a type II censored step-stress accelerated life-testing

sample.

?

−t − τ1

?

,

t ≥ τ1.

(5)

?

r2 ?

i=1

1

θ2

exp

?

−t2,i− τ1

θ2

???

exp

?

−τ2− τ1

θ2

??n−r1−r2

=

(n − r1)!

(n − r1− r2)!θ−r2

2

exp

?

−S2

θ2

?

.

fj(tj,1,...,tj,rj|tj−1,1,...,tj−1,rj−1,τi−1) =

n −

j−1

?

i=1

j?

ri

?

?

!

n −

i=1

ri

!

θ−rj

j

exp

?

−Sj

θj

?

,

j = 3,...,k.

L(θ1,...,θk) =

n!

(n − r1)!θ−r1

1

exp

?

−S1

θ1

?

···

n −

k−1

?

i=1

k ?

ri

?

?

!

n −

i=1

?

ri

!

θ−rk

k

exp

?

−Sk

θk

?

=

n −

k ?

i=1

ri

?

!

k ?

i=1

θ−ri

i

exp

?

−

k ?

i=1

Si

θi

.

Theorem 2. Suppose that (t1,1,...,t1,r1,...,tk,1,...,tk,rk) is a type II censored step-stress accelerated life-testing sample.

Then, under assumption 1, the likelihood function is given by

?

i=1

L(θ1,...,θk) =

n!

k ?

?

n −

i=1

ri

?

!

k ?

i=1

θ−ri

i

exp

−

k ?

Si

θi

?

,

(6)

where

Si=

ri ?

j=1

(ti,j− ti−1,ri−1) +

?

n −

i ?

j=1

rj

?

(ti,ri− ti−1,ri−1)

is the total time on test at stress xi(i = 1,2,...,k), t0,r0= 0.

The proof of Theorem 2 is similar to that of Theorem 1 and only replacing τiby ti,ri.

Becausethelikelihoodfunctions(3)and(6)arederivedbasedonassumption1only,accordingtothelikelihoodprinciple,

the tampered random variable model, the cumulative exposure model and the tampered failure rate model are unnecessary

to analyze type I and type II censored exponential step-stress accelerated life-testing data. Thus Theorems 1 and 2 may

enable us to decide whether or not a given model such as the cumulative exposure model is rational. In other words, if the

likelihood function based on type I and type II censored exponential step-stress accelerated life-testing samples is not equal

to (3) and (6) under a given model, then the given model is irrational.

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B.X. Wang / Computational Statistics and Data Analysis 53 (2009) 2702–2709

2705

Theorem 3. Suppose that (t1,1,...,t1,r1,...,tk,1,...,tk,rk) is a type II censored step-stress accelerated life-testing sample. Let

S1,1= nt1,1,...,S1,r1= (n − r1+ 1)(t1,r1− t1,r1−1),

Si,1= ni(ti,1− ti−1,ri−1),...,Si,ri= (ni− ri+ 1)(ti,ri− ti,ri−1),

where ni= n −?i−1

Proof. Note that

ri ?

and the Jacobian of transformation is given by

∂(S1,1,...,Sk,rk)

∂(t1,1,...,tk,rk)=

n −

i=1

so the joint pdf of (S1,1,...,S1,r1,...,Sk,1,...,Sk,rk) is found from Eq. (6) to be

j=1rj,(i = 2,3,...,k). Then, under assumption 1, Si,j,(i = 1,2,...,k,j = 1,2,...,ri) are independent,

and the distributions of Si,j/θi(i = 1,2,...,k,j = 1,2,...,ri) are the standard exponential distributions.

Si=

j=1

Si,j

n!

k ?

?

ri

?

!

,

f(s1,1,...,s1,r1,...,sk,1,...,sk,rk) =

k ?

i=1

θ−ri

i

exp

−

ri ?

j=1

si,j

θi

.

Hence, random variables Si,j,(i = 1,2,...,k,j = 1,2,...,ri) are independent, and the distributions of Si,j/θi(i =

1,2,...,k,j = 1,2,...,ri) are the standard exponential distributions.

The proof is completed.

?

Remark. Under assumptions 1 and 3, the results of Theorems 1–3 are also derived in Mao and Wang (1997).

Next, we want to test the validity of assumption 1 based on type II censored step-stress accelerated life-testing sample.

Because there are no life data at the design stress level, we can only test the validity of assumption 1 at accelerated stress

levels. In other words, we want to test the hypothesis

H0: T1

d= Exp(θ1),T2

d= Exp(θ2),...,Tk

d= Exp(θk),

(7)

whereTiislifetimeofthetestunitatthestresslevelxi.Thatis,thelifetimeatthestresslevelxihasanexponentialdistribution

with a mean of θi(i = 1,2,...,k).

Although a large number of test procedures have been given for exponentiality based on lifetime data at a single stress

level (see Stephens (1986b) and Henze and Meintanis (2005)), to the best of my knowledge, there is no test procedure given

for H0based on the step-stress accelerated life-testing data.

Since H0 =?k

results in Henze and Meintanis (2005), we know that the G test is one of the most powerful test procedures. Moreover,

we know from Wang (2008) the statistic χ2

statistic given by

i=1Hi

0, where Hi

0: Ti

d= Exp(θi), testing H0is equivalent to testing Hi

0for every i ∈ {1,2,...,k}. For

hypothesis Hi

0, Lawless (1982, pp. 446–447) suggested the G test on which is based the Gini statistic. From the simulation

ias indicated below is more powerful than the G test. Hence we propose a test

χ2

i= 2

ri−1

?

j=1

logSi,1+ ··· + Si,ri

Si,1+ ··· + Si,j

.

(8)

This test statistic is clearly scale invariant, with small and large values ofχ2

to mention here that the test statistics χ2

when the available sample is a type II censored step-stress accelerated life-testing sample.

Let

Si,1+ ··· + Si,j

Si,1+ ··· + Si,ri

Then we obtain from Theorem 3 and Stephens (1986b) that the joint distribution of Zi,1,Zi,2,...,Zi,ri−1is the same as the

joint distribution of the ri−1 order statistics (say, U(i)

the uniform (0,1) distribution (say, U(i)

ileading to the rejection of Hi

0. It is important

iin Eq. (8) is a generalization of a test proposed by IEC 605-6 (1986) to the case

Zi,j=

,

i = 1,2,...,k,j = 1,2,...,ri− 1.

(1),U(i)

ri−1). Hence, we have

(2),...,U(i)

(ri−1)) obtained from a random sample of size ri−1 from

1,U(i)

2,...,U(i)

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B.X. Wang / Computational Statistics and Data Analysis 53 (2009) 2702–2709

χ2

i= 2

ri−1

?

j=1

?−logZi,j

?

d= 2

ri−1

?

j=1

?

−logU(i)

(j)

?

d=

ri−1

?

j=1

?

−2logU(i)

j

?

.

(9)

Therefore, it follows from Eq. (9) and Theorem 3 that the null distribution of the test statistic χ2

with 2ri− 2 degrees of freedom and that χ2

Since χ2

(1986a), p. 358) is given by

iis the χ2distribution

1,...,χ2

kare independent.

1,...,χ2

kare k independent test statistics, a statistic for testing H0based on Fisher’s method (see Stephens

W = −2

k ?

i=1

log(1 − |1 − 2Pi|),

(10)

where ciis the value of χ2

degrees of freedom. The critical region of size α for testing H0based on statistic W is

C =?

where χ2

The p-value of the test is given by

iwhen test is made, and Pi= Pr(χ2

α(2k)?,

i> ci|Hi

0). When H0is true, W has the χ2distribution with 2k

W ≥ χ2

α(v) is the upper α critical value of the χ2distribution with v degrees of freedom.

p = Pr(W ≥ w) = Pr(χ2(2k) ≥ w),

where w is the value of the test statistic W when the test is made.

(11)

3. Simulation study

Because it is difficult (if not impossible) to obtain the power function of the test statistic W by analytical means, a Monte

Carlosimulationstudyisconductedtodeterminethepowerunderdifferentalternatives.Thefollowinglifetimedistributions

are used as alternatives to the exponential distributions:

1. Weibull distributions with shape parameters m = 0.5,2.0 and scale parameters θi = exp(α + βxi). The cdf of the

Weibull distribution with parameters (m,θi) is of the form

Fi(t) = 1 − exp?−(t/θi)m?,

distribution with parameters (µi,σ) is of the form

?log(t) − µi

where Φ(·) is the cdf of the standard normal distribution.

These distributions comprise widely used alternatives to the exponential model and include distributions with

decreasing hazard function, increasing hazard function as well as models with a non-monotone hazard function.

Since the test statistic W is scale invariant, without loss of generality, we chooseα = 4,β = −1,x0= 0,x1= 0.5,x2=

0.75,x3= 1,x4= 1.25. The significance levels of the test are 0.1 and 0.05. Schemes of simulation experiments consider

optimalsimplestep-stressacceleratedlife-testingplansinBaietal.(1989)andcompromisetestplansinKhamisandHiggins

(1996). The following steps are used to generate a type II censored step-stress accelerated life-testing sample from Weibull

distributions based on the cumulative exposure model. A type II censored step-stress accelerated life-testing sample from

Lognormal distributions can also be generated using similar steps.

(1) A sample of n uniform (0,1) random variables is generated and ordered from the smallest to the largest. Let

U(1)≤ U(2)≤ ··· ≤ U(n)denote the ordered sample from uniform distribution U(0,1).

(2) For given values of (m,θ1), ordered observations on a Weibull random variable are generated by Xj= θ1[−log(1 −

U(j))]1/m, j = 1,2,...,n.

(3) For given values of the parameters (m,α,β), let t1,j = Xj, (j = 1,2,...,r1), t2j−R1=

R1+ 1,...,R2),..., tk,j=

ti,j,i = 1,2,...,k,j = 1,2,...,riis the required type II censored step-stress accelerated life-testing sample from Weibull

distributions.

We generated 10,000 sets of data in order to obtain the estimated power values for different choices of sample sizes

and step-stress accelerated life test schemes. According to a referee’s suggestion, the type I error rates of the proposed test

statistic are also studied by a simulation method. These values are tabulated in Tables 1 and 2. A referee pointed out that the

numbers of failures in schemes of simulation experiments are too high and may be unrealistic. In practice, if there are only

a few failures at some stress levels, it is difficult to determine the life distributions at accelerated stress levels unless there

is prior information for the life distributions at accelerated stress levels.

t > 0;

2. Lognormal distributions with shape parametersσ = 0.5,1.0 and scale parametersµi= α+βxi. The cdf of Lognormal

?

Fi(t) = Φ

σ

,

t > 0,

θ2

θ1(Xj− XR1) + t1,r1, (j =

θk

θ1(Xj− XRk−1) + tk−1,rk−1, (j = Rk−1+ 1,...,Rk), where Ri= r1+ ··· + ri,i = 1,...,k. Then

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B.X. Wang / Computational Statistics and Data Analysis 53 (2009) 2702–2709

2707

Table 1

Monte Carlo power estimates for Weibull distributions.

(r1,...,rk)

nkm = 0.5

10%

m = 1

10%

m = 2

10% 5% 5% 5%

30

30

50

50

50

70

70

70

80

80

80

90

90

90

100

100

100

80

80

90

90

100

100

100

100

100

120

120

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

4

4

4

4

( 9, 6)

(12, 8)

(12, 8)

(15, 10)

(18, 12)

(12, 8)

(21, 14)

(24, 16)

(15, 10)

(24, 16)

(30, 20)

(18, 12)

(27, 18)

(33, 22)

(21, 14)

(30, 20)

(39, 26)

(18, 12, 9)

(24, 16, 12)

(18, 12, 9)

(30, 20, 15)

(18, 12, 9)

(24, 16, 12)

(30, 20, 15)

(24, 16, 12, 8)

(24, 20, 16, 12)

(25, 18, 12, 7)

(30, 20, 15, 9)

0.6271

0.7548

0.7267

0.8176

0.8831

0.7163

0.9139

0.9471

0.8046

0.9450

0.9788

0.8689

0.9614

0.9876

0.9058

0.9760

0.9954

0.8265

0.9237

0.8306

0.9654

0.8246

0.9148

0.9643

0.9083

0.9173

0.9006

0.9561

0.5193

0.6675

0.6371

0.7413

0.8234

0.6198

0.8649

0.9136

0.7236

0.9084

0.9612

0.8024

0.9328

0.9755

0.8530

0.9562

0.9891

0.7514

0.8751

0.7512

0.9381

0.7470

0.8649

0.9386

0.8542

0.8681

0.8453

0.9257

0.1056

0.1020

0.1002

0.1008

0.0976

0.0952

0.1041

0.0951

0.0972

0.0981

0.0938

0.0961

0.0980

0.0947

0.0996

0.1018

0.1018

0.0982

0.0996

0.0972

0.0984

0.0992

0.0982

0.0988

0.1106

0.1063

0.1041

0.1137

0.0542

0.0484

0.0497

0.0509

0.0487

0.0461

0.0529

0.0493

0.0497

0.0493

0.0483

0.0510

0.0483

0.0473

0.0474

0.0528

0.0512

0.0513

0.0498

0.0493

0.0465

0.0475

0.0480

0.0495

0.0541

0.0515

0.0520

0.0584

0.4507

0.6068

0.5888

0.7174

0.8104

0.5740

0.8692

0.9206

0.7028

0.9179

0.9711

0.7919

0.9479

0.9831

0.8559

0.9676

0.9930

0.7169

0.8688

0.7118

0.9401

0.7077

0.8546

0.9400

0.8026

0.8217

0.7975

0.8967

0.3054

0.4507

0.4399

0.5699

0.6863

0.4255

0.7644

0.8436

0.5593

0.8378

0.9316

0.6660

0.8881

0.9583

0.7487

0.9298

0.9826

0.5759

0.7627

0.5713

0.8791

0.5674

0.7498

0.8764

0.6801

0.7042

0.6724

0.8101

Table 2

Monte Carlo power estimates for Lognormal distributions.

nk

(r1,...,rk)σ = 0.5

10%

σ = 1

10% 5% 5%

30

30

50

50

50

70

70

70

80

80

80

90

90

90

100

100

100

80

80

90

90

100

100

100

100

100

120

120

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

4

4

4

4

( 9, 6)

(12, 8)

(12, 8)

(15,10)

(18,12)

(12, 8)

(21, 14)

(24, 16)

(15, 10)

(24, 16)

(30, 20)

(18, 12)

(27, 18)

(33, 22)

(21, 14)

(30, 20)

(39, 26)

(18, 12, 9)

(24, 16, 12)

(18, 12, 9)

(30, 20, 15)

(18, 12, 9)

(24, 16, 12)

(30, 20, 15)

(24, 16, 12, 8)

(24, 20, 16, 12)

(25, 18, 12, 7)

(30, 20, 15, 9)

0.9431

0.9894

0.9966

0.9993

0.9998

0.9983

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.9998

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.8500

0.9542

0.9806

0.9954

0.9989

0.9902

0.9998

1.0000

0.9995

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.9993

1.0000

0.9994

1.0000

0.9996

1.0000

1.0000

0.9999

1.0000

1.0000

1.0000

0.3015

0.3306

0.4705

0.5172

0.5468

0.5495

0.6857

0.7022

0.6512

0.7618

0.7923

0.7350

0.8208

0.8366

0.7991

0.8665

0.8863

0.6003

0.6725

0.6481

0.7505

0.6675

0.7542

0.7929

0.6797

0.6876

0.7506

0.7879

0.1825

0.2032

0.3200

0.3564

0.3821

0.3766

0.5278

0.5483

0.4816

0.6155

0.6473

0.5855

0.6851

0.7160

0.6548

0.7473

0.7846

0.4379

0.5088

0.4851

0.6075

0.5060

0.6085

0.6576

0.5167

0.5300

0.6003

0.6490

It is observed from the simulation results of m = 1 in Table 1 that the observed type I error rates are close to the nominal

coverage probabilities. It is quite clear from Tables 1 and 2 that for a fixed sample size, as the failure numbers r1,...,rkat

accelerated stress levels increase the power values of the proposed test procedure increase as expected. It is also clear that

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B.X. Wang / Computational Statistics and Data Analysis 53 (2009) 2702–2709

Fig. 1. The pdfs of the standard lognormal and the standard exponential distributions.

for a fixed sample size and failure numbers r1,...,rkat accelerated stress levels the alternatives with decreasing hazard

function have larger power values than the alternatives with increasing hazard function. It is also observed that there are

low power values for the lognormal alternative with σ = 1. The reason for this is that because the pdf of the standard

lognormal distribution is near to the pdf of the standard exponential distribution for x > 0.5 (see Fig. 1); it is difficult to

distinguish between the lognormal distribution with σ = 1 and the exponential distribution unless the failure numbers

r1,...,rkare all large.

4. An illustrative example

To illustrate the proposed procedure, recall the example of Section 4 of Wang and Fei (2003). This example is to get all

the reliability indices of a kind of electronic component at the normal temperature of S0= 25◦C, Now n = 100 units from a

batch of products are randomly selected for the simple accelerated life test model log(θ) = 1/(273.15+S). The accelerated

temperature levels are S1= 100◦C and S2= 150◦C. At S1= 100◦C, when 30 products have failed, the stress level rises to

S2= 150◦C and the test continues until 20 more products have failed. Their failure times are as follows.

Failure times for the stress level S1: 32, 54, 59, 86, 117, 123, 213, 267, 268, 273, 299, 311, 321, 333, 339, 386, 408, 422,

435, 437, 476, 518, 570, 632, 666, 697, 796, 854, 858, 910.

Failure times for the stress level S2: 16, 19, 21, 36, 37, 63, 70, 75, 83, 95, 100, 106, 110, 113, 116, 135, 136, 149, 172, 186.

These times are the differences t2,j− t1,r1.

According to (10) and (11), the value of the test statistic W is 1.2385. The p-value of the test is given by

p = P(χ2(4) ≥ 1.2385) = 0.1283.

The p-value of the test shows that the life distributions of test units at the stress x1,x2are exponential.

Acknowledgements

The author thanks the Associate Editor and the referees for their detailed comments and suggestions, which helped in

improving the manuscript.

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