comparison between constant-stress and step-stress accelerated life tests under time constraint

Comparison between Constant-Stress and Step-Stress Accelerated Life Testsunder Time Constraint

David Han,1 H.K.T. Ng2

1Department of Management Science and Statistics, University of Texas at San Antonio, Texas 78249

2Department of Statistical Science, Southern Methodist University at Dallas, Texas 75275

Received 20 November 2012; revised 22 July 2013; accepted 22 July 2013DOI 10.1002/nav.21551

Published online 14 September 2013 in Wiley Online Library (wileyonlinelibrary.com).

Abstract: By running life tests at higher stress levels than normal operating conditions, accelerated life testing (ALT) quicklyyields information on the lifetime distribution of a test unit. The lifetime at the design stress is then estimated through extrapolationusing a regression model. In constant-stress testing, a unit is tested at a fixed stress level until failure or the termination time point oftest, whereas step-stress testing allows the experimenter to gradually increase the stress levels at some prefixed time points duringthe test. In this work, the optimal k-level constant-stress and step-stress ALTs are compared for the exponential failure data undercomplete sampling and Type-I censoring. The objective is to quantify the advantage of using the step-stress testing relative to theconstant-stress one. Assuming a log-linear life–stress relationship with the cumulative exposure model for the effect of changingstress in step-stress testing, the optimal design points are determined under C/D/A-optimality criteria. The efficiency of step-stresstesting to constant-stress one is then discussed in terms of the ratio of optimal objective functions based on the information matrix.© 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 541–556, 2013

Keywords: accelerated life testing; constant-stress testing; maximum likelihood estimation; optimal regression design; step-stresstesting; Type-I censoring

1. INTRODUCTION

With increasing reliability and substantially long life-spansof products, it is often difficult for standard life testing meth-ods under normal operating conditions to obtain sufficientinformation about the failure time distribution of the prod-ucts. This difficulty is overcome by accelerated life test (ALT)where the test units are subjected to higher stress levels thannormal for rapid failures. By applying more severe stresses,ALT collects information on the parameters of lifetime distri-butions more quickly. Some key references in the area of ALTare Nelson [17], Meeker and Escobar [14], and Bagdonavi-cius and Nikulin [3]. There are two special classes of ALT:constant-stress testing and step-stress testing. In constant-stress testing, a unit is tested at a fixed stress level until failureoccurs or the life test is terminated, whichever comes first.On the other hand, step-stress testing allows the experimenterto gradually increase the stress levels at some prefixed timepoints during the test.

Correspondence to: D. Han ([email protected])

The optimal ALT design has attracted great attention in thereliability literature. Miller and Nelson [15] initiated researchin this area by considering a simple step-stress model withexponential failure time distribution under complete sam-pling. The fundamental model used was the one proposed bySedyakin [19], which is known as the cumulative exposuremodel. This model was further discussed and generalized byBagdonavicius [1] and Nelson [16]. Bai et al. [4] extendedthe results of Miller and Nelson [15] to the time-censoredsituation, whereas Khamis and Higgins [12] studied the caseof three stress levels. Khamis and Higgins [13] also con-sidered the problem under Weibull lifetime distribution forunits subjected to stress. Their work was later generalizedby Bagdonavicius et al. [2]. Khamis [11] then comparedconstant-stress ALT and step-stress ALT for Weibull failuredata. Yeo and Tang [20] investigated the optimality problemin the situation when a target acceleration factor was prespec-ified. Recently, exact conditional inference for a step-stressmodel with exponential competing risks was studied by Bal-akrishnan and Han [5], Han and Balakrishnan [8]. Ng et al.[18] discussed the problem of determining the optimal sam-ple size allocation for a general k-level model with extreme

© 2013 Wiley Periodicals, Inc.

542 Naval Research Logistics, Vol. 60 (2013)

value regression, whereas Gouno et al. [7], Balakrishnan andHan [6] discussed the problem of determining the optimalstress duration under progressive Type-I censoring; see alsoHan et al. [9] for some related comments. Under completesampling, Hu et al. [10] studied the statistical equivalency ofa simple step-stress ALT to other stress loading designs.

The main focus of this article is to investigate the advan-tage of using step-stress ALT relative to constant-stress ALT.Assuming a log-linear relationship between the mean life-time parameter and stress level, with the cumulative exposuremodel for the effect of changing stress in step-stress ALT,the optimal design point is determined under various opti-mality criteria. In particular, the cases of complete samplingand Type-I censoring are considered under exponential life-time distribution for units subjected to stress. Using the ratioof optimal objective functions as a measure of relative effi-ciency, comparison of k-level step-stress testing to k-levelconstant-stress testing is discussed through a numerical study.

The rest of the article is organized as follows. Section2 describes the model under study, derives the maximumlikelihood estimators (MLE) of the model parameters andthe associated Fisher information for k-level constant-stressALT and step-stress ALT. Section 3 then defines the threeoptimality criteria based on the Fisher information (viz., vari-ance, determinant, and trace) and talks about the existence ofoptimal design points in each case under complete samplingand Type-I censoring. For the purpose of further comparisonbetween constant-stress and step-stress tests, the expectedtermination time of test is also obtained in Section 4. Section5 provides the results of a numerical study and discusses therelative efficiency of these two classes of ALT under con-sideration. Finally, Section 6 is devoted to some concludingremarks and future works in this direction.

2. MODEL DESCRIPTION, MLE, AND FISHERINFORMATION

Let us first define x1 < x2 < · · · < xk to be the orderedstress levels to be used in the test. Then, the following assump-tions form the basis of constructing both the constant-stressmodel and the step-stress model.

ASSUMPTIONS 2.1:

1. For any stress level xi , the lifetime of a test unitfollows an exponential distribution. That is, the prob-ability density function (PDF) and the correspondingcumulative distribution function (CDF) of a test unitat stress level xi are

fi (t) = 1

θi

exp

(− t

θi

), 0 < t < ∞, (2.1)

Fi (t) = 1 − Si (t) = 1 − exp

(− t

θi

), 0 < t < ∞,

(2.2)

respectively;2. At stress level xi , the mean time to failure (MTTF) of

a test unit, θi , is a log-linear function of stress givenby

logθi = α + βxi , (2.3)

where the regression parameters α and β areunknown and need to be estimated.

No notational distinction is made in this article betweenthe random variables and their corresponding realizations.Also, we adopt the usual conventions that

∑m−1j=m aj ≡ 0 and∏m−1

j=m aj ≡ 1.

2.1. k-level Constant-Stress Test under Type-ICensoring

A constant-stress test under Type-I censoring proceeds asfollows. For i = 1, 2, . . . , k, Ni ≡ nπi units are allocated ontest at stress level xi such that

∑ki=1 Ni = n or equivalently,∑k

i=1 πi = 1. πi = Ni/n is the allocation proportion of units(out of total n units under the test) assigned to stress level xi .The allocated units are then tested until time τi at which pointall the surviving items are withdrawn, thereby terminating thelife test. Let ni denote the number of units failed at stresslevel xi in time interval [0, τi) and yi,l denote the l-th orderedfailure time of ni units at xi , l = 1, 2, . . . , ni while Ni − ni

denotes the number of units censored at time τi . Obviously,when there is no right censoring (viz., τi = ∞ and ni = Ni),this situation corresponds to the k-level constant-stress testingunder complete sampling as a special case.

Now, under the assumption (1), the likelihood functionof n = (n1, n2, . . . , nk) and y = (y1, y2, . . . , yk) withyi = (

yi,1, yi,2, . . . , yi,ni

)is obtained as

fJ (y, n) =[

k∏i=1

Ni !(Ni − ni)!

][k∏

i=1

θ−ni

i

]exp

(−

k∑i=1

Ui

θi

),

(2.4)

where

Ui =ni∑

l=1

yi,l + (Ni − ni) τi , i = 1, 2, . . . , k. (2.5)

Note that Ui in (2.5) is the Total Time on Test statistic atstress level xi . Now, using (2.4) and the assumption (2), thelog-likelihood function of (α, β) can be written as

Naval Research Logistics DOI 10.1002/nav

Han and Ng: Constant-Stress and Step-Stress Accelerated Life Tests 543

l (α, β) = −α

k∑i=1

ni − β

k∑i=1

nixi

−k∑

i=1

Uiexp [− (α + βxi)] . (2.6)

On differentiating (2.6) with respect to α and β, theMLEs α̂ and β̂ are obtained as simultaneous solutions tothe following two equations:[

k∑i=1

ni

][k∑

i=1

Uixiexp(−β̂xi

)]

=k∑

i=1

nixi

k∑i=1

Uiexp(−β̂xi

), (2.7)

α̂ = log

⎛⎝∑k

i=1 Uiexp(−β̂xi

)∑k

i=1 ni

⎞⎠ . (2.8)

As shown above, α̂ and β̂ are nonlinear functions of ran-dom quantities and thus, statistical inference with these MLEscan be based on the asymptotic distributional result that the

vector(α̂, β̂

)is approximately distributed as a bivariate nor-

mal with mean vector (α, β) and variance-covariance matrixI−1n (α, β), where In (α, β) is the Fisher information matrix.

By using the following properties of the counts and orderstatistics, In (α, β) is obtained where selected proofs of thesubsequent lemmas and theorems are presented in Appendix.

PROPERTIES 2.1:

1. For i = 1, 2, . . . , k, the random variable ni has abinomial distribution with parameters (Ni , Fi (τi)).

2. For i = 1, 2, . . . , k, given ni , the random variablesyi,l , l = 1, 2, . . . , ni , are distributed jointly as orderstatistics from a random sample of size ni froma right-truncated exponential distribution with PDFfi,τi

(z) = fi (z)

Fi (τi )for 0 ≤ z ≤ τi .

THEOREM 2.1: Under this setup of the constant-stresstest with Type-I censoring, the Fisher information matrix is

In (α, β) = n

⎛⎜⎜⎜⎜⎜⎝

k∑i=1

Ai

k∑i=1

Aixi

k∑i=1

Aixi

k∑i=1

Aix2i

⎞⎟⎟⎟⎟⎟⎠ , (2.9)

where

Ai = πiFi (τi) , i = 1, 2, . . . , k. (2.10)

2.2. k-level Step-Stress Test under Type-I Censoring

For i = 1, 2, . . . , k, let us first define ni to be the num-ber of units failed at stress level xi in time interval

[τi−1, τi)

and yi,l to be the l-th ordered failure time of ni units at xi ,l = 1, 2, . . . , ni . Furthermore, let Ni denote the number ofunits operating and remaining on test at the start of stress levelxi (viz., Ni = n − ∑i−1

j=1 nj ). Then, a step-stress test underType-I censoring proceeds as follows. A total of N1 ≡ n

test units is initially placed at stress level x1 and tested untiltime τ1 at which point the stress is changed to x2. The test iscontinued on remaining N2 = n − n1 units until time τ2 atwhich the stress is changed to x3, and so on. Finally, at timeτk , all the surviving items are withdrawn, thereby terminatingthe life test. Note that the number of surviving items at timeτk is n − ∑k

i=1 ni = Nk − nk . Obviously, when there is noright censoring (viz., τk = ∞ and nk = Nk), this situationcorresponds to the k-level step-stress testing under completesampling as a special case.

Now, under the cumulative exposure model along with theassumption (1), the PDF and CDF of a test unit are

f (t) =⎡⎣i−1∏

j=1

Sj

(�j

)⎤⎦ fi (t − τi−1)

if

{τi−1 ≤ t ≤ τi for i = 1, 2, . . . , k − 1

τk−1 ≤ t < ∞ for i = k, (2.11)

F (t) = 1 −⎡⎣i−1∏

j=1

Sj

(�j

)⎤⎦ Si (t − τi−1)

if

{τi−1 ≤ t ≤ τi for i = 1, 2, . . . , k − 1

τk−1 ≤ t < ∞ for i = k,

(2.12)

where �j = τj − τj−1 is the step duration at stress levelxj , and fi (t) and Fi (t) are as given in (2.1) and (2.2),respectively. Then, using (2.11) and (2.12), the likelihoodfunction of n = (n1, n2, . . . , nk) and y = (y1, y2, . . . , yk)

with yi = (yi,1, yi,2, . . . , yi,ni

)is obtained as in (2.4) where

Ui =ni∑

l=1

(yi,l − τi−1

)+ (Ni − ni) �i , i = 1, 2, . . . , k.

(2.13)

Again, note that Ui in (2.13) is the Total Time on Teststatistic at stress level xi . Using (2.4) and the assumption (2),the log-likelihood function of (α, β) can be written as in (2.6)and as a result, we obtain the MLEs α̂ and β̂ as simultaneoussolutions to (2.7) and (2.8) with Ui given in (2.13).



Just like in the case of constant-stress testing, α̂ and β̂ arenonlinear functions of random quantities and hence, infer-ence using these MLEs are based on the asymptotic dis-

tributional result (viz.,(α̂, β̂

)′ ∼̇BV N((α, β)′ , I−1

n (α, β))).

Again, by using the following properties of the counts andorder statistics (which include Properties 2.1 as a specialcase), we can derive the expression of In (α, β) as well asthe expectation of Ni .

PROPERTIES 2.2:

1. The random variable n1 has a binomial distributionwith parameters (n, F1 (�1)). For i = 2, 3, . . . , k,given n1, n2, . . . , ni−1, the random variable ni has abinomial distribution with parameters (Ni , Fi (�i)).

2. Given n1, n2, . . . , ni , the random variables(yi,l − τi−1

), l = 1, 2, . . . , ni , are distributed jointly

as order statistics from a random sample of sizeni from a right-truncated exponential distributionwith PDF fi,�i

(z) = fi (z)

Fi (�i)for 0 ≤ z ≤ �i and

i = 1, 2, . . . , k.

LEMMA 2.1: For i = 1, 2, . . . , k,

E [Ni] = n

i−1∏j=1

Sj

(�j

)(2.14)

THEOREM 2.2: Under this setup of the step-stress testwith Type-I censoring, the Fisher information matrix is as in(2.9) where

Ai =⎡⎣i−1∏

j=1

Sj

(�j

)⎤⎦Fi (�i) , i = 1, 2, . . . , k. (2.15)

3. OPTIMALITY CRITERIA AND EXISTENCE OFOPTIMAL DESIGN POINTS

In this section, we define different optimality criteria fordetermining the optimal design points, which then can beused to compare between the multilevel constant-stress testand step-stress test. For the k-level constant-stress testing, thefocus is to determine the optimal allocation proportions π∗ =(π∗

1 , π∗2 , . . . , π∗

k

)with π∗

k = 1−∑k−1i=1 π∗

i while it is to deter-mine the optimal stress durations �∗ = (

�∗1, �∗

2, . . . , �∗k

)for the k-level step-stress testing. These objective functionsare purely based on the Fisher information matrix In (α, β)

derived in the preceding section. Unlike Ai defined in Gounoet al. [7], Ai’s in (2.10) and (2.15) are always positive, and this,in turn, eliminates any disconcerting anomalies and ensures apositive determinant of In (α, β) as well as a positive variancefunction. Therefore, there is no particular restriction on thesearch region for the optimal design points in these cases.

3.1. C-optimality

In an ALT experiment, researchers often wish to estimatethe parameters of interest with maximum precision and min-imum variability possible. In both the constant-stress andstep-stress settings under consideration here, such a parame-ter of interest is the mean lifetime of a unit at the use-condition(viz., θ0). For this purpose, we consider an objective functiongiven by

φ (·) = nAVar(

logθ̂0

)= nAVar

(α̂ + β̂x0

)= n (1x0) I−1

n (α, β)

(1x0

)

= 2∑k

i=1 Ai(xi − x0)2∑k

i=1

∑kj=1 AiAj

(xi − xj

)2 , (3.1)

where AVar stands for asymptotic variance and x0 is the nor-mal use-stress (i.e., x0 < x1). The C-optimal design pointsare the ones that minimize φ (·) in (3.1). In the case of k = 2(i.e., the case of a simple stress testing), the objective functionin (3.1) reduces to

φ (·) = A1(x1 − x0)2 + A2(x2 − x0)

2

A1A2(x2 − x1)2

= (1 + ξ0)2

A1+ ξ 2

0

A2, (3.2)

where ξ0 = x1−x0x2−x1

.

THEOREM 3.1: In the case of a simple constant-stress testunder Type-I censoring, the C-optimal allocation proportionsare

π∗1 = 1

1 + and π∗

2 =

1 + ,

where 2 = ξ 20

(1+ξ0)2

F1(τ1)

F2(τ2)=(

x1−x0x2−x0

)2F1(τ1)

F2(τ2).

COROLLARY 3.1: In the case of a simple constant-stresstest under Type-I censoring, the C-optimality allocates anequal number of test units at each stress level (viz., π∗

1 =π∗

2 = 1/2) when (x1 − x0)2F1 (τ1) = (x2 − x0)

2F2 (τ2).

THEOREM 3.2: In the case of a k-level constant-stresstest under complete sampling (i.e., no right censoring by let-ting τi → ∞ for i = 1, 2, . . . , k), the C-optimal allocationproportions are

π∗1 = 1

1 + , π∗

i = 0 for i = 2, 3, . . . , k − 1,

and π∗k =

1 + ,

where = x1−x0xk−x0

.



COROLLARY 3.2: In the case of a simple constant-stress test under complete sampling, the C-optimal allocationproportions are

π∗1 = 1

1 + and π∗

2 =

1 + ,

where = ξ0

1+ξ0= x1−x0

x2−x0.

THEOREM 3.3: In the case of a simple step-stress testunder Type-I censoring with an equal step duration (viz.,�1 = �2 = �), there exists a C-optimal step duration �∗which is the unique solution to the equation φ′ (�) = 0.

THEOREM 3.4: In the case of a simple step-stress testunder complete sampling (i.e., no right censoring by lettingτ2 → ∞), the C-optimal stress change point is

�∗1 = θ1log

(1 + 1

),

where = ξ0

1+ξ0= x1−x0

x2−x0.

3.2. D-optimality

Another optimality criterion often used in planning ALTis based on the determinant of the Fisher information matrix,which equals to the reciprocal of the determinant of theasymptotic variance-covariance matrix. Note that the over-all volume of the Wald-type joint confidence region of (α, β)

is proportional to |I−1n (α, β)|1/2 at a fixed level of confidence.

In other words, it is inversely proportional to |In (α, β)|1/2,the square root of the determinant of In (α, β). Consequently,a larger value of |In (α, β)| would correspond to a smallerasymptotic joint confidence ellipsoid of (α, β) and thus ahigher joint precision of the estimators of α and β. Moti-vated by this, our second objective function is simply givenby

δ (·) = n−2|In (α, β)|

= 1

2

k∑i=1

k∑j=1

AiAj

(xi − xj

)2. (3.3)

The D-optimal design points are obtained by maximizing(3.3) for the maximal joint precision of (α̂, β̂). For k = 2, theobjective function (3.3) reduces to

δ (·) = A1A2(x2 − x1)2. (3.4)

THEOREM 3.5: In the case of a simple constant-stress testunder Type-I censoring, the D-optimality allocates an equalnumber of test units at each stress level (viz., π∗

1 = π∗2 =

1/2).

THEOREM 3.6: In the case of a k-level constant-stresstest under complete sampling (i.e., no right censoring by let-ting τi → ∞ for i = 1, 2, . . . , k), the D-optimal allocationproportions are

π∗1 = 1

2, π∗

i = 0 for i = 2, 3, . . . , k − 1, and π∗k = 1

2.

COROLLARY 3.3: In the case of a simple constant-stresstest under complete sampling, the D-optimality allocatesan equal number of test units at each stress level (viz.,π∗

1 = π∗2 = 1/2).

THEOREM 3.7: In the case of a simple step-stress testunder Type-I censoring with an equal step duration (viz.,�1 = �2 = �), there exists a D-optimal step dura-tion �∗ which is the unique solution to the equationθ1F1 (�) S2 (�) = θ2 [1 − 2S1 (�)] F2 (�).

THEOREM 3.8: In the case of a simple step-stress testunder complete sampling (i.e., no right censoring by lettingτ2 → ∞), the D-optimal stress change point is the medianof the distribution at stress level x1 (viz., �∗

1 = θ1log2).

3.3. A-optimality

Another optimality criterion considered in our study isbased on the trace of the first-order approximation of thevariance-covariance matrix of the MLEs. It is identical to thesum of the diagonal elements of I−1

n (α, β). The A-optimalitycriterion provides an overall measure of the average vari-ance of the parameter estimates and gives the sum of theeigenvalues of the inverse of the Fisher information matrix.The A-optimal design points minimize the objective functiondefined by

a (·) = ntr(I−1n (α, β)

)= 2

∑ki=1 Ai

(1 + x2

i

)∑k

i=1

∑kj=1 AiAj

(xi − xj

)2 . (3.5)

In the case of the simple stress testing (k = 2), the objectivefunction in (3.5) becomes

a (·) = A1(1 + x2

1

)+ A2(1 + x2

2

)A1A2(x2 − x1)

2

= ξ 22

A1+ ξ 2

1

A2, (3.6)

where ξi =√

1+x2i

x2−x1for i = 1, 2.



THEOREM 3.9: In the case of a simple constant-stress testunder Type-I censoring, the A-optimal allocation proportionsare

π∗1 = 1

1 + and π∗

2 =

1 + ,

where 2 = ξ 21

ξ 22

F1(τ1)

F2(τ2)=(

1+x21

1+x22

)F1(τ1)

F2(τ2).

COROLLARY 3.4: In the case of a simple constant-stresstest under Type-I censoring, the A-optimality allocates anequal number of test units at each stress level (viz., π∗

1 =π∗

2 = 1/2) when(1 + x2

1

)F1 (τ1) = (

1 + x22

)F2 (τ2).

THEOREM 3.10: In the case of a k-level constant-stresstest under complete sampling (i.e., no right censoring by let-ting τi → ∞ for i = 1, 2, . . . , k), the A-optimal allocationproportions are

π∗1 = 1

1 + , π∗

i = 0 for i = 2, 3, . . . , k − 1,

and π∗k =

1 + ,

where 2 = 1+x21

1+x2k

.

COROLLARY 3.5: In the case of a simple constant-stress test under complete sampling, the A-optimal allocationproportions are

π∗1 = 1

1 + and π∗

2 =

1 + ,

where 2 = ξ 21

ξ 22

= 1+x21

1+x22.

THEOREM 3.11: In the case of a simple step-stress testunder Type-I censoring with an equal step duration (viz.,�1 = �2 = �), there exists an A-optimal step duration�∗ which is the unique solution to the equation a′ (�) = 0.

THEOREM 3.12: In the case of a simple step-stress testunder complete sampling (i.e., no right censoring by lettingτ2 → ∞), the A-optimal stress change point is

�∗1 = θ1log

(1 + 1

),

where 2 = ξ 21

ξ 22

= 1+x21

1+x22.

REMARK 3.1: It is of interest to note that for a simpleconstant-stress test, the D-optimal design allocates an equalnumber of test units at each stress level regardless of the

stress levels used, the presence of Type-I censoring nor thetime points of censoring at any stress level.

REMARK 3.2: From the results above, we see that in plan-ning a k-level constant-stress test under complete sampling,there is no need to assign the test units at any stress levelother than the lowest (x1) and the highest (xk) levels. Ifthe D-optimality is concerned, one should allocate 50% ofthe units to stress level x1 and the remaining 50% to stresslevel xk . If the C-optimality is considered, then one shouldallocate 100π∗

1 % of the units to stress level x1 and the remain-ing 100

(1 − π∗

1

)% to stress level xk where π∗

1 is definedaccordingly. One should note that the optimal plan is usuallyemployed as benchmarks in practice. Although statisticallyoptimal, using only two extreme stress levels (i.e., a simplestress ALT) does not allow us to verify whether the log-linearlife–stress relationship is a valid model assumption. Also, insome situations, it might be beneficial to use more than twostress levels for more flexibility despite some loss of effi-ciency. In that case, the methodology developed here can bestill used to design a different optimal allocation plan subjectto a particular allocation constraint (e.g., allocation of testunits at some intermediate stress levels); see for example Nget al. [18].

REMARK 3.3: For a multilevel constant-stress test undercomplete sampling, it is observed that under the C-optimalityand A-optimality criteria, 0 < < 1 since x0 < x1 < x2,and therefore π∗

1 = 11+

> 12 . This means that the C-optimal

and A-optimal designs allocate more test units at the lowerstress level than does the D-optimal design.

REMARK 3.4: For a simple step-stress test under com-plete sampling, it is observed that under the C-optimality andA-optimality criteria, 0 < < 1 since x0 < x1 < x2, andtherefore F1

(�∗

1

) = 11+

> 12 . This means that the C-optimal

and A-optimal stress change time points are larger than theD-optimal point, which is the median of the lifetime distrib-ution at stress level x1.

REMARK 3.5: For a simple step-stress test under com-plete sampling, it is noted that for each optimality consideredhere, as θ1 increases, �∗

1 increases in a manner such that theratio of �∗

1 to θ1 stays constant across the values of θ1. This isbecause the optimal stress change point is a fixed percentilefrom the distribution at stress level x1, irrespective of theMTTF at that level. From the numerical study of Balakrish-nan and Han [6], the same feature also prevailed in the caseof a k-level step-stress test under Type-I censoring as well asunder progressive Type-I censoring.

All the optimality criteria considered here, as well as someother information-based criteria, have been used extensivelyin the design selection process for linearly designed experi-ments. From a practitioner’s point of view, the choice of the



optimality criterion will be certainly guided by the objec-tive of the experiment. In cases where the planner is moreinterested in the precise estimation of the MTTF θ0 at nor-mal use-condition, the C-optimality is surely the criterion ofchoice. On the other hand, if one is more concerned aboutestimating the mean function given in assumption (2) orestimating the regression parameters α and β with high pre-cision, a more reasonable criterion of choice should be theD-optimality or A-optimality.

4. EXPECTED TERMINATION TIME OF TEST

To compare the test termination time between the constant-stress test and the step-stress test, the expected terminationtime of test is computed. In general, for a life test under Type-Icensoring with the sample size n and the censoring time pointτ , the expected termination time of test is

T ε = E [Yn:n|Yn:n < τ ] Pr (Yn:n < τ) + τP r (Yn:n ≥ τ) ,(4.1)

where Yn:n is the largest order statistic from a lifetime dis-tribution characterized by, say, the PDF f (t) and the CDFF(t). Note that given Yn:n < τ , Yn:n is distributed as thelargest order statistic from a random sample of size n froma right-truncated distribution with PDF fτ (t) = f (t)

F (τ)for

0 ≤ t ≤ τ . Using this property, the conditional expectationabove is expressed as

E [Yn:n|Yn:n < τ ] = n

∫ τ

0y

[F (y)

F (τ)

]n−1 [f (y)

F (τ)

]dy

= n

[F (τ)]n

n−1∑l=0

(n − 1

l

)(−1)l

∫ τ

0y[S (y)]lf (y) dy.

(4.2)

Since Pr (Yn:n < τ) = [F (τ)]n, we now have

T ε =n∑

l=1

(n

l

)(−1)l+1l

∫ τ

0y[S (y)]l−1f (y) dy

+ τ(1 − [F (τ)]n

). (4.3)

4.1. k-Level Constant-Stress Test under Type-ICensoring

From (4.3), the expected time spent at stress level xi is

T εi = θi

Ni∑l=1

(Ni

l

)(−1)l+1

l

(1 − [Si(τi)]l

)

= θi

Ni∑l=1

(Ni

l

)(−1)l+1

lFi(lτi) (4.4)

for i = 1, 2, . . . , k. When there is only one facility availablefor constant-stress testing, the life tests at different stress lev-els have to proceed in a sequential way. Then, the expectedtermination time of test is simply the sum of the expectedtime spent at each stress level.

THEOREM 4.1: In the case of a k-level constant-stresstest under Type-I censoring, the expected termination time oftest is

T ε =k∑

i=1

T εi =

k∑i=1

θi

Ni∑l=1

(Ni

l

)(−1)l+1

lFi (lτi)

COROLLARY 4.1: In the case of a simple constant-stresstest under Type-I censoring, the expected termination time oftest is

T ε = θ1

N1∑l=1

(N1

l

)(−1)l+1

lF1 (lτ1)

+ θ2

N2∑l=1

(N2

l

)(−1)l+1

lF2 (lτ2)

THEOREM 4.2: In the case of a k-level constant-stresstest under complete sampling (i.e., no right censoring by let-ting τi → ∞ for i = 1, 2, . . . , k), the expected terminationtime of test is

T ε =k∑

i=1

θi

Ni∑l=1

(Ni

l

)(−1)l+1

l.

COROLLARY 4.2: In the case of a simple constant-stresstest under complete sampling, the expected termination timeof test is

T ε = θ1

N1∑l=1

(N1

l

)(−1)l+1

l+ θ2

N2∑l=1

(N2

l

)(−1)l+1

l.

REMARK 4.1: When there are k multiple facilities avail-able for constant-stress testing, the life tests at different stresslevels may proceed in a partially or completely parallel way.If the test proceeds in a completely parallel way with the samestarting time point, the termination time of the entire test isdetermined by the longest time taken among tests at differentstress levels. For a k-level constant-stress test under completesampling, the expected termination time of test is

T ε =∫ ∞

0t fpar (t) dt ,



Table 1. Optimal step durations for a step-stress test under Type-I censoring.

k = 2 k = 3 k = 4

�∗C/θ1 �∗

D/θ1 �∗A/θ1 �∗

C/θ1 �∗D/θ1 �∗

A/θ1 �∗C/θ1 �∗

D/θ1 �∗A/θ1

ρ = 0.1 1.099 0.698 0.847 1.098 0.082 0.842 1.098 0.081 0.842ρ = 0.3 1.119 0.829 0.889 0.565 0.271 0.343 0.544 0.093 0.154ρ = 0.5 1.148 0.941 0.932 0.721 0.450 0.511 0.504 0.231 0.299

where

Fpar (t) =k∏

i=1

[Fi (t)]Ni , 0 < t < ∞,

fpar (t) = d

dtFpar (t)

=k∏

i=1

[Fi (t)]Ni

k∑i=1

Ni

fi (t)

Fi (t), 0 < t < ∞.

For a simple constant-stress test under complete sampling,this is

T ε =2∑

i=1

θi

Ni∑l=1

Ni′∑l′=0

(Ni

l

)(Ni ′

l′

)(−1)l+l′+1

l

(1 + l′θi

lθi ′

)−2

,

where i ′ = 1, 2, and i ′ �= i, obtained by using the standardbinomial expansion.

4.2. k-Level Step-Stress Test under Type-I Censoring

Using (4.3) again, the expected termination time of a step-stress test is derived where proof is presented in Appendix.

THEOREM 4.3: In the case of a k-level step-stress testunder Type-I censoring (or under complete sampling), theexpected termination time of test is

T ε =k∑

i=1

θi

n∑l=1

(n

l

)(−1)l+1

l

⎡⎣i−1∏

j=1

Sj

(�j

)⎤⎦l

Fi (l�i)

COROLLARY 4.3: In the case of a simple step-stress testunder Type-I censoring, the expected termination time of testis

T ε =n∑

l=1

(n

l

)(−1)l+1

l

× (θ1F1 (l�1) + θ2S1 (l�1) F2 (l�2)) .

COROLLARY 4.4: In the case of a simple step-stress testunder complete sampling, the expected termination time oftest is

T ε =n∑

l=1

(n

l

)(−1)l+1

l(θ1F1 (l�1) + θ2S1 (l�1)) .

5. NUMERICAL RESULTS

5.1. Optimal Designs and Relative Efficiency

A numerical study was conducted to investigate the relativeefficiency of step-stress testing compared to constant-stresstesting and to evaluate it as a function of varying parameters(i.e., MTTF and the number of stress levels). For the pur-pose of illustration, we considered equispaced stress levelsas xi = x0 + id with the use-stress level x0 = 10 and thestress increment d = 5. We also chose the ordered MTTF as

θi+1 = ρθi , i = 1, 2, . . . , k − 1, 0 < ρ < 1,

with selected choices of θ1 and ρ. Hence, a decreasing geo-metric sequence of MTTF was simulated with an increasingarithmetic sequence of stress levels.

Although a general k-level step-stress test under Type-Icensoring allows nonuniform step durations, for the purposeof illustration using a univariate optimization, an equal stepduration � was considered in this study. The optimizationwas then carried out by some of the popular techniques suchas the Nelder–Mead method or quasi–Newton method. Table1 presents the values of the optimal step duration �∗

C , �∗D , and

�∗A, which are independent of the sample size n. Rather than

the specific values of the optimal stress durations, the table isintended to provide a qualitative insight into the way the opti-mal choice changes as a function of the relevant parameters.Table 2 presents the corresponding optima of each objectivefunction described in Section 3. From the extensive numeri-cal study, it was found that these optima are independent ofthe values of θ1 as well as the sample size n. From Table 1, itis observed that �∗

C > �∗A > �∗

D except for the simple step-stress case with ρ = 0.5. This order, however, was found tobe a consequence of the specific setting chosen here and didnot necessarily hold for general stress levels in a more elabo-rate study. The behavior of the optimal � as a function of the



Table 2. Corresponding optima of the objective functions for a step-stress test under Type-I censoring.

k = 2 k = 3 k = 4

φ∗ss δ∗

ss a∗ss φ∗

ss δ∗ss a∗

ss φ∗ss δ∗

ss a∗ss

ρ = 0.1 9.00 6.24 49.17 9.00 9.44 49.13 9.00 9.45 49.13ρ = 0.3 9.08 5.76 50.42 7.87 13.00 31.57 7.85 19.65 28.76ρ = 0.5 9.36 5.04 53.63 6.55 13.32 27.50 6.04 23.88 20.55

Table 3. Optimal allocation proportions for a constant-stress test under Type-I censoring.

π∗C π∗

D π∗A

k = 2ρ = 0.1 (0.710, 0.290) (0.500, 0.500) (0.638, 0.362)ρ = 0.3 (0.707, 0.293) (0.500, 0.500) (0.628, 0.372)ρ = 0.5 (0.697, 0.303) (0.500, 0.500) (0.611, 0.389)

k = 3ρ = 0.1 (0.786, 0, 0.214) (0, 0.500, 0.500) (0.688, 0, 0.312)ρ = 0.3 (0.820, 0, 0.180) (0.500, 0, 0.500) (0.753, 0, 0.247)ρ = 0.5 (0.803, 0, 0.197) (0.500, 0, 0.500) (0.711, 0, 0.289)

k = 4ρ = 0.1 (0.830, 0, 0, 0.170) (0, 0.500, 0, 0.500) (0.726, 0, 0, 0.274)ρ = 0.3 (0.861, 0, 0, 0.139) (0, 0.500, 0, 0.500) (0.841, 0, 0, 0.159)ρ = 0.5 (0.863, 0, 0, 0.137) (0.500, 0, 0, 0.500) (0.789, 0, 0, 0.211)

MTTF values is also interesting. From the extensive numeri-cal study, it was found that for fixed k and ρ, as θ1 increases,�∗

C , �∗D , and �∗

A increase in a manner such that the ratios�∗

C/θ1, �∗D/θ1, and �∗

A/θ1 are constant across the values ofθ1. It explains another observation from this numerical studythat for a given k andρ, the ratios�∗

C/�∗D and�∗

D/�∗A remain

constant over varying ranges of θ1. This translates to �∗C , �∗

D ,and �∗

A being fixed percentiles from the stage-1 exponentialdistribution, irrespective of the value of θ1. The same observa-tion was made in Balakrishnan and Han [6] under progressiveType-I censoring. Besides, the dependence of the optimal val-ues on ρ was found to be less noticeable for smaller valuesof k.

Using the optimal step durations obtained in Table 1 asthe censoring time points at each stress level, the allocationproportions π = (π1, π2, . . . , πk) were then optimized for ak-level constant-stress test under Type-I censoring. Table 3presents the values of these optimal allocation proportionsπ∗

C , π∗D , and π∗

A, and Table 4 presents the corresponding

optima of each objective function described in Section 3.From the extensive numerical study, it was shown that theseoptimal allocation proportions and the corresponding optimaare also independent of the values of θ1 as well as the samplesize n. From Table 3, it is observed that π∗

1,C > π∗1,A > π∗

1,Dregardless of the values of k and ρ. It is also interesting to notethat except for the first and last stress levels, the C-optimalityand the A-optimality do not allocate any test units in the inter-mediate stress levels. The same observation was analyticallyproven under complete sampling as stated in Theorems 3.2and 3.10. The D-optimality on the other hand allocates anequal number of test units at two stress levels with no unitsallocated in any other stress levels.

To compare the termination time between the constant-stress test and the step-stress test under the optimal condi-tions, the expected termination time of each test was com-puted based on the results derived in Section 4 with theoptimal step durations in Table 1 and the optimal allocationproportions in Table 3. The results are given in Tables 5 and

Table 4. Corresponding optima of the objective functions for a constant-stress test under Type-I censoring.

k = 2 k = 3 k = 4

φ∗cs δ∗

cs a∗cs φ∗

cs δ∗cs a∗

cs φ∗cs δ∗

cs a∗cs

ρ = 0.1 11.90 3.14 68.97 5.46 3.49 23.24 3.87 13.86 13.36ρ = 0.3 11.90 3.30 68.98 7.75 5.64 38.00 5.72 6.43 39.81ρ = 0.5 12.07 3.23 70.81 6.80 7.57 31.01 6.03 9.79 24.88



Table 5. Expected termination time of the optimal step-stress test under Type-I censoring.

k = 2 k = 3 k = 4

T εC/θ1 T ε

D/θ1 T εA/θ1 T ε

C/θ1 T εD/θ1 T ε

A/θ1 T εC/θ1 T ε

D/θ1 T εA/θ1

n = 5ρ = 0.1 1.185 0.852 0.980 1.185 0.176 0.976 1.185 0.174 0.976ρ = 0.3 1.430 1.217 1.266 0.996 0.625 0.731 0.975 0.294 0.425ρ = 0.5 1.617 1.463 1.455 1.363 1.062 1.143 1.151 0.738 0.865

n = 10ρ = 0.1 1.279 0.920 1.055 1.279 0.184 1.050 1.279 0.182 1.050ρ = 0.3 1.637 1.381 1.440 1.123 0.686 0.807 1.097 0.315 0.462ρ = 0.5 1.888 1.673 1.662 1.583 1.195 1.299 1.321 0.817 0.973

n = 20ρ = 0.1 1.349 0.987 1.122 1.348 0.191 1.117 1.348 0.189 1.117ρ = 0.3 1.817 1.513 1.587 1.220 0.734 0.867 1.190 0.332 0.490ρ = 0.5 2.097 1.809 1.795 1.763 1.285 1.414 1.463 0.873 1.056

n = 30ρ = 0.1 1.388 1.026 1.162 1.388 0.194 1.157 1.388 0.193 1.157ρ = 0.3 1.915 1.572 1.656 1.265 0.759 0.899 1.233 0.341 0.503ρ = 0.5 2.185 1.852 1.836 1.853 1.319 1.463 1.536 0.896 1.095

6, respectively. It is observed that in either testing schemesalone, the expected termination time of test does not changemuch as k increases. However, the step-stress testing takeslonger than the constant-stress testing as k increases althoughthey are quite similar when k = 2. In both cases, it takeslonger to complete the test as the sample size n increases.From Tables 5 and 6, it is also observed that T ε

C > T εA > T ε

D

regardless of the values of other parameters. As observed sim-ilarly in Table 1, for fixed k and ρ, as θ1 increases, T ε

C , T εD , and

T εA increase in a manner such that the ratios T ε

C/θ1, T εD/θ1,

and T εA/θ1 are constant across the values of θ1, irrespective

of the value of n.

To formally assess the efficiency of the step-stress testingcompared to the constant-stress testing under the optimal con-dition discussed in this section, pair-wise ratios of the optimaunder each criterion were computed using the results obtainedin Tables 2 and 4 (viz., effC = φ∗

cs/φ∗ss , effD = δ∗

ss/δ∗cs ,

effA = a∗cs/a

∗ss). The values are tabulated in Table 7 and

the number greater than 1 indicates higher efficiency of thestep-stress test compared to the constant-stress one since theobjective of the D-optimality is to maximize (3.3), whereasthe objectives of the C-optimality and the A-optimality areto minimize (3.1) and (3.5), respectively. As expected, theseratios are invariant across the values of θ1 and the sample

Table 6. Expected termination time of the optimal constant-stress test under Type-I censoring.

k = 2 k = 3 k = 4

T εC/θ1 T ε

D/θ1 T εA/θ1 T ε

C/θ1 T εD/θ1 T ε

A/θ1 T εC/θ1 T ε

D/θ1 T εA/θ1

n = 5ρ = 0.1 1.137 0.778 0.947 1.047 0.086 0.807 1.038 0.072 0.819ρ = 0.3 1.346 1.135 1.250 0.650 0.392 0.430 0.567 0.129 0.180ρ = 0.5 1.674 1.399 1.465 0.944 0.724 0.725 0.623 0.378 0.412

n = 10ρ = 0.1 1.270 0.921 1.050 1.106 0.104 0.857 1.092 0.082 0.841ρ = 0.3 1.634 1.414 1.448 0.699 0.455 0.474 0.571 0.150 0.194ρ = 0.5 1.906 1.736 1.688 1.068 0.845 0.878 0.626 0.432 0.464

n = 20ρ = 0.1 1.343 0.990 1.106 1.119 0.111 0.866 1.100 0.084 0.844ρ = 0.3 1.812 1.542 1.567 0.752 0.495 0.539 0.594 0.164 0.203ρ = 0.5 2.106 1.845 1.808 1.188 0.886 0.957 0.726 0.455 0.517

n = 30ρ = 0.1 1.382 1.028 1.149 1.123 0.115 0.870 1.100 0.084 0.844ρ = 0.3 1.906 1.593 1.645 0.769 0.513 0.563 0.601 0.171 0.215ρ = 0.5 2.188 1.870 1.843 1.255 0.896 0.988 0.755 0.460 0.544



Table 7. Efficiency of the step-stress testing to the constant-stresstesting under Type-I censoring.

k = 2 k = 3 k = 4

Optimality C D A C D A C D A

ρ = 0.1 1.32 1.99 1.40 0.61 2.70 0.47 0.43 0.68 0.27ρ = 0.3 1.31 1.75 1.37 0.99 2.30 1.20 0.73 3.06 1.38ρ = 0.5 1.29 1.56 1.32 1.04 1.76 1.13 1.00 2.44 1.21

size n. It is also observed from Table 7 that the highestefficiency is achieved by the D-optimality, followed by theA-optimality, and then by the C-optimality in general. FromTable 7, the step-stress test is empirically shown to be moreefficient compared to the corresponding constant-stress onein many cases under the optimal situations. As k increasesand ρ decreases; however, the efficiency seems to get lower,especially for the C-optimality.

5.2. Sensitivity Analysis

Because the model parameters α and β are unknown atthe planning stage of a test, they need to be pre-estimatedfrom experience, by using similar data, or from a prelimi-nary pilot test. These estimates can, however, substantiallydeviate from their true values and such deviations generatenonoptimal designs which may result in poor precision ofthe model parameters and MTTF at the use-condition. In thissection, the sensitivity of the optimal designs to the parameteruncertainty is investigated to examine the overall robustnessof the optimal designs as well as to identify any parameterwhich needs to be estimated with special care for minimizingthe risk of obtaining an erroneous optimal plan.

For an illustration purpose, the same parameter setting asin the previous section was considered. Using the log-linearrelationship in (2.3), θ1 and ρ can be uniquely reparame-terized into the regression parameters α and β. That is,α = logθ1 − βx1 = logθ1 − (

x0d

+ 1)

logρ with β = logρ

d.

Reanalyzing Table 1 given the use-stress level x0 = 10 andthe stress incrementd = 5, it was found that with the esti-mate of β held constant, a unit change in the estimate of α

corresponds to a unit difference in log�∗ regardless of thevalues of k and the optimality criteria. Hence, a small devia-tion of the estimate of α from its true value could cause a largechange in the optimal stress duration for a k-level step-stresstest under Type-I censoring. Reanalyzing Table 2, however,it revealed that a change in the estimate of α alone does notinduce any change in the corresponding optima of the objec-tive functions. This is because the optima are independent ofthe values of θ1 across the values of k and the optimality cri-teria. Reanalyzing Tables 3 and 4, the same observation wasmade for a k-level constant-stress test under Type-I censor-ing. With the estimate of β fixed, a change in the estimate ofα alone did not induce any change in the optimal allocation

proportions π∗C , π∗

D , and π∗A as well as the corresponding

optima of each objective function because they were bothfound to be independent of θ1 regardless of the values of kand the optimality criteria.

On the other hand, with the estimate of α kept constant,as the relative change in the estimate of β from its true valueincreased up to 50%, the optimal � could change very rapidlyand drastically for a k-level step-stress test under Type-Icensoring (e.g., up to 10 times for �∗

C , up to 20 times for�∗

D , and up to 40 times for �∗A). As k increased, the fold

change seemed to be smaller for the C-optimality relative tothe D-optimality and the A-optimality. The changes in theoptima corresponding to each optimal � were also found tobe significant (e.g., −30–3% for the C-optimality, −15–75%for the D-optimality, and −40–7% for the A-optimality). Asthe value of k increased, there seemed to be a reduction inthe optima for the C-optimality and the A-optimality but anincrease for the D-optimality. For a k-level constant-stresstest under Type-I censoring, as the relative change in the esti-mate of β from its true value increased up to 50% or the valueof k increased with the estimate of α fixed, the optimal allo-cation proportions π∗ exhibited a small degree of change ornone (e.g., up to 5% for π∗

C and π∗A, and no change for π∗

D).The changes in the optima corresponding to each optimalallocation proportion were, however, found to be quite sub-stantial (e.g., up to 45% for the C-optimality, up to −70% forthe D-optimality, and up to −65% for the A-optimality). As kincreased, there seemed to be a reduction in the optima for theD-optimality while an increase for the C-optimality and theA-optimality. Nevertheless, these observations were a conse-quence of the specific setting chosen in this study and did notnecessarily hold in other situations in a more elaborate studywe have conducted. Overall, the sensitivity analysis indicatedthat the optimal ALT plans are relatively insensitive to smalldepartures of the initial estimate of α while misspecificationof β has a large effect on the selection of the optimal designpoints. Hence, the proposed optimal plans are more robustagainst misspecification of the intercept parameter α than theslope parameter β.

6. SUMMARY AND CONCLUSION

In this work, the optimal k-level constant-stress ALT andstep-stress ALT were compared for the exponential fail-ure data under Type-I censoring. One of the objectives ofthe study was to quantify the advantage of using the step-stress ALT compared to the constant-stress one. A log-linearrelationship was assumed between the mean lifetime para-meter and the stress level, and the cumulative exposuremodel was assumed for the effect of changing stress lev-els in the step-stress ALT. Based on the Fisher informationmatrix, the optimal design points were determined under



the C-optimality, D-optimality, and A-optimality criteria.The relative efficiency of the k-level step-stress ALT to thek-level constant-stress ALT was then measured via pair-wiseratios of the optima under each criterion, and some interest-ing observations were made from the results of the numericalstudy under a particular setup. A sensitivity analysis alsorevealed that these optimal ALT designs are less sensitiveto misspecification of the intercept parameter than the slopeparameter. For further and generalized studies, it would beof interest to consider comparison between the constant-stress ALT and the step-stress ALT under different censoringschemes with the lifetime data from other popular distribu-tions belonging to a log-location-scale family (e.g., Weibulland log-normal). Moreover, a log-linear link was assumedbetween the MTTF and the stress level in this study becauseit is the most commonly used model in ALT representingseveral well-known life–stress relationships based on somephysical principles such as Arrhenius, inverse power law, andEyring; see Miller and Nelson [15]. Nevertheless, verificationof the model assumption and study of the inferential sensitiv-ity to it are other important tasks to be accomplished when thelife–stress relationship is not clear or questionable. Researchin these directions is under progress and we hope to reportthese findings in a future paper.

APPENDIX: PROOF OF LEMMAS AND THEOREMS

PROOF OF THEOREM 2.1: Using Properties 2.1, the expected value ofUi in (2.5) is

E [Ui ] = E

[ni∑

l=1

yi,l

]+ (Ni − E [ni ]) τi

= E [ni ]

(θi − τi

Si (τi )

Fi (τi )

)+ (Ni − NiFi (τi )) τi

= NiFi (τi ) θi = NiFi (τi ) exp (α + βxi) ,

for i = 1, 2, . . . , k. The matrix In (α, β) can then be expressed as

In (α, β) =(

Iαα Iαβ

Iαβ Iββ

),

where

Iαα = E

[− ∂2

∂α2l (α, β)

]=

k∑i=1

E [Ui ] exp [− (α + βxi)]

=k∑

i=1

NiFi (τi ) = n

k∑i=1

Ai ,

Iαβ = E

[− ∂2

∂α∂βl (α, β)

]=

k∑i=1

E [Ui ] exp [− (α + βxi)] xi

=k∑

i=1

NiFi (τi ) xi = n

k∑i=1

Aixi ,

Iββ = E

[− ∂2

∂β2l (α, β)

]=

k∑i=1

E [Ui ] exp [− (α + βxi)] x2i

=k∑

i=1

NiFi (τi ) x2i = n

k∑i=1

Aix2i ,

with Ai defined as in (2.10). �

PROOF OF LEMMA 2.1: The expression in (2.14) can be verified byinduction. Following the usual convention, we have E [N1] = n for i = 1from (2.14). Now, suppose that (2.14) holds for i = i′. Using the Properties2.2 (1) and the fact that Ni+1 = Ni − ni , we have

E[Ni′+1

] = E [Ni′ − ni′ ] = E [Ni′ ] − E [ni′ ]

= E [Ni′ ] Si′ (�i) = n

i′∏j=1

Sj

(�j

),

which is precisely the expression in (2.14) for i = i′ + 1. Hence, theresult. �

PROOF OF THEOREM 2.2: Using Properties 2.2, the expected value ofUi in (2.13) is

E [Ui ] = E

[ni∑

l=1

(yi,l − τi−1

)]+ (E [Ni ] − E [ni ])�i

= E [ni ]

(θi − �i

Si (�i)

Fi (�i)

)+ (E [Ni ] − E [Ni ] Fi (�i))�i

= E [Ni ] Fi (�i) θi = E [Ni ] Fi (�i) exp (α + βxi) ,

for i = 1, 2, . . . , k. The matrix In (α, β) can then be expressed as

In (α, β) =(

Iαα Iαβ

Iαβ Iββ

),

where

Iαα = E

[− ∂2

∂α2l (α, β)

]=

k∑i=1

E [Ui ] exp [− (α + βxi)]

=k∑

i=1

E [Ni ] Fi (�i) = n

k∑i=1

Ai ,

Iαβ = E

[− ∂2

∂α∂βl (α, β)

]=

k∑i=1

E [Ui ] exp [− (α + βxi)] xi

=k∑

i=1

E [Ni ] Fi (�i) xi = n

k∑i=1

Aixi ,

Iββ = E

[− ∂2

∂β2l (α, β)

]=

k∑i=1

E [Ui ] exp [− (α + βxi)] x2i

=k∑

i=1

E [Ni ] Fi (�i) x2i = n

k∑i=1

Aix2i ,

with Ai defined as in (2.15) by using Lemma 2.1. �



PROOF OF THEOREM 3.2: Directly using In (α, β) in (2.9) withAi (π) = πi , (3.1) can be re-expressed as

φ (π) =∑k

i=1 πi(xi − x0)2

∑ki=1 πix

2i −

(∑ki=1 πixi

)2.

Then, its first partials are obtained as

∂φ (π)

∂πj

=(xj − xk

){∑k

i=1 πix2i −

(∑ki=1 πixi

)2}2

×[

2k∑

i=1

πixi (xi − x0) − (xj + xk

) ( k∑i=1

πixi − x0

)]

×(

k∑i=1

πixi − x0

)

for j = 1, 2, . . . , k − 1 since πk = 1 − ∑k−1i=1 πi . When j = 1, ∂φ(π)

∂π1=

0 implies 2∑k

i=1 πixi (xi − x0) = (x1 + xk)(∑k

i=1 πixi − x0

)since∑k

i=1 πixi > x0 (i.e., x0 < x1). Hence, for j = 2, 3, . . . , k − 1,

∂φ (π)

∂πj

=(x1 − xj

) (xj − xk

){∑k

i=1 πix2i −

(∑ki=1 πixi

)2}2

(k∑

i=1

πixi − x0

)2

> 0.

Therefore, φ (π) monotonically increases in πj for j = 2, 3, . . . , k − 1.Because 0 ≤ πj ≤ 1, φ (π) is minimized at π∗

j = 0 for j = 2, 3, . . . , k − 1.

As a result, the only nonzero πj for j = 1, 2, . . . , k − 1 is π1. Now, ∂φ(π)∂π1

issimplified to

∂φ (π1)

∂π1

= (x1 − xk)2 (xk − x0) [π1ρ − (1 − π1)] [π1x1 + (1 − π1) xk − x0]{π1x

21 + (1 − π1) x2

k − [π1x1 + (1 − π1) xk]2}2.

Solving ∂φ(π1)∂π1

= 0 gives π∗1 = (1 + ρ)−1 as the only viable solution and

consequently, π∗k = 1 − π∗

1 . �

PROOF OF THEOREM 3.3: The first two derivatives of (3.2) withrespect to � are

φ′ (�) = −(1 + ξ0)2 A′

1 (�)

[A1 (�)]2− ξ2

0A′

2 (�)

[A2 (�)]2,

φ′′ (�) = −(1 + ξ0)2 A1 (�) A′′

1 (�) − 2[A′

1 (�)]2

[A1 (�)]3

− ξ20

A2 (�) A′′2 (�) − 2

[A′

2 (�)]2

[A2 (�)]3,

where

A1 (�) = F1 (�) = 1 − S1 (�) , A′1 (�) = 1

θ1S1 (�) ,

A′′1 (�) = − 1

θ21

S1 (�) ,

A2 (�) = F2 (�) S1 (�) = (1 − S2 (�)) S1 (�) ,

A′2 (�) =

[1

θ2S2 (�) − 1

θ1F2 (�)

]S1 (�) ,

A′′2 (�) = −

[1

θ22

S2 (�) + 2

θ1θ2S2 (�) − 1

θ21

F2 (�)

]S1 (�)

Since A1 (�) > 0 and A′′1 (�) < 0 for all � > 0, A1 (�) A′′

1 (�) −2[A′

1 (�)]2

< 0, making the first term of φ′′ (�) positive for all � > 0.Furthermore, A2 (�) > 0 and

A2 (�) A′′2 (�) − 2

[A′

2 (�)]2

= −F2 (�)

[1

θ22

S2 (�) + 2

θ1θ2S2 (�) − 1

θ21

F2 (�)

][S1 (�)]2

− 2

[1

θ2S2 (�) − 1

θ1F2 (�)

]2

[S1 (�)]2

= −{

1

θ22

S2 (�) F2 (�) + 2

θ22

[S2 (�)]2

− 2

θ1θ2S2 (�) F2 (�) + 1

θ21

[F2 (�)]2

}[S1 (�)]2

= −{

1

θ22

S2 (�) +[

1

θ2S2 (�) − 1

θ1F2 (�)

]2}

[S1 (�)]2 < 0

for all � > 0, making the second term of φ′′ (�) positive, too. There-fore, φ (�) is convex as φ′′ (�) > 0 for all � > 0. Hence, �∗ satisfyingφ′ (�∗) = 0 minimizes φ (�) for k = 2. �

PROOF OF THEOREM 3.4: The first two derivatives of (3.2) withrespect to �1 are as given in the proof of Theorem 3.3 where

A1 (�1) = F1 (�1) = 1 − S1 (�1) , A′1 (�1) = 1

θ1S1 (�1) ,

A′′1 (�1) = − 1

θ21

S1 (�1) ,

A2 (�1) = S1 (�1) , A′2 (�1) = − 1

θ1S1 (�1) , A′′

2 (�1) = 1

θ21

S1 (�1)

Since A1 (�1) > 0 and A′′1 (�1) < 0 for all �1 > 0, A1 (�1) A′′

1 (�1) −2[A′

1 (�1)]2

< 0, making the first term of φ′′ (�1) positive for all �1 > 0.Furthermore, A2 (�1) > 0 and

A2 (�1) A′′2 (�1) − 2

[A′

2 (�1)]2 = 1

θ21

[S1 (�1)]2 − 2

θ21

[S1 (�1)]2 < 0

for all �1 > 0, making the second term of φ′′ (�1) positive as well. There-fore, φ (�1) is convex as φ′′ (�1) > 0, and �∗

1 satisfying φ′ (�∗1

) = 0minimizes φ (�1). Now, φ′ (�∗

1

) = 0 implies

(1 + ξ0)2[S1

(�∗

1

)]2 = ξ20

[1 − S1

(�∗

1

)]2.

Solving this quadratic equation for S1(�∗

1

)gives ρ

1+ρas the only viable

solution. �




δ (π) =k∑

i=1

πix2i −

(k∑

i=1

πixi

)2

.


∂δ (π)

∂πj

= x2j − x2

k − 2(xj − xk

) k∑i=1

πixi

for j = 1, 2, . . . , k − 1 since πk = 1 − ∑k−1i=1 πi . When j = 1, ∂δ(π)

∂π1= 0

implies 2∑k

i=1 πixi = x1 + xk . Hence, for j = 2, 3, . . . , k − 1,

∂δ (π)

∂πj

= x2j − x2

k − (xj − xk

)(x1 + xk) = − (

xk − xj

) (xj − x1

)< 0.

Therefore, δ (π) monotonically decreases in πj for j = 2, 3, . . . , k − 1.Since 0 ≤ πj ≤ 1, δ (π) is maximized at π∗

j = 0 for j = 2, 3, . . . , k − 1.As a result, the only nonzero πj for j = 1, 2, . . . , k − 1 is π1. Now, solving

∂δ (π1)

∂π1= x2

1 − x2k − 2 (x1 − xk) [π1x1 + (1 − π1) xk] = 0,

π∗1 = 1/2 results and consequently, π∗

k = 1 − π∗1 = 1/2. �

PROOF OF THEOREM 3.7: Differentiating (3.4) with respect to �, weobtain

δ′ (�) = {A′

1 (�) A2 (�) + A1 (�) A′2 (�)

}(x2 − x1)

2

= 1

θ1F1 (�) F2 (�)

{QD

1 + θ1

θ2QD

2 − 1

}S1 (�) (x2 − x1)

2,

where QD1 = S1(�)

F1(�)and QD

2 = S2(�)F2(�)

. Now, we observe that δ (�) monoton-

ically increases when QD1 > 1 or equivalently when � < θ1log2 as QD

1 > 1implies δ′ (�) > 0. In other words, when the chance of a unit to survive thefirst stress level exceeds 50%, δ (�) increases. Besides, we see that θ1 > θ2

according to (2.3) from Assumptions 2.1 (ii) because the MTTF decreasesas the stress increases (i.e., x1 < x2). Then, the following is true:

θ1 > θ2 ⇔ S1 (�) > S2 (�) ⇔ F1 (�) < F2 (�)

It is then obvious that QD1 > QD

2 and so, QD1 + θ1

θ2QD

2 < QD1 +

θ1θ2

QD1 = QD

1

(1 + θ1

θ2

). If QD

1

(1 + θ1

θ2

)< 1 or equivalently if � >

θ1log(

2 + θ1θ2

), δ′ (�) < 0 and δ (�) monotonically decreases. Since δ′ (�)

is absolutely continuous in �, there is �∗ ∈(θ1log2, θ1log

(2 + θ1

θ2

))such

that δ′ (�∗) = 0. As δ′ (�) changes the sign around �∗, �∗ maximizesδ (�). �

PROOF OF THEOREM 3.8: Differentiating (3.4) with respect to �1, weobtain

δ′ (�1) = 1

θ1[S1 (�1) − F1 (�1)] S1 (�1) (x2 − x1)

2,

δ′′ (�1) = − 1

θ21

[3S1 (�1) − F1 (�1)] S1 (�1) (x2 − x1)2.

Setting δ′ (�∗1

) = 0, �∗1 satisfies S1

(�∗

1

) = F1(�∗

1

) = 1/2. Sinceδ′′ (�∗

1

)< 0, �∗

1 maximizes δ (�1). �


a (π) = 1 +1 +

(∑ki=1 πixi

)2

∑ki=1 πix

2i −

(∑ki=1 πixi

)2.


∂a (π)

∂πj

=(xj − xk

){∑k

i=1 πix2i −

(∑ki=1 πixi

)2}2

×⎧⎨⎩2

(1 +

k∑i=1

πix2i

)(k∑

i=1

πixi

)− (

xj + xk

)⎡⎣1 +(

k∑i=1

πixi

)2⎤⎦⎫⎬⎭

for j = 1, 2, . . . , k − 1 since πk = 1 − ∑k−1i=1 πi . When j = 1, ∂a(π)

∂π1= 0

implies 2(

1 +∑ki=1 πix

2i

)(∑ki=1 πixi

)= (x1 + xk)

[1 +

(∑ki=1 πixi

)2]

.

Hence, for j = 2, 3, . . . , k − 1,

∂a (π)

∂πj

=(x1 − xj

) (xj − xk

){∑k

i=1 πix2i −

(∑ki=1 πixi

)2}2

⎡⎣1 +

(k∑

i=1

πixi

)2⎤⎦ > 0.

Therefore, a (π) monotonically increases in πj for j = 2, 3, . . . , k − 1.Since 0 ≤ πj ≤ 1, a (π) is minimized at π∗

j = 0 for j = 2, 3, . . . , k − 1.

As a result, the only nonzero πj for j = 1, 2, . . . , k − 1 is π1. Now, ∂a(π)∂π1

issimplified to

∂a (π1)

∂π1= (x1 − xk)

2 (1 + x2k

) [(ρ2 − 1

)π2

1 + 2π1 − 1]

{π1x

21 + (1 − π1) x2

k − [π1x1 + (1 − π1) xk]2}2.

Solving ∂a(π1)∂π1

= 0 gives π∗1 = (1 + ρ)−1 as the only viable root of the

quadratic equation and consequently, π∗k = 1 − π∗

1 . �

PROOF OF THEOREM 3.11: The first two derivatives of (3.6) withrespect to � are

a′ (�) = −ξ22

A′1 (�)

[A1 (�)]2− ξ2

1A′

2 (�)

[A2 (�)]2,

a′′ (�) = −ξ22

A1 (�) A′′1 (�) − 2

[A′

1 (�)]2

[A1 (�)]3

− ξ21

A2 (�) A′′2 (�) − 2

[A′

2 (�)]2

[A2 (�)]3.

Using similar arguments as in the proof of Theorem 3.3, it can be shownthat a (�) is convex for all � > 0. Hence, �∗ satisfying a′ (�∗) = 0minimizes a (�) for k = 2. �

PROOF OF THEOREM 3.12: The first two derivatives of (3.6) withrespect to �1 are as given in the proof of Theorem 3.11. Using similararguments as in the proof of Theorem 3.4, it can be shown that a (�1) isconvex for all �1 > 0, and �∗

1 satisfying a′ (�∗1

) = 0 minimizes a (�1).Now, a′ (�∗

1

) = 0 implies

ξ22

[S1(�∗

1

)]2 = ξ21

[1 − S1

(�∗

1

)]2.

Solving this quadratic equation for S1(�∗

1

)gives ρ

1+ρas the only viable

solution. �



PROOF OF THEOREM 4.3: According to (4.3) with τk in place of τ , theexpected termination time of a k-level step-stress test under Type-I censoringis obtained as follows by using (2.11) and (2.12) along with (2.1) and (2.2).

T ε =n∑

l=1

(n

l

)(−1)l+1l

∫ τk

0y[S (y)]l−1f (y) dy + τk

(1 − [F (τk)]n

)

=n∑

l=1

(n

l

)(−1)l+1l

k∑i=1

∫ τi

τi−1

y

θi

⎡⎣i−1∏

j=1

Sj

(�j

)⎤⎦l

× [Si (y − τi−1)]ldy + τk

(1 − [F (τk)]n

)(∵ θifi (t) = Si (t))

=n∑

l=1

(n

l

)(−1)l+1l

k∑i=1

⎡⎣i−1∏

j=1

Sj

(�j

)⎤⎦l

×∫ �i

0

(y + τi−1)

θi

[Si (y)]ldy + τk

(1 − [F (τk)]n

)

=n∑

l=1

(n

l

)(−1)l

l

k∑i=1

[S (τi−1)]l{(lτi + θi ) [Si (�i)]l − (lτi−1 + θi )}

+ τk

(1 − [F (τk)]n

)(∵ S (τi−1) Si (�i) = S (τi ))

=n∑

l=1

(n

l

)(−1)l

l

k∑i=1

[S (τi )]l (lτi + θi )

−n∑

l=1

(n

l

)(−1)l

l

k−1∑i=0

[S (τi )]l (lτi + θi+1) + τk

(1 − [F (τk)]n

)

=n∑

l=1

(n

l)(−1)l

l[S (τk)]

l (lτk + θk)

+n∑

l=1

(n

l)(−1)l+1

l

k−1∑i=0

[S (τi )]l (θi+1 − θi )

+ τk

(1 − [F (τk)]

n)(∵ by setting θ0 ≡ 0)

= τk

n∑l=1

(n

l

)(−1)l [S (τk)]l + θk

n∑l=1

(n

l

)(−1)l

l[S (τk)]l

+n∑

l=1

(n

l

)(−1)l+1

l

k−1∑i=0

[S (τi )]l (θi+1 − θi )

+ τk

(1 − [F (τk)]n

)= τk

n∑l=0

(n

l

)[−S (τk)]l − τk + θk

n∑l=1

(n

l

)(−1)l

l[S (τk)]l

+n∑

l=1

(n

l

)(−1)l+1

l

k∑i=1

[S (τi−1)]l (θi − θi−1)

+ τk

(1 − [F (τk)]n

)= τk[1 − S (τk)]n − τk + θk

n∑l=1

(n

l

)(−1)l

l[S (τk)]l

+n∑

l=1

(n

l

)(−1)l+1

l

k∑i=1

[S (τi−1)]l (θi − θi−1)

+ τk

(1 − [F (τk)]n

)= θk

n∑l=1

(n

l

)(−1)l

l[S (τk)]l

+n∑

l=1

(n

l

)(−1)l+1

l

k∑i=1

[S (τi−1)]l (θi − θi−1)

=k∑

i=1

(θi − θi−1)

n∑l=1

(n

l

)(−1)l

l[S (τk)]l

+n∑

l=1

(n

l

)(−1)l+1

l

k∑i=1

[S (τi−1)]l (θi − θi−1)

=n∑

l=1

(n

l

)(−1)l+1

l

k∑i=1

{[S (τi−1)]l − [S (τk)]l} (θi − θi−1)

=n∑

l=1

(n

l

)(−1)l+1

l

k∑i=1

{[S (τi−1)]l − [S (τi )]l}θi

=k∑

i=1

θi

n∑l=1

(n

l

)(−1)l+1

l[S (τi−1)]l

× (1 − [Si (�i)]l

)(∵ S (τi−1) Si (�i) = S (τi ))

=k∑

i=1

θi

n∑l=1

(n

l

)(−1)l+1

l[S (τi−1)]lFi (l�i)

(∵ [Si (t)]l = Si (lt)

)

=k∑

i=1

θi

n∑l=1

(n

l

)(−1)l+1

l

⎡⎣i−1∏

j=1

Sj

(�j

)⎤⎦l

Fi (l�i)

which completes the proof. �

ACKNOWLEDGMENTS

The authors are grateful to the associate editor and theanonymous reviewers for their critical and valuable com-ments on the previous version of the manuscript, which ledto a considerable improvement of the paper. The secondauthor was supported by a grant from the Simons Foundation(#280601 to Tony Ng).

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comparison between constant-stress and step-stress accelerated life tests under time constraint

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