Design of Accelerated Life Test Plans—Overview and Prospect
Chen et al. Chin. J. Mech. Eng.
Design of Accelerated Life Test Plans- Overview and Prospect
Wen‑Hua Chen 0
Jun Pan 0
Ping Qian 0
Qing‑Chuan He 0
0 Key Laboratory of Reliability Technology for Mechanical and Electrical Product of Zhejiang Province, Zhejiang Sci‐ Tech University , Hangzhou 310018 , China
Accelerated life test (ALT) is currently the main method of assessing product reliability rapidly, and the design of efficient test plans is a critical step to ensure that ALTs can assess the product reliability accurately, quickly, and economically. With the promotion of the national strategy of civil‑ military integration, ALT will be widely used in the research and development (R&D) of various types of products, and the ALT plan design theory will face further challenges. To aid engineers in selecting appropriate theories and to stimulate researchers to develop the theories required in engineering, with focus on the demands for theory research that arise from the implementation of ALT, this paper reviews and summarizes the development of ALT plan design theory. The development of the theory and method for planning optimal ALT for location‑ scale distribution, which is the most applied and mature theory of designing the optimal ALT plan, are described in detail. Taking this as the center of radiation, some problems that ALT now faces, such as the verification of the statistical model, limitation of sample size, solutions of resource limits, optimization of the test arrangement, and management of product complexity, are discussed, and the general ideas and methods of solving these problems are analyzed. Suggestions for selecting appropriate ALT plan design theories are proposed, and the urgent solved theory problems and opinions of their solutions are proposed. Based on the principle of convenience for engineers to select appropriate methods according to the problems found in practice, this paper reviews the development of optimal ALT plan design theory by taking the engineering problems arising from the ALT implementation as the main thread, provides guidelines on selecting appropriate theories for engineers, and proposes opinions about the urgent solved theory problems for researchers.
Accelerated life test; Test plan; Optimal design; Reliability assessment
(1) Assess the reliability indices of product. For
example, assess the reliability level, reliable life, mean
time between failures (MTBF), and failure rate.
(2) Improve and perfect products. For example,
eliminate product defects and screen unqualified
The former is mainly concerned with how to test or
estimate the reliability indices of products accurately,
quickly, and economically. Determining a suitable
statistical theory, method, and technology for the experiment
design and data analysis is key for conducting this type
of test; this is often called statistics-based reliability
testing (SRT). The latter mainly focuses on how the processes
of design, material selection, manufacture, assembly, and
application affect the storage, performance, and
maintenance of the product. The key factor of achieving goals
is the profound understanding of the performance
evolution law of a particular product throughout its life cycle.
This type of test has a higher requirement for
engineering experience, and it is often called the
engineeringbased reliability test (ERT). This classification is only to
emphasize the different focus of the two types of tests. In
practice, to carry out an SRT correctly, the engineering
elements, such as usage conditions, failure mode,
failure mechanism, test equipment, and cost limits, should
be specified; in an ERT, a large amount of data should be
collected and analyzed based on statistics.
In practice, ERT and SRT are often used in
combination, and play different roles in various stages throughout
the life cycle of the product. According to the testing
purposes, the reliability test can be further classified into the
reliability growth test (RGT), reliability qualification test
(RQT), reliability screening test (RST), reliability
acceptance test (RAT), and reliability determination test (RDT).
According to the relationship of the test stress and the
normal work stress, the reliability test can be divided into
traditional test and accelerated test (AT). The types of the
major reliability tests are shown in Figure 1.
In general, as shown in Figure 1, corresponding to the
design, finalization of design, production, delivery, and
use phases of the entire product life cycle, the major
reliability tests implemented are RGT, RQT, RST, RAT, and
RDT, respectively. Among them, RGT and RST belong
to ERT; RQT, RAT, and RDT belong to SRT. The main
statistical inference method used in RQT and RAT is
the hypothesis test, so they are often collectively called
the reliability verification test (RVT); the main statistical
inference method used in RDT is parameter estimation.
In the initial period of reliability formation, products
usually have low reliability and short lifespan, and
reliability tests can be carried out by simulating the actual
usage conditions. However, with the improvement in
product reliability, it gradually becomes difficult to
induce product failure effectively using this type of
reliability test, and it cannot be used to obtain adequate
failure data within an acceptable test time and sample size.
To solve this problem, the AT method was developed: the
sample was tested within an environment more severe
than it would have experienced during normal use. Data
was collected at high stress levels and was used to predict
the product life at the normal stress level and to improve
the product reliability. Among the ATs, ERT mainly
include the accelerated RGT , highly accelerated life
test (HALT) , and highly accelerated stress
screening test (HASS)  (the latter two are often referred to
as the reliability enhancement test, RET [
]); SRT mainly
includes the accelerated life test (ALT) [
] and the
accelerated degradation test (ADT) [6–8].
ALT is currently the most widely used method of
assessing product reliability rapidly in practice. To ensure
that the reliability of products can be assessed accurately,
quickly, and economically, the design of an efficient plan
is critical before ALT is conducted, and this requires the
support of relevant statistical theories. With the
promotion of the national strategy of civil-military integration,
ALT will be widely used in the research and
development (R&D) of various types of products, and the ALT
plan design theory will also face more challenges. To aid
engineers in selecting appropriate theories and to
stimulate researchers to develop the theories required in
engineering, following the principle that make engineers be
convenient for selecting appropriate theories and
methods in accordance with the problems found in practice,
this paper provides a review of the development of ALT
plan design theory. Section 2 gives a brief description of
the types, statistical essentials and development
overview of ALT. Section 3 describes the development of the
design theories and methods for planning optimal ALT
with location-scale distribution. From the viewpoint of
engineering application, except for some most simple
product which life follows exponential distribution, one
always gives priority to the location-scale distribution to
describe the product life distribution, and prepares the
test as far as possible to reach the requirements of the
test form and sample size according to statistical
theories. Then, one has a relatively mature and programmed
method for designing test plan and analyzing data.
However, there are many problems encountered in practice
that cannot be solved by using the method provided in
Section 3. Section 4 discusses the current views and
possible methods of addressing these problems, and puts
forward some opinions on their development trend.
Section 5 proposes suggestions of selecting appropriate
ALT plan design theories, and gives viewpoints about
the urgent solved theory problems and opinions of their
2 Overview of ALT
2.1 Types of ALT
In an ALT, with the premise that the failure modes and
failure mechanisms of product are the same as those
under normal stress, the samples are tested at stress
levels higher than normal. Then, the product life at the
normal stress level can be estimated by extrapolating the life
information of samples at high stress levels to the normal
level based on the stress-life relationship. The types of
ALT can be classified according to four characteristics
(1) The mode of stress loading, which includes the
main stresses of constant stress, step stress, and
progressive stress. The three kinds of loading mode
correspond to the constant stress ALT (CSALT),
step stress ALT (SSALT) and progressive stress ALT
(2) The criteria for stopping the test, which includes
the main criteria of time-censored (type-I
censoring) and failure-censored (type-II censoring).
(3) The strategies of performance inspection for the
test unit, which include continuous inspection and
periodic inspection. These two strategies generate
the life data and group data, respectively.
(4) The number of accelerated stresses, which includes
single stress, double stresses and multiple stresses
(the number of test stresses greater than or equal to
2.2 Statistical Essentials of ALT
The statistical model, statistical analysis method, and test
plan design method are the three key elements of ALT.
They are closely related to the type of ALT, primarily as
(1) For CSALT, the statistical model includes the life
distribution and the stress-life relationship;
however, for SSALT and PSALT, the statistical model
also includes the equivalent principle of stress level
(2) The different types of stress loading, test stopping
criteria and performance inspection methods
correspond to different data types and data analysis
(3) The stress-life relationships of single stress, double
stress and multiple stress ALT are single, binary,
and multivariate functions, respectively. The ALT
plan design method is dependent on the stress
2.3 Current Situation of ALT
In terms of test purpose, most current ALTs carried out
belong to RDT. In terms of test mode, the most widely
used test in practice is the CSALT with single stress and
time-censoring. With the increasing requirements of
product reliability, increasing complexity of operational
conditions, and improvement in the technical level of
both the instruments and equipment, the multiple stress
ALT (MSALT) and PSALT with time-censoring are used
in an increasing number of engineering applications.
Corresponding to the classification of statistical
thoughts in statistics, the statistical models and methods
of ALT can be classified into descriptive statistics,
parametric statistics, nonparametric statistics, and Bayesian
]. In general, the most widely used models
and methods almost all belong to the parametric
statistics. Among them, the most widely used and advanced
theory is the maximum likelihood estimation (MLE)
theory for the location-scale distribution and the
linear stress-life relationship. With the wide application of
ALTs to an increasing number of engineering objects,
problems such as difficultly in determining the type of
product life distribution and the lack of test data become
sharp, and non-parametric and Bayesian methods are
The problem of planning ALTs is the opposite to that
of data analysis. Finding the optimal test plan is a
common pursuit of engineers and statisticians, and is the
most attractive research subject in the application
theories of ALT. In principle, a stress loading mode, statistical
model, data analysis method, testing condition limitation,
and design objective should correspond to a respective
problem of designing an optimal ALT plan. Therefore,
the test method, statistical models, and data analysis
methods of ALT correspond to the following studies of
planning optimal ALT covers:
(1) Various test forms, such as CSALT, SSALT, PSALT,
type-I censoring ALT, type-II censoring ALT, single
stress ALT, double stress ALT, and multiple stress
(2) Various statistical models, such as descriptive,
parametric, nonparametric, and Bayesian statistics
(3) Various optimization objectives and constraints
such as V-optimization (to obtain the optimal plan
by minimizing the asymptotic variance of the MLE
of the pth quantile of the product life distribution
under normal stress), D-optimization (to obtain the
optimal plan by maximizing the determinant of the
Fisher information matrix of MLE), single objective,
multiple objectives, cost limits, and resource limits.
Among the studies mentioned above, the most widely
used and researched ALT plan is the V-optimal
continuous-inspection type-I censoring CSALT plan for
statistical models with location-scale distribution and linear
3 Design Theory and Method of ALT
with Location‑Scale Distribution
3.1 Development Thread
The study of the statistical theory of ALT began with
exponential distribution [
]. From an engineering
point of view, ALT was applied initially for electronic
products, and the exponential distribution was used
widely as “standard distribution” in the reliability
analysis of electronic products; from a statistical point of view,
the exponential distribution led to numerous simple and
beautiful analytic conclusions, beloved by theory
statisticians. However, the research focus began to change as
more engineers and applied statisticians found that it was
more appropriate to describe the product life distribution
of most products as a function belonging to the
locationscale distribution, such as normal distribution, Weibull
distribution and log-normal distribution, than
Nelson et al. [
7, 8, 14–19
] researched the data analysis
method and the optimal plan design method for CSALT
with single stress and type-I censoring for lifetimes
following the Weibull distribution (or extreme value
distribution) and log-normal distribution (or normal
distribution). This was a milestone in the development of
the statistical theory and methods of ALT. Thus far, the
modes of ALT have been extended, the statistical models
generalized, and the statistical methods improved;
however, the basic ideas and methodological frameworks of
Nelson et al. have not yet been exceeded, and the results
of their researches are widely used in engineering, and
as the important basis and reference for promoting the
development of research and comprising the efficiency of
the optimal ALT design method. Their studies are briefly
described in Section 3.2.
Following Nelson et al. from the aspect of data analysis
methods, the dominance of MLE is difficult to shake.
Further development of the ALT theory is focused mainly on
the optimal design of the test plan, which can be divided
into two directions overall:
(1) Following the design ideas and methodology of
Nelson et al., expand the optimization model, stress
loading modes, and statistical models; and develop
the studies of other optimization objectives,
constraints, test modes, and statistics models, such as
type-II censoring, group data, multiple stresses,
SSALT, PSALT, competition failure, and nonlinear
(2) Attempt to use design ideas and methodology that
are different from those of Nelson et al. to address
the problems of unknown model parameters,
model deviations and limited sample size during
the optimal design, and to explore better methods
Table 1 summarizes the evolution of the design of the
optimal CSALT plan in the first direction. In Table 1,
there are some studies beyond the category of the
location-scale distribution and parametric statistics, or those
involved in the development of the second direction;
however, to facilitate the narrative, they are still listed in
the table. This paper focuses mainly on the study
development of the first direction in the MSALT (Section 3.3)
and SSALT (Section 3.4), and the study development of
the second direction (Section 3.5).
3.2 Classical Model and Method for Planning Single Stress CSALTs
3.2.1 Statistical Model
(1) The logarithm life of the product follows an
extreme value distribution or normal distribution
(that is the product life follows the Weibull
distribution or lognormal distribution, respectively). The
cumulative distribution function (CDF) is F(y; μ,
σ) = Φ[(y − μ)/σ], where μ is the location
parameter, σ is the scale parameter, and Φ(•) is the standard
extreme value or the standard normal distribution
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(2) The location parameter μ is a linear function of the
standardized stress ξ (0 ≤ ξ ≤ 1), that is the
stresslife relationship is μ(ξ) = γ0 + γ1ξ, where γ1 < 0. The
Arrhenius model and the inverse power law model,
which are the most widely used in engineering,
could both be transformed into linear stress-life
7, 8, 14–19
(3) The scale parameter σ is constant and independent
of ξ [
(4) For each test unit, the failure time is statistically
(5) The type-I censoring CSALT is considered, and the
censoring time at each stress level is τ [
3.2.2 Statistical Method
The estimation method is MLE. Assume that there are k
stress levels, then the sample size on the ith level ξi (i = 1,
2,…, K) is Ni, and the lifetime of the jth (j = 1, 2,…, Ni)
samples on ξi is (tij, δij) (if the sample fails, then δij = 1; if
the sample is censored, then δij = 0 and tij = τ). The log
likelihood function of the MLE is
ln L =
δij ln f (tij; μ(ξi), σ ) + (1 − δij)
× ln(1 − F (τ ; μ(ξi), σ ))],
where f(tij; μ(ξi), σ) is the probability density function
(pdf ) of the extreme value or normal distribution [
3.2.3 Method of Designing Optimal ALT Plans
Theoretically, the problem of designing optimal ALT
plans could be expressed as [
7, 8, 14, 16
]: given the prior
estimate values γ0,e, γ1,e, and σe of the model parameters
γ0, γ1, and σ, respectively, and given the censoring time
τ and the failure probability p at the normal stress level,
find the number of stress levels K*, the stress level ξi*, and
the sample location ratio pi* that minimize the asymptotic
variance of the MLE for the pth quantile yp of product life
distribution at the normal stress level.
Nelson and Meeker [
] studied the solutions of
this problem (called the statistically optimal plan), and
drew the following key conclusions:
(1) If the censoring time is not too long, then K* is
always equal to two, and the optimal maximum
stress level ξH∗ is always one;
(2) The optimal minimum stress level ξL∗ and the
sample location ratio pL∗ of ξL∗ are functions of
ae = (lnτ − γ0,e)/σ, be = − γ1,e/σ and p, and
generally pL∗ is greater than 0.5. This means that
allocating more samples on the lowest stress level helps to
improve the accuracy of the estimation;
(3) For a given value of ae, the greater the value of be,
the smaller the value of the optimal variance factor
V ∗ . This means that for the same censoring time,
the greater the acceleration factor, the higher the
maximum stress level, and thus the higher the
(4) For the same value of be, the greater the value of
ae, the smaller the V ∗
K value, meaning that with the
same acceleration factor and highest stress level,
the longer the censoring time and the shorter the
product life at the normal stress level, the higher the
(5) In particular, if the censoring time is too long, the
ALT will degenerate into the censored test at the
normal stress level, and then K* = 1, ξL∗ = 0, pL = 1,
and VK∗ approaches a constant.
However, Nelson and Meeker considered that the
statistically optimal plan might not have good performance
in application because [
7, 8, 14–19
(1) The optimal plan depends on the values of
unknown model parameters γ0, γ1, and σ.
During the optimization, by substituting γ0, γ1, and σ
for their prior estimates γ0,e, γ1,e, and σe, the errors
in the prior values may cause the efficiency of the
optimal statistical plan to differ substantially from
that of actual optimal plan; this may be worse than
that of some traditional empirical plans;
(2) If the statistically optimal plan is adopted, the
errors in the prior estimates may lead to test failure
because the actual test may lack sufficient failure
samples at the lowest stress level;
(3) If the true distribution of the product life is not a
Weibull distribution or a lognormal distribution,
the efficiency of the optimal statistical plan may
(4) The statistically optimal plan only has two stress
levels, and cannot test the correctness of the
(5) If the true stress-life relationship deviates from a
linear relationship, the performance of the
statistically optimal plan may likely reduce greatly;
(6) If the sample size cannot meet the requirements for
using an asymptotic theory, the plan performance
may not be guaranteed.
To solve these problems, Nelson [
] suggested the
use of a “compromise plan” with three or four stress
levels, to enable the middle stress level to test the stress-life
relationship, prevent test failures, and improve the plan
robustness to the deviations in the statistical model and
model parameters. Furthermore, Meeker  proposed
several compromise plans, and the criterion that
evaluates the ALT plan robustness to deviations in the model
parameters and product life distribution. Through
computer experiment, over a wide range of values of γ0,e,
γ1,e, and σe, and considering the deviations in the model
parameters and life distribution, Meeker studied the
actual efficiency of statistically optimal plans and
compromise plans. He concluded that the best plan of
considering the estimation accuracy and robustness is the
optimal compromise plan with three equally spaced test
stresses, and the sample location ratio of the middle
stress level is 10% or 20% (the ratio at minimum stress
level should be determined via optimization; this plan
was furtherly simplified by Meeker and Hahn [
become the “4:2:1 plan”, which has three equally spaced
levels and with a sample allocation ratio of 4:2:1 at the
lowest, middle, and highest stress levels, respectively).
The optimal compromise plan proposed by Meeker was
widely used in engineering and became the “benchmark”
for most of the subsequent improved plans, and the
method of comparing and selecting the ALT plan with
a combined consideration of robustness and estimation
accuracy became the basis method followed by almost all
studies of designing optimal ALT plans.
3.3 Design of Multiple CSALT Plans
The multiple constant stress ALT (MCSALT) loads two
or more accelerated stresses on the product
simultaneously. Compared with the single stress ALT, MCSALTs
are closer to the real usage conditions for most
products and can make products fail faster. With the rapid
development of environment simulation technology,
the multiple-stress-test-equipment, which can load two
or more environment stresses (such as temperature &
humidity, temperature & vibration, thermal & vacuum,
and temperature & humidity & vibration) on products
has been gradually becoming available in the market, and
MCSALTs have gradually started to have wide
applications in engineering. However, theoretically, when the
number of accelerated stresses is greater than one, the
stress-life relationship changes into a binary or
multivariate function that leads to problems that are different from
those of planning single stress ALTs.
Escobar and Meeker [
] carried out the earliest study
on the theory and method of planning the optimal
MCSALT for location-scale distribution. They used the
assumptions mentioned in Section 3.2.1, and generalized
the stress-life relationship into a binary linear function
μ(ξ1, ξ2) = γ0 + γ1ξ1 + γ2ξ2 (where γi < 0, 0 ≤ ξi ≤ 1, and
i = 1, 2), and proved the following important conclusions
(1) The V-optimal MCSALT plan is not unique.
(2) There is a type of V-optimal plan with stress level
combinations ξ*i = (ξi*1, ξi*2) (i = 1, 2, …, K*)
distributed on a straight line connecting the normal stress
level (0, 0) and the highest stress level (1, 1). Such
plans cannot determine all parameters of the
stresslife relationship, and are defined as the optimal
(3) Each optimal degenerated plan corresponds to an
infinite number of optimal non-degenerated plans
(all parameters of the stress-life relationship can be
determined). The stress level combinations of
optimal non-degenerated plans distribute on the
stresslife relationship contour through the point ξi* = (ξi*1,
ξi2), and can be related to the stress level
combinations of the optimal degenerated plan by some
Because the optimal plan is not unique, to obtain a
determined plan, one should restrict the arrangement
mode of the stress level combinations (called test points)
in the feasible region of the test (called test region), and
restrict the sample location ratio on test points. Escobar
and Meeker [
] proposed a method of obtaining the
optimal non-degenerated plan (called splitting plan): find
the test point ξi* and the sample location ratios pi* thereof
for the optimal degenerated plan by solving the
optimization problem of single stress ALT; then, find the two
intersection points ξi*,1 and ξi*,2 of the stress-life
relationship contour through the point ξi* and the boundary of
the test region; and make the sample location ratio of ξi*,1
and ξi*,2 inversely proportional to their distance to ξi*.
The splitting plan is the V-optimal plan, and is also the
D-optimal plan among all V-optimal plans [
However, in consideration of the uncertainty of the actual
efficiency of the theoretical optimal plan, which is due to the
errors of prior estimates on the model parameters, and
the need to examine the model and analyze the effect of
accelerated stress through ALT, some researchers
proposed other arrangement modes of test points. For
example, Park, et al. [
], Yang [
] and Guo et al. 
arranged the test points via orthogonal designs (called
an orthogonal plan). Chen et al. [
] arranged the test
points based on uniform designs (called a uniform plan).
Over a wide value range of model parameters, Gao et al.
 compared these plans through computer experiment
from three aspects, namely the estimation accuracy of
the pth quantile, robustness to the deviation of the model
parameters, and the estimation accuracy of the model
parameters; they concluded that the splitting plan was
the best in terms of comprehensive performance.
Another problem in designing the optimal MCSALT
plan is that the stresses at the highest level may not be
loaded on the product simultaneously, thus resulting in
a non-rectangular test region [
]. To solve this
problem, Chen et al. [
] demonstrated that the
conclusions drawn by Escobar and Meeker remained valid for
simply-connected test regions with convex boundaries.
In this situation, the test points of the optimal
degenerated plan were distributed along the line connecting the
normal stress level (0, 0) and the highest stress level (ξH1,
ξH2), where (ξH1, ξH2) was the point at which the value of
the stress-life relationship reached the minimum on the
boundary of the test region. Based on this, they
generalized the splitting plan to simply-connected test regions
with convex boundaries. Later, Gao et al. [
that if the test objective was only to estimate the pth
quantile of the product life distribution, it was not
necessary to estimate all the parameters in the stress-life
relationship. Therefore, the optimal degenerated plan
also has practical value, and it can be applicable for both
rectangular and non-rectangular test regions. The
comparison results from the computer experiment show that
for the double-stress test and on the different shapes of
the test regions, the optimal degenerated plan has
better actual efficiency on average than the corresponding
splitting plan, over a wide value range of model
parameters and in consideration of the effects of the model
parameter error. Furthermore, for a splitting plan for
the MCSALT with more than two stresses, the number
of test points and the difficulty in finding them increases
sharply with the increase in the number of stresses; in
addition, the sample allocations at the test points are
reduced accordingly, and this increases the risk of test
failure. However, the degenerated plans are almost
irrelevant to the dimension [
Finally, when the number of accelerated stresses is
greater than one, the interaction effect between the
stresses causes the stress-life relationship to be a
nonlinear function. In principle, with a little generalization,
the splitting, orthogonal, and uniform plans are all
applicable to the nonlinear stress-life relationship [
However, Gao et al.  found that one could achieve a
better plan by using a line segment (chord) to connect
the highest and lowest points of the curve or surface that
corresponds to the nonlinear stress-life relationship, and
by considering the chord as a new stress-life
relationship from which to design the test plan and extrapolate
the pth quantile. Accordingly, they proposed a “chord
method” for planning the V-optimal CSALT with time
censoring and continuous inspection. For the problem of
planning optimal MCSALT, whether the stress-life
relationship is univariate or multivariate, linear or nonlinear,
and whether the test region is rectangular or
non-rectangular, the method could transform it into the problem
of planning a single stress ALT with a linear stress-life
3.4 Design of SSALT Plans
The step stress test was originally used in ERT to detect
the working limits and defects of products. Nelson [
introduced the hypothesis of cumulative damage, which
states that the development of product damage under
the same type of stress and failure mechanism was only
related to the current state and current stress level, and
was independent of the history of stress loading. Based
on this hypothesis, Nelson established a rule of
equivalent conversion between the life distributions and test
times at different stress levels, and proposed the theory
and method of applying the step stress test to the SRT. In
SSALT, the main method of estimating the model
parameters is still the MLE [
]. Bai [
] and Khamis
] gradually established the theory and method of
planning the V-optimal SSALT with type-I censoring.
Although it is theoretically possible to find the optimal
stress levels and stress switching times on each step of a
SSALT with finite multiple steps, the “simple SSALT” that
has only two steps was generally used in practice [
However, if one needs to check whether the stress-life
relationship is linear or not, he should use a three-step
To assess the product reliability only, it is not
necessary to apply incremental stress. The loading order of
stress levels can also be optimized to improve the
estimation accuracy and reduce the test cost. For the
exponential distribution, Miller and Nelson [
] referred to a
test with step-down stress (aptly named the step-down
test; similarly, the test with gradually increasing stress is
called the step-up test, and both the step-up and
stepdown tests belong to the SSALT), and proved that the
statistically optimal CSALT plan, step-down test plan
and step-up test plan all have equivalent variance factors.
Afterwards, for the Weibull distribution, Khamis [
compared the variance factors of the optimal CSALT and
step-up test plan under the fixed scale parameters and
lowest stress level. The results showed that the step-up
test was superior. Zhang [
] first studied the step-down
test for the Weibull distribution, and pointed out that the
effects of the step-down and step-up tests were different
owing to the influence of the scale parameters, and the
step-down test was superior most of the time in that it
need a smaller sample size and shorter test time to reach
the same estimation accuracy as the step-up test.
Furthermore, Wang et al. [
] demonstrated that if the stress
levels and stress switching times were all the same for
both the step-down and step-up tests, the step-down test
was better than the latter in terms of the estimation
accuracy, failure sample size, etc. For the Weibull distribution
and lognormal distribution, in the estimation accuracy of
pth quantile and the robustness to the deviation of model
parameters, Ma and Meeker [
] made comprehensive
comparisons on the optimal CSALT, optimal simple
stepup and step-down test. They drew the following major
(1) The relationship between the estimation accuracy
and robustness of the three plans varied with the
scale parameters, but without a simple rule that is
applicable to all values of the model parameters.
(2) If ranked by the estimation accuracy of the pth
quantile from high to low, when the scale
parameter is less than one, the order is the CSALT, step-up
test and step-down test; when the scale parameter
is greater than one, the order is the step-down test,
step-up test, and CSALT.
(3) If ranked by the robustness from high to low, the
order followed is the step-down test, CSALT and
step-up test when the scale parameter is less than
one, and is the step-down test, step-up test and
CSALT when the scale parameter is greater than
Although the studies mentioned above are still not
sufficient to determine the best ALT mode among the
CSALT, step-up test, and step-down test, they prove the
following at least: in some cases, in addition to inciting
product failure faster than CSALT, the optimal SSALT
has a higher estimation accuracy and robustness than
the optimal CSALT. This is sufficient to make SSALT a
strong competitor to CSALT. Furthermore, one of the
major expenses of ALTs in practice is the site cost
(calculated according to the number of test equipment and
time occupied). The SSALT could reduce test costs: when
the total sample size and censoring time are the same, if
only one test device is available, then the test time for a
SSALT is τ, and 2τ to 4τ for a CSALT; If there is no limit
on the number of test devices used simultaneously, then
a SSALT only needs one device, but a CSALT needs two
to four devices.
3.5 Solutions to Robustness and Limited Sample Size
In the 8th to 14th paragraphs of Section 3.2.3, six doubts
about the actual effect of the statistically optimal plan are
mentioned, and they can be summarized into the
following three aspects of the problem:
(1) Unknown parameters. The 1st and 2nd doubts
arose from the query of the actual effect of the
statistically optimal plan, considering the errors of the
prior estimate value of model parameters;
(2) Model deviations. The 3rd to 5th questions are
arose from the concern regarding the inferred
errors caused by the assumptions of the statistical
model not in line with the actual conditions.
(3) Limited sample size. The asymptotic variance was
used as the objective function in designing the
optimal plan. If the actual sample size cannot meet the
requirements of a large sample size, the efficiency of
the optimal plan based on the asymptotic variance
is in doubt, which is pointed out in the 6th doubt.
Among these three problems, model deviations and
limited sample sizes are very common in many
statistical methods, and are not unique to the statistics of the
ALT. However, the problem of unknown parameters
comes from the “censoring,” which leads to the
correlation between the optimization objective function and the
unknown model parameters; this is unique to the optimal
ALT plan design, and increases the difficulty in
processing the problems of model deviations and limited sample
size. In practice, only after these problems are explained
rationally or solved, do the engineers use the optimal
Nelson and Meeker used the “compromise plan” to
solve these problems; it is a simple and effective solution.
However, there is no precise theory to support whether it
was the best way for solving these problems, and
therefore some researchers are still trying to find the optimal
solution in theory.
Regarding the problem of unknown parameters,
Chaloner et al. [
] and Ginebra et al.  proposed
methods of designing the optimal ALT plan, which use
prior distributions and intervals, respectively, to describe
the unknown model parameters. Their optimal
objectives are to minimize the mathematical expectation of
the asymptotic variance of the MLE of the pth quantile
over the prior distributions, and minimize the maximum
value of the asymptotic variance of the MLE of the pth
quantile over the parameter interval, respectively. The
two objectives include the estimation of the parameter
range, and the obtained optimal plan was the plan taking
into account the parameter estimation errors. In
addition, Tang and Liu [
40, 41, 59
] attempted to consider and
control the process of giving the prior estimates in the
framework of sequential tests during the plan design. It
was perhaps a more complete approach.
For the problem of model deviations, Chaloner et al.
] and Pascual [
] established objective
functions containing the effects of model deviation; therefore,
they could obtain the optimal plan directly by solving the
optimization problems with regard to the model
deviations. Specifically, there are two situations:
(1) There are several types of life distributions and
stress-life relationships for selection, but the model
cannot be determined yet. Therefore, it was
necessary to make a test plan with better performance in
all candidate models. Chaloner  provided the
corresponding optimization model based on the
Bayesian method. The objective of the
optimization is to minimize the mathematical expectation
of asymptotic variances of the p-th quantile
corresponding to each possible model; Pascual [
] replaced the asymptotic variance in Chaloner’s
objective function with an index called
asymptotic sample ratio (ASR). The ASR in each possible
model is regarded as a component of a vector, and
the objective functions are defined by the norms in
different forms. In particular, Pascual studied the
optimal model under the ∞-norm.
(2) The statistical model was determined during the
plan design, but it is expected that the test plan
could achieve the best possible estimation
accuracy even if the wrong model is selected. For
situations where the wrong form of life distribution and
stress-life relationship might be chosen, Pascual
] proposed optimization models that take the
asymptotic bias (ABias) and the asymptotic mean
squared error (AMSE) as the objective functions
20, 21, 62–64
] demonstrate that the ALT plan
obtained by the above method is more robust in terms of
the model deviation than the compromise plan.
For the problem of limited sample size, Escobar and
] used the Monte Carlo (MC) method to
simulate the implementation of the optimal plan based on
the asymptotic variance, calculate the sample variance of
the pth quantile, and investigate the applicability of the
theoretical optimal plan by the approximation degree of
the sample variance and asymptotic variance. To
investigate the approximation degree of the optimal
solutions, over the whole feasible area of a one-dimensional
optimization problem, Pascual [
] compared the
objective function based on the asymptotic variance with the
objective function that corresponds to a limited sample
size and is calculated by the MC method. The
calculation results show that the two objective functions are
close to each other in pattern, and without great
difference in terms of optimal solutions. Ma and Meeker [
studied how the sample size and model parameter errors
effect the test success rate and estimation accuracy, and
introduced a constraint into the optimization model to
assure the success rate of ALT with a limited sample size.
By combining the optimization based on the asymptotic
variance with the graphical method and the stimulation
evaluation based on the MC method, they put forward a
method of designing the optimal compromise plan with
the comprehensive consideration of the effects of
limited sample size and unknown parameters. In addition,
] suggested that to design the optimal ALT
plan based on the objective function obtained by the MC
method, rather than the objective function based on the
asymptotic variance; Wang [
] conducted a systematic
research on this topic, and this type of method is called
the “simulation based optimization”, from which the
optimal plan corresponding to the limited sample size can be
Overall, under some conditions, these studies indeed
produce plans better than the compromise plan.
However, these methods have not yet been widely used in
engineering, because they are sometimes complicated
for engineers, and there is still no sufficient evidence to
support their superiority to the compromise plans in all
aspects of practice.
4 Additional Problems of ALT
4.1 Statistical Model Test
Whether the statistical model is suitable for
engineering practices largely determines the actual effectiveness
of the statistical inference. Therefore, it is necessary to
test the statistical model before it is used in reliability
assessment. The current theories provide methods for
testing various common life distributions [
]. For the
ALT, the verification on the stress-life relationship and
the assumptions of cumulative damage (for SSALT and
PSALT) is more applicable in engineering. In this respect,
although the traditional theories on regression diagnosis
are rich [
7, 8, 85, 86
], few of them are applicable to the
censored data and non-normal distribution. In addition
to the graphics method in Refs. [7, 8], to meet the need of
the ALT for electrical connectors, based on the principle
of failure physics, hypothesis testing, and regression
theory, Qian et al. [
] and Liu  proposed methods
of testing the multivariate stress-life relationship and the
assumptions of cumulative damage of a time-censored
ALT with Weibull distribution, respectively. However, its
effectiveness lacks support from precise statistical theory.
Recently, Pan et al. [
] proposed a method of designing
an optimal ALT plan for the verification of the
stresslife relationship in the framework of a generalized linear
In general, testing a model needs more sample sizes
and stress levels than estimating model parameters. For
the ALT, it is difficult to validate the extrapolation effect
of the model directly because of the long product life and
high reliability. Perhaps, new ideas are required to deal
with the uncertainty of the acceleration model. Besides,
for some products with complicated structures, their life
distribution and stress-life relationship may be difficult to
describe in a simple form.
If the model cannot be determined, there are two main
(1) Find a robust plan with good performance in a
variety of alternative models [
20, 21, 62–64
studies were introduced in Section 3.5.
(2) Use the nonparametric methods to reduce the
reliance of statistical inference on the specific form
of the models. Refs. [
] provide a
comprehensive description of the nonparametric
models and methods commonly used in the life tests,
such as the proportional risks model; Refs. [
are researches on planning the optimal ALT with
nonparametric models. To some extent, the
nonparametric method can avoid the problem of the
uncertainty of the model, as well as deal with the
life distributions and stress-life relationship that are
difficult to express with analytic functions;
however, they generally require a large sample size [
which is another main problem faced by ALT.
4.2 Restrictions on Sample Size
To enable the MLE to achieve an acceptable accuracy,
the number of failure samples in a test is at least 8 to 10,
and typically 30 to 40, or even hundreds. However, the
ALT for many products cannot meet this requirement.
On one hand, with the improvement in product
reliability and lifespan, in some products with long life and
high reliability, it is difficult to induce failure within the
acceptable test time and sample size (for example, some
military components require the 99% reliable life to be up
to 24 years). On the other hand, for some very expensive
and low-yield parts, equipment, devices, and machines,
the number of samples used for the reliability test may be
only four to five, one to two, or even zero, and it is
difficult to carry out the statistical inference generally.
Solving the two aspects of the problem may be beyond the
scope of the general ALT and parameter statistics. The
statistics for the minimal sample size, Bayesian method,
and ADT can reduce the requirements of sample size to a
certain extent; thus, this is currently receiving increasing
4.3 Resource Limits
In practice, the limits on the test conditions, sites, and
cost are considered for the implementation of ALTs.
The limits on test conditions mainly refers to the
situation in which the performance parameters of the product
could only be periodically inspected. Periodic
inspection generates group data. In theory, there have special
methods of data analysis and optimal plan design for
group data [
5–11, 34–39, 88
]. However, from the
application point of view, the method of the data analysis and
plan design for life data are relatively simple. Therefore,
in practice, the engineers tend to use the method for
life data to deal with the group data, by transferring the
group data into life data by interpolation methods [
]. However, a systematic study on the accuracy of this
method has not yet been reported.
The limits on test sites mainly refer to the restriction
in the number of equipment that can be used
simultaneously within a certain time. The limits on test costs are
mainly due to the sample size, test methods, test time,
equipment number and test stress levels. Many
studies introduced some constraint on test sites and costs
in the optimization model [
65, 66, 68
]; however, we are
not aware of a general model or a systematic study on
the characteristics of the solutions of such optimization
4.4 Optimal Mode of Stress Loading
In addition to CSALT, SSALT and PSALT, the researchers
proposed other test modes, such as the group ramp test
], trapezoid test [
], and ramp-constant
stress mixing test (RCSMT) [
]. At present, there are
still doubts regarding the implementation methods and
effects of the SSALT and PSALT, and other methods of
stress loading are limited. However, as the performances
of SSALT and RCSMT are sometimes superior to that
of CSALT [
80, 82, 83, 108
], at least in theory, it is
possible that the modes of stress loading can still be
optimized, but no theoretical research on this topic has been
4.5 Optimal Arrangement of Reliability Test
Besides estimating the pth quantile, it is often expected in
engineering to estimate the model parameters and
compare or verify the reliability indexes of the product by the
ALT (see Table 1). From a greater perspective, in
product reliability engineering, many kinds of ATs are carried
out with different objectives and in different stages. The
question is how to arrange these tests most efficiently. No
quantitative studies on this topic have been reported yet.
4.6 Product Complexity
Thus far, in general, successful ALTs are mainly aimed
at the materials, components, and parts under
simple stresses (such as constant temperature, voltage, and
vibration), with simple failure mechanisms (such as
single failure due to oxidation, electrical aging, and wears)
and in the single-failure mode. However, these products
account for only a small percentage in practice. The
complexity in product structure, work stress, failure
mechanism, and failure mode increase the challenges of the
From the aspect of work stress, there are three main
problems. (1) The functions of materials and parts in
the components and systems are subject to the general
environment and other parts in the system. If the ALT
on the materials and parts lacks sufficient consideration
of these factors, the accuracy and credibility of the test
results will be affected. In general, the same material,
parts, or components are often used in different
environments, and it is impossible to give assessments for
all working conditions. The methods of applying the
results of ALT to wider environments and assessing the
application scope of ALT are the major problems to be
solved. Few quantitative research projects are currently
addressing this topic. (2) A large number of
electromechanical components are loaded with stresses
dependent on time, such as alternating current and force.
Only a few studies on the ALT under time-dependent
stress have been reported [
] (this does not refer to
the varying stresses applied in SSALT and PSALT, but
to the varying working stresses on the product). (3) A
large number of products undergo different stages in
their life cycles, including transportation, storage, and
application; however, most of the current ALTs are only
focused on one stage, and researches on multiple stages
are rarely reported.
The aspects of failure mechanism and failure mode
are mainly concerned with the theory and method of
ALT for products with competitive failures. Related
research on this aspect started very early [
general strategy is to transform the ALT of
competitive failure into ALTs of several single failures under the
assumption that each failure mode and mechanism are
not related to each other [
], but few cases meet
this assumption. From the aspect of engineering
application, it is not easy to determine the life distribution
and stress-life relationship of a specific product with
competitive failure, because once there is more than
one failure mode and failure cause, the data collection
and physical analysis of failure becomes more difficult.
For products with multiple failure mechanisms and
modes, or even with relevance to each other (usually
referred to some components or systems), it is
doubtful that their statistical models can be described with a
simple and analytic life distribution and stress-life
relationship. Thus, it is more difficult to develop the
corresponding method for the ALT.
The structure and components of products mainly
involves the ALT on system-level products (including
parts, components, machine, and equipment group). For
these products, various problems, such as minimal
sample size, competitive failure, complex stress and
component relevance, often arise simultaneously; and are
followed by technical problems. For example, the product
size and weight are beyond the range of the test
equipment. At present, except for some special systems, there
is a lack of effective ideas and methods for dealing with
ALTs of system-level products.
5 Conclusions and Outlook
(1) The ALT theory that deals with the location-scale
distribution and the linear stress-life relationship
is suggested as a first choice in practice, because it
is relatively mature and has numerous successful
applications in the fields of materials and
(2) Regarding the problems of selecting test modes
between CSALTs and SSALTs (Sections 3.3 & 3.4),
robust plan design (Section 3.5), limited sample size
(Sections 3.5 & 4.2), model verification (Section 4.1)
and resource limits (Section 4.3), current researches
can provide heuristic guidance and case reference
for engineering practice. However, the relevant
theory still needs further improvement. Among these
problems, the weakest research in present studies is
the model verification problem. There is an urgent
need to establish statistical theories and methods
that can deal with the test of stress-life relationship
with censored data and non-normal distribution,
research the relationship between the ALT used for
testing models and that used for estimating model
parameters, and study the method of designing
optimal plan for testing model.
(3) For ALTs with phased-missions, complex stresses,
and competitive failures or for system-level
products (Section 4.6), the statistical models, modeling
methods and test methods still fall short in terms
of sufficient cases and theoretical support. This
requires more collaboration among engineers and
statisticians to promote its development. These
problems could possibly be solved in three stages.
Firstly, the AT theories and methods that deal with
the phased-mission, time varying stress, and
competitive failures should be established gradually by
conducting research on the AT
estimation-technology of the operational reliability of the components
and material with single failure mode and failure
mechanism in practice. Then, the AT methods that
aim at estimating the reliability of the components
and material with competitive failures could be
further studied and determined. Finally, the AT
methods applied to a system level product can be
studied. These three stages have increasing difficulty,
and are extremely challenging in terms of
estimation theories, methods, and technologies. It is
possible that the framework of ALT alone cannot solve
these problems completely; ALTs need to be
combined with the ADT, dynamic model, and
simulation technology based failure physics, and so on.
WC put forward the framework and content of the paper, completed sec‑
tions 1, 2 and 5, and did the global modification and adjustment of the
manuscript; LG drafted the paper, and completed sections 3.1,3.2, 3.3 and
3.5; JP gave suggestions about the framework and content, and completed
sections 3.4, 4.5and 4.6; PQ completed sections 4.1, 4.2 and 4.3; QH completed
section 4.4, and checked the English draft. All authors read and approved the
1 Key Laboratory of Reliability Technology for Mechanical and Electrical
Product of Zhejiang Province, Zhejiang Sci‑ Tech University, Hangzhou 310018,
China. 2 College of Mechanical and Electrical Engineering, Sichuan Agricultural
University, Ya’an 625014, China.
Wen‑Hua Chen, born in 1963, is currently a professor and a PhD candidate
supervisor at Zhejiang University and Zhejiang Sci‑ Tech University, China; and
also the Director at Key Laboratory of Reliability Technology for Mechanical
and Electrical Product of Zhejiang Province, Zhejiang Sci‑ Tech University,
China. He received his PhD degree from Zhejiang University, China, in 1997.
He is mainly engaged in research on reliability design, testing, and statistical
Liang Gao, born in 1981, is currently an associate professor at College of
Mechanical and Electrical Engineering, Sichuan Agricultural University, China.
Jun Pan, born in 1974, is currently a professor at Key Laboratory of Reli‑
ability Technology for Mechanical and Electrical Product of Zhejiang Province,
Zhejiang Sci‑ Tech University, China.
Ping Qian, born in 1983, is currently an associate professor at Key Labora‑
tory of Reliability Technology for Mechanical and Electrical Product of Zhejiang
Province, Zhejiang Sci‑ Tech University, China.
Qing‑ Chuan He, born in 1984, is currently a lecturer at Key Laboratory
of Reliability Technology for Mechanical and Electrical Product of Zhejiang
Province, Zhejiang Sci‑ Tech University, China.
Supported by National Natural Science Foundation of China (Grant No.
51275480, 51305402, 51405447) and International Science & Technology
Cooperation Program of China (Grant No. 2015DFA71400).
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Springer Nature remains neutral with regard to jurisdictional claims in pub‑
lished maps and institutional affiliations.
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