3.1 Development Thread
The study of the statistical theory of ALT began with exponential distribution [
5‐
11]. 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 location-scale distribution, such as normal distribution, Weibull distribution and log-normal distribution, than exponential distribution.
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 stress-life relationship;
(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 and plans.
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).
Table 1
Classical optimal design model of CSALT and extended researches
Test purpose | To estimate the pth quantile y
p
(0) of lifetime under normal stress | To estimate multiple quantiles [ 20‐ 22]; for acceptance of product [ 23, 24]; to select the optimal product [ 25]; to estimate the quantile of the different products under normal stress [ 26]; to estimate the other parameters [ 27‐ 30] |
Test forms | Type of test censoring | Type-I censoring | Failure censored [ 23, 25, 31, 32]; time failure mixed censored [ 33] |
Data collection method | Continuous testing | |
Strategy for testing the samples | All samples are put in from the start. | Sequential test method [ 40‐ 43] |
Number of stress levels | Singe stress level | Double or multiple stress levels [ 44‐ 55] |
Statistical models | Failure mode | Single failure mode | Multiple failure modes [ 28, 29] |
Types of life distribution | Weibull/lognormal distribution | Two parameter exponential distribution [ 56]; Rayleigh distribution [ 36]; Burry type XII [ 38]; nonparametric method [ 27, 57] |
Parameters relating to stress | Location parameter of location- scale distribution | Location and scale parameter of location-scale distribution [ 24, 58] |
Form of stress-life relationship | Linear function | Polynomial [ 26] /nonparametric [ 27] |
Data form | Life data | |
Data analysis method | MLE | Unbiased estimation [ 56]; Bayesian [ 40, 59, 60] |
Optimal design models of test plan | Optimization objective | Minimize the asymptotic variance of estimated y
p
(0) | |
Design Variables | Stress level | Minimum stress | |
Allocation of sample size | Proportion in which samples are allocated to the lowest stress level. | Proportion in which samples are allocated to the intermediate stress level [ 27, 57, 65‐ 68, 72, 73] |
Allocation of test time | Not optimized | Optimize test time allocated to each stress level [ 65, 66, 68] |
Constraints | Total sample size | Large sample size | |
Total test time | Specify the censoring time allocated to each stress level | Specify the total test time [ 68] |
Space between stress levels | Unconstrained/Equal difference | Unequal difference [ 21, 27, 34, 36, 57, 63, 64, 66‐ 68, 70, 72, 73] |
Allocation of sample size | Unconstrained/equal allocation/Specified proportion | Minimize the expected number of failures [ 27, 39, 57, 66‐ 68, 74] |
Allocation of test time | Equal | |
Other | / | Acceleration factor [ 49]; probability of failure at each stress level [ 65, 67, 72, 73], etc. |
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 [
44] 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
γ
i
< 0, 0 ≤
ξ
i
≤ 1, and
i = 1, 2), and proved the following important conclusions [
44]:
(1)
The V-optimal MCSALT plan is not unique.
(2)
There is a type of V-optimal plan with stress level combinations ξ
i
*
= (ξ
i1
*
, ξ
i2
*
) (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 stress-life relationship, and are defined as the optimal degenerated plan.
(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 stress-life relationship contour through the point ξ
i
*
= (ξ
i1
*
, ξ
i2
*
), and can be related to the stress level combinations of the optimal degenerated plan by some equations.
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 [
44] proposed a method of obtaining the optimal non-degenerated plan (called splitting plan): find the test point
ξ
i
*
and the sample location ratios
p
i
*
thereof for the optimal degenerated plan by solving the optimization problem of single stress ALT; then, find the two intersection points
ξ
i,1
*
and
ξ
i,2
*
of the stress-life relationship contour through the point
ξ
i
*
and the boundary of the test region; and make the sample location ratio of
ξ
i,1
*
and
ξ
i,2
*
inversely proportional to their distance to
ξ
i
*
.
The splitting plan is the V-optimal plan, and is also the D-optimal plan among all V-optimal plans [
44]. 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. [
45], Yang [
46,
47] and Guo et al. [
48] arranged the test points via orthogonal designs (called an orthogonal plan). Chen et al. [
49,
50] arranged the test points based on uniform designs (called a uniform plan). Over a wide value range of model parameters, Gao et al. [
51] 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 [
52]. To solve this problem, Chen et al. [
52] 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. [
53] proposed 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 [
54].
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 [
44,
48]. However, Gao et al. [
55] 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 relationship.
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 [
7,
75] 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 [
7,
75]. Bai [
76,
77] and Khamis [
78,
79] 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 [
76]. However, if one needs to check whether the stress-life relationship is linear or not, he should use a three-step SSALT [
78].
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 [
80] 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 step-down 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 [
79] 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 [
81] 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. [
82] 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 [
83] made comprehensive comparisons on the optimal CSALT, optimal simple step-up and step-down test. They drew the following major conclusions [
83]:
(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 one.
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 plans.
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. [
20,
61] and Ginebra et al. [
70] 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. [
20] and Pascual [
21,
62‐
64] 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 [
20] 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 [
62,
63] 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 [
21,
64] proposed optimization models that take the asymptotic bias (ABias) and the asymptotic mean squared error (AMSE) as the objective functions respectively.
Refs. [
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 Meeker [
44] 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 [
64] 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 [
74] 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, Meeker [
8] 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 [
84] 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 obtained.
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.