Bibliografia de Testes Acelerados

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An Updated Bibliography of Accelerated Test Plans 
Wayne B. Nelson, Fellow, IEEE 
Key Words: Accelerated testing, test plans, overstress, life, degradation, extrapolation, regression models, minimum variance. 
1 I TRODUCTIO 
 Purpose. This paper describes the following bibliography 
of statistical plans for accelerated tests. This introduction 
provides background on accelerated testing and test plans. 
The body of this paper briefly surveys the literature on theory 
for accelerated test plans and the considerations that go into 
choosing or developing such plans. Many plans need to be 
developed, especially for degradation models. The author 
welcomes additions to and corrections of the bibliography, 
which is available on request as a searchable updated Word 
file, which includes email addresses of authors. 
 Overview. This introduction briefly describes accelerated 
testing, the history of test plans, accelerated test (AT) models, 
the simple model, AT plans, and considerations in developing 
such plans. 
 Accelerated testing. Accelerated testing consists of high 
stress testing that shortens product life or hastens degradation 
of product performance. Here "stress" is used in a general 
sense to mean any accelerating variable. The usual purpose of 
such overstress testing is to estimate the life or degradation of 
the product at lower stress levels encountered in normal use. 
This is done by statistically fitting an acceleration model to the 
test data and extrapolating the fitted model to estimate product 
life or degradation at a normal stress level. Tested under 
normal use stress levels, such products last so long or degrade 
so little that their life or performance cannot be estimated. 
Accelerated testing yields such estimates in a much shorter 
time. In this article, the purpose of accelerated testing is to 
measure product reliability or degradation. Nelson [94,97] 
and Meeker and Escobar [82] present in detail many models, 
statistical methodologies, and applications of accelerated 
testing. An entirely different purpose of accelerated testing is 
to force the product to fail to discover failure modes that 
would occur in actual use. Then the product or process is 
improved to reduce those failure modes. Such testing is used 
during product development or debugging of the production 
process and includes HALT, HASS, and environmental stress 
screening. This bibliography does not include references to 
such techniques as they are nonstatistical and do not yield 
estimates of product life. 
History of test plans. Test plans for analysis-of-variance 
and regression models have a long history, starting well before 
the work of R.A. Fisher, and such plans are still being 
developed. A number of general references on such test 
planning and optimization are included in the bibliography. 
Virtually all of those test plans are intended to estimate the 
product "response" within the test region. That is, they are 
plans for interpolation. In contrast, AT plans are used for 
extrapolation outside the range of test stress levels. Chernoff 
[27] developed the first accelerated life test plans; these 
involved the simple exponential distribution, two life-stress 
relationships, and censored data. Nelson and coworkers 
[67,87,98,99] were first to develop plans for the more realistic 
lognormal and Weibull life distributions. Nelson [94,97] 
provided the first book chapter on AT plans and briefly 
summarized the limited literature then. Since then work on 
AT plans has produced the 245 references in this updated 
bibliography. AT plans are included in the bibliography only 
if they include the (possibly asymptotic) variance of some 
estimate and preferably reduce that variance. Also, some 
general references on optimum design are included. Since 
HALT, HASS, and environmental stress screening are used 
only to reveal failure modes and do not yield estimates of 
product life, references to them are excluded. 
 AT models. The following view of accelerated life test 
(ALT) models is used throughout this article. Such a model 
consists of a theoretical life distribution whose parameters are 
functions of accelerating variables (and possibly other 
engineering variables) and unknown coefficients to be 
estimated from the test data. For example, the Arrhenius-
lognormal model consists of the lognormal distribution, an 
Arrhenius temperature relationship for the lognormal median, 
and a constant parameter. An ALT plan is designed to 
perform well for a particular model and test situation and may 
perform poorer for other models. An accelerated degradation 
test (ADT) model consists of a distribution of performance 
whose parameters are a function of accelerating variables and 
the time on test. Comprehensive references on statistical 
models and analyses for AT data are Nelson [94,97] and 
Meeker and Escobar [82]. 
Simple model. Most ALT plans have been developed for 
the following simple model for constant stress testing. The life 
distribution for y = ln(t), the log of life t, has a location 
parameter and a scale parameter . That is, the cumulative 
distribution function (cdf) for the population fraction failed by 
log age y is 
F(y) = G[(y )/ ] ; 
here G[ ] is the standardized cdf with = 0 and = 1. This 
includes the exponential, Weibull, lognormal, and some other 
distributions as special cases. The location parameter is a 
simple linear function of the (possibly transformed) 
accelerating variable x. That is, 
 (x) = 0 + 1 x ; 
here 0 and 1 are unknown coefficients to be estimated from 
test data. Such a life-stress relationship is usually based on 
physical theory; examples include the Arrhenius, inverse 
power, exponential, and other widely used relationships. The 
cdf can then be expressed as 
 
978-1-4799-6703-2/15/$31.00©2015 IEEE
F(y; x, 0, 1, ) = G[(y 0 1x)/ ]. 
Usually one seeks to accurately estimate a specified F fractile 
yF of the distribution at a specified low stress level x0 (an 
actual use level below the test stress levels), namely, 
 yF(x0) = (x0) + zF = 0 + 1 x0 + zF ; 
here zF is the F fractile of the standardized distribution G[ ]. 
yF(x0) is a function of the three unknown model parameters 0, 
1, and , which all must be estimated from the test data. This 
simple model and its special cases are presented in detail in 
Nelson [94,97] and Meeker and Escobar [82]. 
 Test plans. An accelerated life test (ALT) plan with 
constant stress consists of the chosen test conditions, the 
number of specimens at each test condition, and the prechosen 
censoring time of each specimen. For example, for the simple 
model, there may be three test stress levels x1, x2, and x3, 
corresponding numbers of specimens n1, n2, and n3, and 
corresponding chosen censoring times 1, 2, and 3. An ALT 
test can involve accelerating and other engineering variables 
(e.g., materials, designs, environments, process variables, 
etc.). In a test with constant accelerating stresses, a test 
condition is a specified set of constant values for the 
accelerating and engineering variables. In a test with time-
varying accelerating stresses, the test condition is a specified 
varying pattern for each stress and specified constant values 
for the engineering variables. The references present such 
plans that have good performance, which has been evaluated 
using analytic theory or simulation. Test plans for traditional 
design of experiments for analysis-of-variance models and for 
least-squares regression models usually have equal allocation 
of specimens to test conditions. This is appropriate to get 
accurate estimates within the test region. In contrast, AT plans 
are used for extrapolation outside the test region, and unequal 
allocation of specimens yields more accurate extrapolated 
estimates. Then more specimens are tested at low stress than 
at high. The bibliography has a few references on plansfor 
accelerated degradation tests (ADT). 
 Considerations. Such plans deal with considerations 
including 
 life distributions (exponential, Weibull, lognormal, etc.), 
 relationships (life-stress or degradation-stress), 
 optimization criteria (minimum variance, determinant, cost, 
etc.), 
 data censoring (right and interval), 
 estimation method (maximum likelihood, least squares, 
BLUEs, etc.), 
 types of stress loading (constant. step, ramp, cyclical, 
specified profile, stochastic, field use), 
 constraints (on test region, nonsimultaneous testing, etc.) 
 robustness (to model errors, loss of specimens, etc.) 
 specimens (size and geometry), 
The following section briefly discusses these considerations 
and cites some relevant articles. 
 
2 CO SIDERATIO S 
 Purpose. This section briefly discusses the preceding 
considerations for AT plans. 
 Distributions. Most ALT plans have been developed for 
the exponential, Weibull, and lognormal life distributions. 
References on a particular life distribution are easy to locate in 
the bibliography, since the distribution usually appears in the 
title. In practice, the exponential distribution is rarely suitable, 
but plans for it may provide insight into plans suitable for 
more realistic distributions like the Weibull and lognormal. 
There is a need for plans suited to other life distributions used 
in practice. 
 Relationships. Most ALT plans have been developed for 
models with a simple relationship between the life distribution 
mean (or a percentile) and a single accelerating stress (no 
other engineering variables). Such simple relationships are 
widely used in practice, and special cases include the 
Arrhenius, inverse power, exponential, and other relationships. 
Most of these are log-linear relationships between the 
distribution typical life and the (possibly transformed) 
accelerating variable. Few references deal with test plans for 
relationships with more than a single accelerating variable. 
These few include Nelson [97, Ch. 6, Sec. 3] and Escobar and 
Meeker [82]. More complicated ALT models can involve a 
second relationship for the other distribution parameter, such 
as the Weibull shape or the lognormal . Meeter and 
Meeker [88] and Kim and Bai [68] deal with AT plans for 
such relationships. As most accelerated degradation models 
have relationships that are unique to the application, and the 
relationship is rarely appears in the title of a reference. 
 Optimization criteria. Various optimization criteria have 
been used to design ALT plans. Most references develop 
plans that reduce or minimize the asymptotic variance of the 
maximum likelihood (ML) estimator of a specified fractile at a 
specified low stress condition. Some plans minimize the 
asymptotic variance of an ML estimator for a model parameter 
or a function of the parameters, such as the population fraction 
failing by a specified age at a specified low design stress. In 
contrast, for a model with two variables, Escobar and Meeker 
[44] minimize the determinant of the covariance matrix of the 
model parameter estimates, an attempt to globally estimate 
well. Yum and Kim [157] optimize the operating 
characteristic function of a reliability demonstration test. 
Little work has been done on cost optimization of AT plans. 
Some references optimize plans subject to constraints 
discussed below. 
 Censoring. Some ALT plans require that all specimens be 
run to failure, which is poor practice for reasons stated by 
Nelson [96]. Most recent references deal with exact failure 
times and Type I (prechosen fixed) censoring times. That is, 
the units at each test condition start at the same time, run 
together, and have a prechosen common censoring time. 
Some deal with exact failure times and Type II (failure) 
censoring times; that is, the units running together at test 
condition i are stopped at the random time yr of the r-th 
failure. Type II censoring is rare in practice but has easier and 
exact statistical theory compared with Type I censoring. A 
few references deal with interval data (also called read-out 
data), which result from periodic inspection of the units for 
failure. These include Yum and Choi [156], Schatzoff and 
coworkers [116-118], Bai, Kim, and Lee [16], and Seo and 
Yum [119]. 
 Estimation method. The statistical method used to fit an 
AT model to test data determines the accuracy of estimators of 
model parameters and functions of the parameters. 
Consequently, test plans are designed to provide accurate 
estimators with a particular method of fitting. Of course, such 
a plan generally provides good estimators when other fitting 
methods are used. Various fitting methods are described by 
Nelson [94,97] and include maximum likelihood, least 
squares, best linear unbiased, and graphical methods. Most 
recent references use ML fitting, as it is straightforward, 
applies to most AT models, has asymptotically normally 
distributed estimators, and yields approximate variances and 
confidence limits for estimators. Also, many statistical 
packages perform ML fitting of AT models to test data; 
Nelson [97, Chap. 5] describes such current packages. A 
drawback of ML theory is that the approximate variances and 
confidence limits for estimators are accurate only for tests 
with many specimen failures. Small sample properties of ML 
estimators have been investigated using simulation by 
McSorley, Lu, and Li [80], Nelson and Kielpinski [98], and 
Zhang and Meeker [158]. 
 Stress loading. Most AT plans involve constant-stress 
testing. Work on step-stress plans includes Miller and Nelson 
[90], Nilsson [100], Schatzoff and Lane [118], DeGroot and 
Goel [34], Bai, Kim, and Lee [15,16], Bai and Chun [6], and 
Bai, and Kim [12,13] among many others. For a ramp test 
(linearly increasing stress), Bai, Cha, and Chung [5] developed 
plans. In practice, constant-stress tests are easier to carry out 
but may need to run a long time at low stress to yield failures. 
Step-stress and ramp tests quickly yield failures, which 
provide information in a short time. However, the cumulative 
damage models have not been confirmed in most applications, 
the accuracy of estimates is usually poor, as test time is short 
and the failures generally occur at very high stress far from the 
design stress level, requiring much extrapolation. In general, 
the standard error of AT estimates is roughly inversely 
proportional to the total test time, and step-stress and other 
short tests yield poor accuracy. 
 Test constraints. AT plans generally involve constraints 
on the censoring times and on the test region (allowed values 
of accelerating and engineering variables). 
 Most plans have a single common censoring time for all test 
conditions. This usually presumes that all specimens can be 
started and run together. For example, in a temperature-
accelerated test, this requires a separate oven for each test 
temperature. An exception, Disch [37] develops a plan where 
there is one test oven, and different temperature groups of 
specimens must be run one after another. He optimizes the 
censoring time for each group, subject to a specified total oven 
time. Another exception is Bai, Chun, and Kim [8]. 
 Most plans require that the maximum test stress level must 
not exceed a specified value, determined from practical 
considerations. There can be constraints on the engineering 
variables in the AT model; Escobar and Meeker [44] deal with 
this. 
 In practice, there may be cost constraints on a test. Little 
cost modeling has been done for AT plans. 
 Robustness. Researchers have investigated plans that are 
robust to certain practical problems. 
 Distribution error. Any theoretical distribution used in an 
AT model differs from the actual physical distribution. In 
many applications, the erroneous fitted distribution is 
extrapolated to estimatethe lower tail of the physical 
distribution, usually to estimate low percentiles at a design 
stress level. To minimize distribution error, some plans have 
early censoring times at each test condition. Then data come 
from the lower tail of the distribution at each stress level, and 
data from the upper tail do not distort the fit. Nelson [96] 
explains distribution error in detail. 
 Relationship error. Any theoretical relationship used in an 
AT model differs from the actual physical relationship. In 
applications, the erroneous fitted relationship is extrapolated 
to a design stress level. To minimize relationship error, some 
plans use lower test stress levels. Barton [18] develops such a 
plan. Nelson [96] explains relationship error in detail. 
 Loss of specimens. Test specimens can be damaged and 
lost when a test has problems, say, an oven temperature runs 
amok. For example, when this happens and there just two test 
stress levels, then the simple life-stress relationship cannot be 
estimated. Also, no specimens may fail at the lower of just 
two test stress levels; then, too, the life-stress relationship 
cannot be estimated. To be robust, many plans use three or 
more test stress levels to assure that there are failures at two 
stress levels. For discussions of this issue, see Nelson [94,97 
Ch. 6] and Meeker and Hahn [86]. 
 Distribution and parameter uncertainty. In some 
applications, it is not known whether a Weibull or lognormal 
distribution should be used to design the test plan. Bayesian 
work on this problem involves using a prior distribution (a 
weighted average) of the two distributions. Also, test plans 
require knowledge of the true unknown model parameter 
values. Bayesian work replaces the unknown values with a 
priori distributions for the parameter values. Bayesian 
references include DeGroot and Goel [34,35], Chaloner and 
Larntz [24,25], Verdinelli, Polson, and Singpurwalla [137], 
and Pascual and Montepiedra [110,111]. ReliaSoft [114] 
provides sensitivity analysis for uncertain parameter values. 
 Compromise plans. Various authors have developed 
compromise plans that are robust and have good accuracy. 
Examples include Meeker and Hahn [86], Yang and Jin [144], 
Pascual and Montepiedra [110,111]. 
 Specimen size. For some products, the test specimens of 
different sizes may be used. Examples include lengths of 
cable insulation and lengths of aluminum conductors in 
microcircuit test structures. Bai and Yun [17] provide some 
plans for this situation. 
 Software. There is little commercially available software 
for evaluating or optimizing AT plans. Software of Meeker 
and Escobar [83] in S-Plus evaluates plans for exact failure 
data with right (Type I) censoring at specified times. Their 
software optimizes plans for models with a single accelerating 
variable and evaluates the performance of completely 
specified plans for models with more than one variable. 
ReliaSoft [114] handles similar situations and provides 
sensitivity analysis of the plans. Software of Schatzoff and 
coworkers [116,118] evaluates plans for the simple model, 
inspection data, and step-stress testing. JMP (2015, Ch. 15) 
has features for evaluating plans specified by the user and for 
basic optimum plans. 
 Concluding remarks. As stated above, this bibliography 
is intended to aid practitioners in choosing test plans and to 
stimulate researchers to develop needed plans. An updated 
bibliography is available in a searchable Word file from 
WNconsult@aol.com and includes email addresses of many 
authors. Please provide corrections and missing references 
and author email addresses. A search reveals that few 
references or their abstracts appear on the Web. 
 
ACK OWLEDGME T 
 Many authors in the bibliography kindly provided 
information on their publications and those of others. The 
author is indebted to Dr. Guangbin Yang and Prof. Hoang 
Pham, who stimulated him to publish the original 
bibliography, and to Prof. J.C. Lu for helpful suggestions. 
BIOGRAPHY 
 Dr. Wayne Nelson is a leading expert on analysis of 
reliability and accelerated test data. He consults on 
applications, gives training courses for companies and 
professional societies, and works as an expert witness. For 24 
years he consulted across the General Electric Co. and 
received the Dushman Award of GE Corp. R&D for 
developments and applications of product reliability data 
analysis. He was elected a Fellow of the Amer. Statistical 
Assoc. (1973), the Amer. Soc. for Quality (1983), the Institute 
of Electrical and Electronics Engineers (1988) for his 
innovative developments. He was awarded the 2003 Shewhart 
Medal and the 2010 Shainin Medal of ASQ and the 2005 
Lifetime Achievement Award of IEEE for outstanding 
developments of reliability methodology and contributions to 
reliability education. He authored three highly regarded books 
Applied Life Data Analysis (Wiley 1982, 2004), Accelerated 
Testing (Wiley 1990, 2004), Recurrent Events Data Analysis 
(SIAM 2003), two ASQ booklets, and 130 journal articles. 
Contact him at WNconsult@aol.com 
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 
A UPDATED BIBLIOGRAPHY OF 
ACCELERATED TEST PLA S Sep. 20, 2014 
 
This is an updated version of the bibliography published in 
IEEE Trans. on Reliability . You can request an updated 
bibliography at WNconsult@aol.com. References with ** 
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