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Efficient System Reliability Demonstration Tests Using the Probability of Test Success
... Because the power of the test corresponds to the detection of the fulfillment of the requirement, the alternative hypothesis, H 1 , represents the reliability requirement in terms of the lifetime. Therefore, the hypotheses to be used in a reliability demonstration test are defined as the following [38][39][40][41]: ...
... The approximations of (27) and (41) are approximate in several ways: first, (27) and (41) approximate the desired lifetime quantile distribution via a normal distribution, which does not hold for small sample sizes; second, the failure times used in (41) are approximated using order statistics in addition to the Benard approximation to the median of the beta distribution. However, the approximate calculations can be implemented effortlessly, as even most spreadsheet programs of the popular operating systems of personal computers have implementations of the normal distribution. ...
... It can be seen that the general method and the approximate method (see (27)) show very good agreement if parameter-free quantile estimation is used. On the other hand, the approximation based on the MLE (see (41)) shows very good agreement with the general method if the MLE is used for quantile estimation. Therefore, the approximation solely based on the sample quantiles (27) is to be used if empirical sample quantile estimation is used in the test. ...
Statistical power analyses are used in the design of experiments to determine the required number of specimens, and thus the expenditure, of a test. Commonly, when analyzing and planning life tests of technical products, only the confidence level is taken into account for assessing uncertainty. However, due to the sampling error, the confidence interval estimation varies from test to test; therefore, the number of specimens needed to yield a successful reliability demonstration cannot be derived by this. In this paper, a procedure is presented that facilitates the integration of statistical power analysis into reliability demonstration test planning. The Probability of Test Success is introduced as a metric in order to place the statistical power in the context of life test planning of technical products. It contains the information concerning the probability that a life test is capable of demonstrating a required lifetime, reliability, and confidence. In turn, it enables the assessment and comparison of various life test types, such as success run, non-censored, and censored life tests. The main results are four calculation methods for the Probability of Test Success for various test scenarios: a general method which is capable of dealing with all possible scenarios, a calculation method mimicking the actual test procedure, and two analytic approaches for failure-free and failure-based tests which make use of the central limit theorem and asymptotic properties of several statistics, and therefore simplify the effort involved in planning life tests. The calculation methods are compared and their respective advantages and disadvantages worked out; furthermore, the scenarios in which each method is to be preferred are illustrated. The applicability of the developed procedure for planning reliability demonstration tests using the Probability of Test Success is additionally illustrated by a case study.
... In order to establish a broader statistical context, Grundler et al. [4] defined the Probability of Test Success as the statistical power of a reliability demonstration test, since all reliability demonstration tests can be approached as hypothesis tests. By making use of this statistical context, new calculation procedures could be developed [22][23][24][25][26] e. g. using the asymptotic variance of the maximum likelihood estimation in [4]. Although several studies have been conducted in order to enable the application of the Probability of Test Success for systems with multiple failure modes [22,25,26], a proper procedure facilitating a holistic view is still necessary for an efficient planning procedure of reliability demonstration tests. ...
... By making use of this statistical context, new calculation procedures could be developed [22][23][24][25][26] e. g. using the asymptotic variance of the maximum likelihood estimation in [4]. Although several studies have been conducted in order to enable the application of the Probability of Test Success for systems with multiple failure modes [22,25,26], a proper procedure facilitating a holistic view is still necessary for an efficient planning procedure of reliability demonstration tests. In addition to the studies regarding the consideration of uncertainty [23] as well as the combined approaches for using Bayes' theorem [20,21,24,27] and the concept of the Probability of Test Success [4], the combination of all three aspects in a single holistic procedure has not been tackled yet. ...
Empirical life tests are used for reliability demonstration and determination of the actual reliability of the product. Therefore, engineers are faced with the challenge of selecting the most suitable test strategy out of the possible many and also the optimal parameter setting, e.g. sample size, in order to realize reliability demonstration with limited costs, time and with their available testing resources. It becomes even more challenging due to the stochastic nature of failure times and necessary cost and time being dependent on those. The considerations and guidelines in this paper are intended to simplify this process. Even simple products can fail due to several causes and mechanisms and usually have several components and subsystems. Therefore, this paper provides test planning options for single critical failure mechanisms as well as for systems with multiple failure mechanisms. For this purpose, the Probability of Test Success (Statistical Power of a life test) is used as a central, objective assessment metric. It is capable of indicating the probability of a successful reliability demonstration of a test and thus allows, for example, to answer the question of the required sample size for failure-based tests. The main planning resource is prior knowledge, which is mandatory due to the stochastic lifetime, in order to provide estimates for the Probability of Test Success at all. Therefore, it is also shown how to deal with uncertain prior knowledge and how the underlying information can additionally be used to increase the Probability of Test Success using Bayes’ theorem. The guidelines show how the most efficient test can be identified in the individual case and for individual boundary conditions.
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