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Reliability, IEEE Transactions on

Issue 3 • Date Aug. 1972

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  • [Front cover]

    Page(s): c1
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  • IEEE Reliability Group

    Page(s): nil1
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  • [Breaker page]

    Page(s): nil1
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  • Preface

    Page(s): 127 - 128
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  • Editorial

    Page(s): 129
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  • List of referees

    Page(s): 130
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  • The Philosophy and Mathematics of Bayes' Equation

    Page(s): 131 - 135
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    The rudiments of Bayesian philosophy are introduced, and the mathematics of its application are surveyed; there is no uniformity of thought concerning either. The more extreme Bayesian philosophy, which allows subjective probabilities, is a means of plausible reasoning, or of making inferences, through inductive logic. Because inferences concerning reliability concepts are important in the decision process, this philosophy has a place in the reliability field. The difficulties of interpreting the probability function and of assigning prior distributions restrict the presentation of a unified philosophy. Thus, only techniques for describing prior probabilities under various circumstances can be given. Application of the Bayesian approach depends on how the reliability decision is conceptualized. View full abstract»

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  • Bayes' Equation, Reliability, and Multiple Hypothesis Testing

    Page(s): 136 - 147
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    The rudiments of applying Bayes' Equation to hypotheses concerning reliability are introduced in a simple manner. The application is a means of obtaining posterior probabilities, for the reliability hypotheses, which are consistent with the prior beliefs and the available test results. The posterior distributions, from which decision theory could formally arrive at optimal estimates, are greatly dependent on the prior distributions. Thus, the discussion centers about the desired properties of a prior and its effects on the posterior for various data situations. Formulations for both continuous-conjugate and discrete representations of the prior beliefs are discussed and contrasted. The use of discrete priors offers many advantages over the use of continuous-conjugate priors. View full abstract»

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  • Prior Distributions Fitted to Observed Reliability Data

    Page(s): 148 - 154
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    This paper describes methods of fitting prior distributions to equipment MTBF = ¿, shows the priors fitted to different equipments, establishes data criteria for fitting prior distribution to ¿, and presents the results of a robustness analysis performed on the fitted priors. Systematic procedures for fitting priors are shown for Type 1 data (number of failures in fixed time T) and Type 2 data (observed MTBF, number of failures not the same for all equipments), and specific data criteria, in the form of minimum values of n (number of equipments) and K (number of failures) are presented. The inverted-gamma prior-distributions were derived from operational failure data obtained from Tinker AFB. The equipments are primarily electronic, therefore, the time-to-failure distribution was assumed to be exponential; however, the methods are generally applicable whatever the form of the conditional distribution. The robustness analysis shows the effects of errors in estimating the parameters of the prior on the posterior distribution. In general, the effect of errors in estimating parameters of the prior was practically negligible for large values of K. View full abstract»

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  • An Empirical Bayes Approach to Reliability

    Page(s): 155 - 158
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    A Bayesian reliability estimation technique known as the ``empirical Bayes approach'' is developed which uses previous experience nce to get a Bayesian point estimator. The techniques require no knowledge of the form of the unknown prior distribution and are robust to assumptions about its form. Empirical Bayes techniques are applicable to situations in which prior, independent observations of the random variable X from the random couple (¿, X) are available where ¿ is the observed parameter of interest distributed in accordance with the unknown prior distribution. Performance comparisons of the empirical Bayes and other well established techniques are developed by examples for the binomial, exponential, Normal, and Poisson situations which often occur in reliability problems. In all cases the empirical Bayes estimator performed better than the classical estimator in minimizing the average squared error. View full abstract»

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  • Bayesian Decision Analysis of the Hazard Rate for a Two-Parameter Weibull Process

    Page(s): 159 - 169
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    A two-parameter Weibull distribution is assumed to be the appropriate statistical life-model of an engineering device. The hazard rate of this device is the relevant quantity in terms of which statistical decisions are to be made. A Bayesian decision model is constructed around a conjugate probability density function for the Weibull hazard rate. Prior, posterior, and preposterior analysis of this decision model are discussed. The results indicate the rational decision before and after sampling, and permit the optimization of sequential single-item sampling schemes. Such schemes are of particular importance in the reliability testing of high-cost equipment. View full abstract»

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  • Determining Optimum Burn-In and Replacement Times Using Bayesian Decision Theory

    Page(s): 170 - 175
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    An important problem facing a manufacturer is the determination of the amount of time to burn-in items (in order to eliminate early failures) and the age at which to replace items (to avoid failures due to wearout). This problem becomes difficult to solve if the time-to-failure distribution of an item is unknown and must be estimated from test and operational data. This paper describes a method of statistical data analysis which is readily applied to the solution of this decision problem under a realistic but general loss (or gain) function. The method is a multiparameter Bayesian analysis which requires multiple integration of the (multivariate) posterior of the parameters of the time-to-failure distribution to obtain the expected loss (or gain) resulting from a particular choice of burn-in time and item replacement age. This integration is performed by a Monte Carlo Procedure using importance sampling. An example demonstrates the flexibility of this method of analysis. The data are a mixture of ``point'' and truncated data, which often create difficulties when using conventional methods of decision analysis. In addition, since the method permits up to ten parameters for the family of time-to-failure distributions, a ``bathtub'' hazard rate function is used to generate the data for the example. The results are presented in the form of Bayesian confidence intervals for the true hazard rate function and a presentation of the expected loss as a function of burn-in time and age at replacement. View full abstract»

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  • Bayesian Acceptance Sampling

    Page(s): 176 - 180
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    This tutorial paper shows, via an example, how optimum sampling plans can be derived using Bayesian decision theory. Both nonsequential and sequential sampling plans are described. The Bayesian method is contrasted with classical methods. The salient difference between Bayesian and classical methods is that the Bayesian approach allows one formally to pose and answer the question: ``How many should I test?''. View full abstract»

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  • Experience with Bayesian Reliability Measurement of Large Systems

    Page(s): 181 - 185
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    Experience with two methods of Bayesian reliability measurement is described. An aerospace subsystem was evaluated assuming continuous gamma-distributed component failure rates. Priors were developed by conventional reliability prediction methods based on handbook data. The ``strength'' of the prior was expressed in terms of variance about a predicted mean. Comparative evaluation was also made by a classical technique during a test program extending over 10 months. The Bayesian method was preferred though problems inherent in the method were apparent. More recently, a complex marine system was evaluated over a one-year period using a Bayesian formulation in which the failure rate is described by a discrete probability distribution with nonuniform cell widths. This technique avoids some of the operational problems of continuous formulations. Experience with Bayesian methods leaves little doubt of their utility as evaluation tools. The philosophical problems, however, remain as intransigent as ever. View full abstract»

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  • A Personal View of the Bayesian Controversy in Reliability and Statistics

    Page(s): 186 - 194
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    There appear to be two important developing trends in reliability. One is away from the use of statistics by reliability engineers; the other is toward increased use of Bayesian techniques. One source of the latter may be disillusionment with what is regarded as classical statistics; however, one source of the former may be dissatisfaction with what has been called Bayesian statistics. The purpose of this paper is to discuss these trends and to present a personal view of the issues between classical and Bayesian statistics. The objective is to show that there is some substance to the classical statistician's opposition to Bayesian inference and that the issues are pertinent and meaningful to the reliability engineer. View full abstract»

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  • Book Reviews

    Page(s): 197 - 198
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  • Information for Readers

    Page(s): nil2
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  • [Front cover]

    Page(s): c2
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Aims & Scope

IEEE Transactions on Reliability is concerned with the problems involved in attaining reliability, maintaining it through the life of the system or device, and measuring it.

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Meet Our Editors

Editor-in-Chief
Way Kuo
City University of Hong Kong