Bayesian Concepts for the Estimation of Reliability in the Weibull Life-Testing Model

Abstract

Since in many life-testing situations it is not unlikely to note the unpredictable fluctuation of the scale parameter in a failure model, it is justifiable to consider such a parameter as a random variable and, thus, appeal to a Bayesian analysis. In specific, let 0 denote the random variable associated with the scale parameter and 0 its realization. Obviously, a Bayesian analysis depends on the utilization of prior information which, in this case, we assume to exist either in the form of a prior distribution of 0 or a sequence of sufficient statistics from past experiments. For the ordinary Bayes approach, we appeal to a well-known transformation to generalize the results of Bhattacharya (1967) for the one-parameter exponential model so as to include the flexibility provided by the shape parameter ? in the Weibull distribution. In fact, the Weibull failure model has an increasing ( > 1) or decreasing ( < 1) failure rate and, thus, is likely to describe the life-span of items with variable failure rates. The empirical Bayes estimation technique was largely motivated by Robbins (1955), who assumed the existence of a prior distribution for an unknown parameter but not the knowledge of its form. Instead, he substitutes past information which he assumes to exist as a result of the repetitive nature in the problem of estimation. Thus, in the absence of knowledge concerning the form of the prior distribution, we appeal to an empirical Bayes approach to estimate the scale parameter. By using this estimate, an estimate of the reliability function is made possible.