Empirical Bayes Estimation of Reliability

Assessment of the reliability of various types of equipment relies on statistical inference about characteristics of reliability such as reliability function, mean lifetime of the devices, or failure rate. General techniques of statistical inference (estimation and hypotheses testing) are reviewed in Estimation; Least-Squares Estimation; Maximum Likelihood; Nonparametric Tests; Hypothesis Testing; and Significance Level, respectively. Consider the following model. Let T = (T1, T2, . . . , Tm), Ti ∈ T, be independent identically distributed (i.i.d.) observations with the joint probability density function (pdf) f (t |θ), t ∈ T , depending on an unknown parameter θ ∈ . If we are interested in estimating θ or a known function u(θ) of θ , we can, for example, use the maximumlikelihood estimation (MLE) of θ described in Maximum Likelihood. This technique is, however, of very little help if we want to accommodate some prior information about θ which may be available from previous experiments, expert opinions, or other sources of knowledge. To take advantage of this information, we assume that parameter θ is also a random variable or vector, and the particular value of θ associated with our sample comes from a pool of possible values of θ , which have a prior pdf g(θ). Using Bayes formula one can obtain the updated posterior pdf of θ given the sample T as g(θ |T ) = [ f (T |θ)g(θ)] [∫ f (T |θ)g(θ) dθ ] (1)