On Analytic Empirical Bayes Estimation of Failure Rates

The estimation of plant accident rates and component failure rates is addressed within the framework of a parametric empirical Bayes approach. The observables, the numbers of failures recorded in various similar systems, obey the Poisson probability law. The parameters of a common gamma prior distribution are determined by a special moment matching method such that the results are consistent with classical (fiducial) confidence limits. Relations between Bayesian, classical, and Stein's estimation are discussed. The theory of the method is fully developed, although the suggested procedure itself is relatively simple. Solutions exist and they are in allowed ranges for all practical cases, including small samples and clustered data. They are also unbiased for large samples. Numerical examples are analyzed to illustrate the method and to allow comparisons with other methods.