Fitting of failure rate data to gamma-Poisson distribution utilizing method of moments

Because of the responsibility and gravity involved in estimating nuclear power plant failure rates, a consistent and accurate methodology is required for predicting the likelihood that an event will occur. However, the methodology currently employed can vary depending on the data used, as well as subjective conjecture from the expert performing the analysis. The current implementation of the empirical Bayes method to a gamma-Poisson (GaP) distribution utilizes algorithms to solve for the parameters that do not provide consistent answers. Additionally, to achieve a distribution of the likelihood, the variance of each parameter of the GaP distribution must be determined. There is no exact solution to one of the parameter's variance, and it is typically estimated using techniques like the Kass-Steffey adjustment. Thus, a new approach to the problem is proposed, built upon the method of moments for a negative binomial. Using the method of moments approach, we are able to achieve a closed-form estimation of the mean and variance for each parameter in the negative binomial distribution. Due to the relationship between the negative binominal and GaP distribution, comparisons can be made between the distributions. The hyper-priors defined assume a beta prime distribution that is appropriately informed; the results of this application translate back to the gamma distribution for easy utilization in SAPHIRE. Additionally, two cases are explored using publically available data from the Nuclear Regulatory Commission that consider zero-inflated, over-dispersed Poisson data.