Robust Empirical Bayes Analyses of Event Rates

Abstract

A collection of I similar items generates point event histories; for example, machines experience failures or operators make mistakes. Suppose the intervals between events are modeled as iid exponential (λ _i , or the counts as Poisson (λ _i t _i ,) for the ith item. Furthermore, so as to represent between-item variability, each individual rate parameter, λ _i , is presumed drawn from a fixed (super) population with density g _λ (·; θ), θ being a vector parameter: a parametric empirical Bayes (PEB) setup. For g _λ, specified alternatively as log-Student t(n) or gamma, we exhibit the results of numerical procedures for estimating superpopulation parameters ll and for describing pooled estimates of the individual rates, λ _i , obtained via Bayes's formula. Three data sets are analyzed, and convenient explicit approximate formulas are furnished for λ _i estimates. In the Student-t case, the individual estimates are seen to have a robust quality.

KEY WORDS:

• Empirical Bayes
• Robustness
• Poisson process
• Reliability