Robust empirical Bayes analyses of event rates. (English) Zbl 0611.62124

Gaver, Donald P.; O’Muircheartaigh, I. G.

Technometrics 29, 1-15 (1987).

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 \((\lambda_ i)\), or the counts as Poisson \((\lambda_ it_ i)\), for the ith item. Furthermore, so as to represent between-item variability, each individual rate parameter, \(\lambda_ i\), is presumed drawn from a fixed (super) population with density \(g_{\lambda}(\cdot;\theta)\), \(\theta\) being a vector parameter: a parametric empirical Bayes setup.
For \(g_{\lambda}\), specified alternatively as log-Student t(n) or gamma, we exhibit the results of numerical procedures for estimating superpopulation parameters \(\theta\) and for describing pooled estimates of the individual rates, \(\lambda_ i\), obtained via Bayes’s formula. Three data sets are analyzed, and convenient explicit approximate formulas are furnished for \(\lambda_ i\) estimates. In the Student-t case, the individual estimates are seen to have a robust quality.

MSC:

62N05	Reliability and life testing
62F15	Bayesian inference

Keywords:

robustness; Poisson process; parametric empirical Bayes; log-Student; gamma; estimating superpopulation parameters; pooled estimates; Bayes’s formula