Empirical Bayes Estimators with Uncertainty Measures for NEF-QVF Populations

The paper proposes empirical Bayes (EB) estimators for simultaneous estimation of means in the natural exponential family (NEF) with quadratic variance functions (QVF) models. Morris (1982, 1983a) characterized the NEF-QVF distributions which include among others the binomial, Poisson and normal distributions. In addition to the EB estimators, we provide approximations to the MSE’s of these estimators. Our approach generalizes the findings of Prasad and Rao (1990) for the random effects model where only area specific direct estimators and covariates are available. The EB estimators are derived using the theory of optimal estimating functions as proposed by Godambe and Thompson (1989). This is in contrast to the approach of Morris (1988) who found some approximate EB estimators for this problem. Also, unlike Morris (1988), we allow unequal number of observations in different clusters in the derivation of the EB estimators. In finding approximations to the MSE’s, we apply a bias-correction technique as proposed in Cox and Snell (1968). We illustrate our methodology by reanalyzing the toxoplasmosis data of Efron (1978, 1986).