Extension of the stein estimating procedure through the use of estimating functions

This article extends the Stein method through the use of estimating functions (Godambe 1960) to address the simultaneous estimation of k population parameters, θ₁, …, θ_k, in a mixed model setting. The procedure generalizes the Stein method by (a) allowing us to deal effectively with complications, such as inequality of population variances, that may arise in non-Gaussian mixed models; (b) being appropriate for estimating θ_i in populations of varying sizes and, in particular, populations of small sizes; and (c) applying to situations where it cannot be assumed that the θ_i’s have unbiased estimators or even estimators of finite moment. The focus of the article is on the quadratic variance function exponential family (Morris 1983b). Estimators for the parameters of the mixed model are developed in a regression model setting in which the θ_i’s are allowed to vary with a vector of covariates. An application to incidence rates for the Iceland Breast Cancer Incidence Data is presented for illustrative purposes.