All Admissible Linear Estimators of the Vector of Gamma Scale Parameters with Application to Random Effects Models

Abstract

The paper is devoted to the problem of simultaneous estimation of scale and natural parameters of the multiparameter gamma distribution under a quadratic loss. The vector of the scale parameters is assumed to range over a certain subset of the Cartesian product $\mathscr{R}^n_+$ of $n$ positive half lines. We identify the class of all linear admissible estimators for the scale parameters and show that all linear estimators of the natural parameters are inadmissible. Since the problem of invariant quadratic estimation of variance components in balanced random effects normal models leads to a problem of linear estimation of parametric functions of gamma scale parameters restricted to subsets of $\mathscr{R}^n_+$ being considered in this paper, some results on admissibility of invariant quadratic estimators of variance components are also established.