Simultaneous estimation in the multiparameter gamma distribution under weighted quadratic losses

A new class of solutions to a general differential inequality often encountered in multiparameter estimation problems is obtained. Using these solutions as guidelines, improved estimators for the scale parameters as well as the natural parameters of independent gamma distributions are obtained for a large class of weighted quadratic losses. The improved estimators have an empirical Bayes interpretation. They also permit an exact analytical representation of the risk improvement. For the ordinary squared-error loss, a larger class of improved estimates is obtained which may allow for incorporation of prior information in choosing an alternative estimate. Numerical results are given which indicate the extent of risk improvement in certain situations.