A note on second-order admissibility of the Graybill-Deal estimator of a common mean of several normal populations

Abstract

Consider the problem of estimating the common mean of several normal populations with unknown and possibly unequal variances under the squared error loss function. The well-known unbiased estimator of the common mean is the Graybill-Deal estimator. The general admissibility (or otherwise) of this estimator is still an open problem even though Sinha and Mouqadem ( Comm. Statist. Theory Methods 11 (1982) 1603–1614) proved some restricted admissibility results. In this paper, we follow the asymptotic decision-theoretic approach of ‘second-order admissibility’ as adopted by Ghosh and Sinha ( Ann. Statist. 9 (1981) 1334–1338), and prove that the Graybill-Deal estimator is second-order admissible.