Minimax estimates of a normal mean vector for arbitrary quadratic loss and unknown covariance matrix
The purpose of this paper is to find a class of estimators which dominate the usual estimator X of the mean vector \(\mu\) of a multivariate normal distribution under arbitrary quadratic loss in the case where the covariance matrix is unknown. The integration-by-parts methods for both the multivariate normal and Wishart distributions are combined to yield unbiased estimates of risk difference (versus X) for certain classes of estimators, defined indirectly through a ''seed'' function dependent upon X and an independent Wishart matrix. In this way we obtain a new class of minimax estimators of \(\mu\) which is closely related to classes of estimators considered in \textit{J. Berger} and \textit{L. R. Haff}, Stat. Decis. 1, 105-129 (1983; Zbl 0562.62005) and by the author, Ann. Stat. 7, 838-846 (1979; Zbl 0418.62004).
scientific article; zbMATH DE number 3992613