## Abstract

Let X = (X_{1},…, X_{K}), where X_{i} are mutually independent p-variate (K > p+1) normal vectors with unknown means θ_{i}and unknown positive definite variance-covariance matrix V. Assume the statistic V is available for estimating V, where V has a Wishart distribution W_{P}(n, V)/(n+p+1), n> p+1, and is independent of X. It is desired to estimate θ = (θ_{1},…,θ_{K}) under the quadratic loss L_{Q}*(θ, θ) = tr{(θ - θ)^{1}Q*(θ - θ)}, where Q* = V^{-1/2}QV^{-1/2}, V =V^{1/2}V^{1/2}, and Q is a known positive definite matrix chosen by the researcher. The L_{Q*} loss includes the widely used loss L(θ, θ) = tr{(θ - θ)^{1}V^{-1}(θ - θ)} as a special case. It is shown that under some specifications of τ(V,S), a symmetric pxp matrix, the proposed empirical Bayes estimator (Ip - (VS^{-1} τ(V, S))X dominates the maximum likelihood estimator X and is minimax under the L_{Q*} loss. Unlike the previous work on the estimation of vector normal means under quadratic losses with a weight matrix Q, the proposed empirical Bayes minimax estimators are structurally free of Q and the minimaxity holds for a class of quadratic loss functions L_{Q*}. The simulated risks of several competing EB estimators are considered, and the risk improvement of these estimators over the sample mean is calculated.

Original language | English |
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Pages (from-to) | 317-342 |

Number of pages | 26 |

Journal | Statistics and Risk Modeling |

Volume | 11 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 1993 |

## Keywords

- Wishart identity
- empirical Bayes
- frequentist risks
- matrix normal means
- minimax
- quadratic loss