Minimax Estimation of Location Parameters for Spherically Symmetric Unimodal Distributions Under Quadratic Loss

Abstract

Families of minimax estimators are found for the location parameter of a $p$-variate $(p \geqq 3)$ spherically symmetric unimodal distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by $L(\delta, \theta) = (\delta - \theta)'D(\delta - \theta)$ where $D$ is a known $p \times p$ positive definite matrix, one main result, for one observation, $X$, on a multivariate s.s.u. distribution about $\theta$, presents a class of minimax estimators whose risk dominate the risk of $X$, provided $p \geqq 3$ and trace $D \geqq 2d_L$ where $d_L$ is the maximum eigenvalue of $D$. This class is given by $\delta_{a,r}(X) = (1 - a(r(\|X\|^2)/\|X\|^2)) X$ where $0 \leqq r(\bullet) \leqq 1, r(\|X\|^2)$ is nondecreasing, $r(\|X\|^2)/\|X\|^2$ is nonincreasing, and $0 \leqq a \leqq (c_0/E_0(\|X\|^{-2}))((\operatorname{trace} D/d_L) - 2)$, where $c_0 = 2p/((p + 2)(p - 2))$ when $p \geqq 4$ and $c_0 \approx .96$ when $p = 3$.