Estimating the Location Vector for Spherically Symmetric Distributions

Abstract

When a $p\times 1$ random vector $\mathbf{X}$ has a spherically symmetric distribution with the location vector ${\varvec{\theta }},$ Brandwein and Strawderman [7] proved that estimators of the form $\mathbf{X}+a\mathbf{g}(\mathbf{X})$ dominate the $\mathbf{X}$ under quadratic loss if the following conditions hold: (i) $||\mathbf{g}||^2/2\le -h\le -\triangledown \circ \mathbf{g},$ where $-h$ is superharmonic, (ii) $E[-R^2h(\mathbf{V})]$ is nondecreasing in R,  where $\mathbf{V}$ has a uniform distribution in the sphere centered at $\varvec{\theta }$ with a radius $R=||\mathbf{X}-{\varvec{\theta }}||,$ and (iii) $0<a\le 1/[pE(R^{-2})].$ In this paper we not only use a weaker condition than their (ii) to show the dominance of $\mathbf{X}+a\mathbf{g}(\mathbf{X})$ over the $\mathbf{X},$ but also obtain a new bound $E(R)/[pE(R^{-1})]$ for a,  which is always better than bounds obtained by Brandwein and Strawderman [7] and Xu and Izmirlian [24]. The generalization to concave loss function is also considered. In addition, estimators of the location vector are investigated when the observation contains a residual vector and the scale is unknown.