# Estimating the Location Vector for Spherically Symmetric Distributions

• Jian-Lun Xu
Chapter

## 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.

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