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Estimating the Location Vector for Spherically Symmetric Distributions

  • Jian-Lun XuEmail author
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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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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Biometry Research GroupNational Cancer InstituteBethesdaUSA

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