Estimating the Location Vector for Spherically Symmetric Distributions

Jian-Lun Xu ()
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Jian-Lun Xu: National Cancer Institute, Biometry Research Group

Chapter Chapter 13 in Contemporary Experimental Design, Multivariate Analysis and Data Mining, 2020, pp 201-215 from Springer

Abstract: 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

Date: 2020
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DOI: 10.1007/978-3-030-46161-4_13

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