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September, 1991 Generalizations of James-Stein Estimators Under Spherical Symmetry
Ann Cohen Brandwein, William E. Strawderman
Ann. Statist. 19(3): 1639-1650 (September, 1991). DOI: 10.1214/aos/1176348267


This paper is primarily concerned with extending the results of Stein to spherically symmetric distributions. Specifically, when $X \sim f(\|X - \theta\|^2)$, we investigate conditions under which estimators of the form $X + ag(X)$ dominate $X$ for loss functions $\|\delta - \theta\|^2$ and loss functions which are concave in $\|\delta - \theta\|^2$. Additionally, if the scale is unknown we investigate estimators of the location parameter of the form $X + aVg(X)$ in two different settings. In the first, an estimator $V$ of the scale is independent of $X$. In the second, $V$ is the sum of squared residuals in the usual canonical setting of a generalized linear model when sampling from a spherically symmetric distribution. These results are also generalized to concave loss. The conditions for domination of $X + ag(X)$ are typically (a) $\|g\|^2 + 2\nabla \circ g \leq 0$, (b) $\nabla \circ g$ is superharmonic and (c) $0 < a < 1/pE_0(1/\|X\|^2)$, plus technical conditions.


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Ann Cohen Brandwein. William E. Strawderman. "Generalizations of James-Stein Estimators Under Spherical Symmetry." Ann. Statist. 19 (3) 1639 - 1650, September, 1991.


Published: September, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0741.62058
MathSciNet: MR1126343
Digital Object Identifier: 10.1214/aos/1176348267

Primary: 62C99
Secondary: 62F10 , 62H99

Keywords: concave loss , James-Stein estimation , location parameters , minimaxity , spherical symmetry , squared error loss , superharmonic

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 3 • September, 1991
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