## The Annals of Statistics

### Generalizations of James-Stein Estimators Under Spherical Symmetry

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 19, Number 3 (1991), 1639-1650.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348267

Digital Object Identifier
doi:10.1214/aos/1176348267

Mathematical Reviews number (MathSciNet)
MR1126343

Zentralblatt MATH identifier
0741.62058

JSTOR