Generalizations of James-Stein Estimators Under Spherical Symmetry

Brandwein, Ann Cohen ; Strawderman, William E.

Ann. Statist., Tome 19 (1991) no. 1, p. 1639-1650 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1991-09-14
Classification:  Spherical symmetry,  minimaxity,  squared error loss,  concave loss,  James-Stein estimation,  superharmonic,  location parameters,  62C99,  62F10,  62H99