Improving on the MLE of a Bounded Mean for Spherical Distributions

• Activity Number: 285
• Type: Contributed
• Date/Time: Wednesday, August 14, 2002 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Francois Perron*+
• Affiliation(s): Université de Montréal
• Keywords: Maximum likelihood estimator ; restricted parameter space ; spherically symmetric distribution ; scale mixture of normals ; multivariate student distribution ; Bayes estimator

Abstract

For the problem of estimating under squared error loss the mean of a $p$-variate spherically symmetric distribution where the mean lies in a ball of radius $m$, a sufficient condition for an estimator to dominate the maximum likelihood estimator is obtained. We use this condition to show that the Bayes estimator, with respect to a uniform prior on the boundary of the parameter space, dominates the maximum likelihood estimator whenever $m\leq\sqrt{p}$ in the case of a multivariate student distribution with $d$ degrees of freedom, $d \geq p$. The sufficient condition $m\leq\sqrt{p}$ matches the one obtained by Marchand and Perron (2001) in the normal case with identity matrix. Furthermore, we derive a class of estimators which, for $m< \sqrt{p}$, dominates the maximum likelihood estimator simultaneously for the normal distribution with identity matrix and for all multivariate student distributions with $d$ degrees of freedom, $d\geq p$. The family of distributions where dominance occurs includes the normal case; and includes all student distributions with $d$ degrees of freedom, $d\geq 1$, for the case $p=1$.