All estimates with a given risk, Riccati differential equations and a new proof of a theorem of Brown

Abstract

For the canonical problem of estimating the mean of a multivariate normal distribution with a known covariance matrix using a squared error loss, we give a general method for finding estimates that have risk functions identical to that of a given inadmissible estimate. In the case of more than one dimension, the estimates considered are spherically symmetric, but in one dimension no such assumption is made. Generally speaking, we characterize all estimates which have the risk duplication property. It is proven that every James-Stein estimator except $(1 - (p - 2)/ \parallel X \parallel^2) \rm X$ can be duplicated in risk by infinitely many estimators. A general theorem is presented from which a principal inadmissibility result for spherically symmetric estimates in Brown follows. This and the other results all basically depend on solving Riccati differential equations of an appropriate kind. A curious result is that two smooth estimates whose graphs intersect cannot have identical risk. Several results for the entire discrete exponential family and the binomial case demonstrate that the phenomena are fundamentally different in continuous and discrete cases. Our results indicate a new method for constructing explicit dominating estimates that may work in many problems.