Minimum risk equivariant estimator in linear regression model

Abstract

The minimum risk equivariant estimator (MRE) of the regression parameter vector β in the linear regression model enjoys the finite-sample optimality property, but its calculation is difficult, with an exception of few special cases. We study some possible approximations of MRE, with distribution of the errors being known or unknown: A finite-sample approximation uses the Hájek–Hoeffding projection or the Hoeffding–van Zwet decomposition of an initial equivariant estimator of β, a large-sample approximation is based on the asymptotic representation of the same. A nonparametric approximation uses the expected value with respect to the conditional empirical distribution function, developed by Stute (1986). The only possible approximation avoiding a difficult calculation of conditional expectations is the asymptotic approximation, based on the score function of the underlying distribution of the errors.