Equivariant estimation functions and normal distribution assumption for the model of linear regression

Abstract

When estimating a linear functionalβ →g(β) in a linear modelM=(Y, Xβ, σ2I), it is well known that, for convex loss, the OLS estimator minimizes the risk uniformly in the class ℰ(M, g) of all unbiased estimators providedY is normally distributed. For squared error loss andX a (n×2)-matrix we identify allX andg for which, in some sense, the converse holds:Y is necessarily normally distributed if the OLS estimator minimizes the risk uniformly in the class of equivariant estimators in ℰ(M, g).