Linear models without moments

Abstract

In models y = Xβ + e having spherical errors without moments, the usual estimator β is shown to be (a) median-unbiased for β, (b) most concentrated around β among all median-unbiased linear estimates, and (c) consistent for β in a sequence of identical but dependent experiments possibly having Cauchy errors. It is shown for all spherical error laws that the joint null distributions of variance ratios have exactly their normal theory forms, and for unimodal error laws that the power functions are monotone and the usual tests are unbiased.