Asymptotic theory of least distances estimate in multivariate linear models

Abstract

We consider the multivariate linear model where Yi is a p-vector random variable, Xi is a q X p matrix, βÀ0 is an unknown q-vector parameter and {si} is a sequence of iid p-vector random variable with median vector zero. The estimate βÀn of βÀ0 such that is called the least distances (LD) estimator. It may be recalled that the least squares (LS) estimator is obtained by minimizing the sum of norm squares. In this paper, it is shown that the LD estimator is unique, consistent and has an asymptotic q-variate normal distribution with mean β0 and co variance matrix V which depends on the distribution of the error vectors {εi}. A consistent estimator of V is proposed which together with βÀn enables an asymptotic inference on β0. In particular, tests of linear hypotheses on β0 analogous to those of analysis of variance in the GATJSS-MARKOFF linear model are developed. Explicit expressions are obtained in some cases for the asymptotic relative efficiency of the LD compared to the LS estimator.