Asymptotic theory of least distances estimate in multivariate linear models

Abstract

We consider the multivariate linear model Y_i=X'_iβ₀ + εi, i = 1,...,n where Y_i is a p-vector random variable, X_i is a q × p matrix, βÀ₀ is an unknown q-vector parameter and {s_i} is a sequence of iid p-vector random variable with median vector zero. The estimate βÀ_n of βÀ₀ such that min βΣⁿ_i=1||Y_i-X'_iβ||=Σⁿ_i=1||Y_i-X'_iβ|| 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 {ε_i}. A consistent estimator of V is proposed which together with βÀ_n 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.