M-estimation of multivariate linear regression parameters under a convex discrepancy function

Abstract

There is vast literature on M-estimation of linear regression parameters. Most of the papers deal with special cases by choosing particular discrepancy functions to be minimized or particular estimating equations. A few discuss general results, but prove results under heavy assumptions which seem to exclude important special cases. In this paper, a general theory of M-estimation is developed using a convex discrepancy function under what appear to be a necessary set of assumptions to develop a satisfactory asymptotic theory. Detailed proofs are given for establishing the asymptotic normality of the distribution of M-estimates and the results are applied to several particular cases. Appropriate criteria are developed for tests of hypotheses concerning regression parameters. The problem is discussed in the multivariate situation which includes the univariate case.