AFFINE EQUIVARIANT MULTIVARIATE RANK-BASED AND GENERALIZED RANK REGRESSION

Abstract

Abstract : Affine equivariant estimates for the regression coefficient matrix of the mul- tivariate linear model are proposed. These estimates are based on a transformation and retransformation technique that uses Tyler's (1987) M-estimator of scatter. The proposed estimates are obtained by retransforming the componentwise rank- based estimate due to Davis and McKean (1993) and a componentwise generalized rank estimate. Asymptotic properties of the estimates are established under some regularity conditions. It is shown that both estimates have a multivariate nor- mal limiting distribution. The influence function of the retransformed generalized rank estimate has a bounded influence in both factor and response spaces. It is shown through a simulation study that the transformed-retransformed R and GR estimates are highly efficient compared to the non-equivariant componentwise R, GR and least absolute deviations estimates. Also, it is shown that the new esti- mates perform better than the least squares estimate when the errors have a heavy tailed distribution. Based on the new estimates quadratic procedures for testing the general hypothesis H0 : HBK = 0 are developed. Examples illustrating the estimation and testing procedures are presented.