MANOVA type tests under a convex discrepancy function for the standard multivariate linear model

Abstract

We provide the M-theory for the standard multivariate linear model Y = XB + E, where Y is n X p matrix of observations, X is n X m design matrix, B is m X p matrix of unknown parameters and E is n X p matrix of errors with the row vectors independently distributed. Two test criteria based on the roots of determinantal equations are proposed for testing linear hypotheses of the form P'B = C₀, where P is a matrix of rank q. The tests are similar to those considered in MANOVA using least squares techniques. One of them is the Wald type statistic and another is the Rao's score type statistic. The asymptotic distributions of these test statistics are derived. Consistent estimates of nuisance parameters are obtained for use in computing the test statistics. The M-method of estimation considered is the minimization of Σϱ(e_i), where ϱ is a convex function and e_i is the i-th row vector in (Y-XB). All results are derived under a minimal set of conditions.