Test of multivariate linear models using spatial concordances

Kendall’s τ for univariate samples is extended to the multivariate case using unit direction vectors in place of signs. Depending on whether the design matrix is fixed, this statistic yields a test for the multivariate analysis of variance or multivariate regression. Tests of independence based on this statistic are easy, efficient, and robust. Its asymptotic properties and its Pitman efficiency are studied. In addition, some useful algebra in multiple dimensions is presented.