Rank scores tests of multivariate independence

Abstract

New rank scores test statistics are proposed for testing whether two random vectors are independent. The tests are asymptotically distribution-free for elliptically symmetric marginal distributions. Recently, Gieser and Randles (1997), Taskinen, Kankainen and Oja (2003) and Taskinen, Oja and Randles (2005) introduced and discussed different multivariate extensions of the quadrant test, Kendall's tau and Spearman's rho statistics. In this paper, standardized multivariate spatial signs and the (univariate) ranks of the Mahalanobis-type distances of the observations from the origin are combined to construct ranks cores tests of independence. The limiting distributions of the test statistics are derived under the null hypothesis as well as under contiguous sequences of alternatives. Three different choices of the score functions, namely the sign scores, the Wilcoxon scores and the van der Waerden scores, are discussed in greater detail. The small sample and limiting efficiencies of the test procedures ara compared and the robustness properties are illustrated by an example. It is remarkable that, in the multinormal case, the limiting Pitman efficience of the van der Waerden scores test equals to that of the classical parametric Wilks’s test.