Multivariate Nonparametric Tests of Independence

• Published in: Journal of the American Statistical Association Vol. 100; no. 471; pp. 916 - 925
• Main Authors: Taskinen, Sara, Oja, Hannu, Randles, Ronald H
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.09.2005; American Statistical Association; Taylor & Francis Ltd
• Subjects: Affine invariance; Asymptotic methods; Correlations; Exact sciences and technology; Gaussian distributions; Kendall's tau; Mathematical vectors; Mathematics; Matrices; Multivariate analysis; Nonparametric inference; Nonparametric tests; Null hypothesis; P values; Pitman efficiency; Probability and statistics; Quadrant test; Quadrants; Robustness; Sciences and techniques of general use; Spearman's rho; Statistics; T distribution; Tests; Theory; Theory and Methods; Correlation coefficient; Non parametric test; Spearman's rho; Kendall's tau; Multivariate distribution; Pitman efficiency; Multivariate analysis; Aftine invariance; Statistical method; Ouadrant test: Robustness; Covariance; Correlation analysis; Asymptotic normality; Independence test; Kendall tau; Rank correlation; Robustness; Spearman rho
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214505000000097

New test statistics are proposed for testing whether two random vectors are independent. Gieser and Randles, as well as Taskinen, Kankainen, and Oja have introduced and discussed multivariate extensions of the quadrant test of Blomqvist. This article serves as a sequel to this work and presents new multivariate extensions of Kendall's tau and Spearman's rho statistics. Two different approaches are discussed. First, interdirection proportions are used to estimate the cosines of angles between centered observation vectors and between differences of observation vectors. Second, covariances between affine-equivariant multivariate signs and ranks are used. The test statistics arising from these two approaches appear to be asymptotically equivalent if each vector is elliptically symmetric. The spatial sign versions are easy to compute for data in common dimensions, and they provide practical, robust alternatives to normal-theory methods. Asymptotic theory is developed to approximate the finite-sample null distributions as well, as to calculate limiting Pitman efficiencies. Small-sample null permutation distributions are also described. A simple simulation study is used to compare the proposed tests with the classical Wilks test. Finally, the theory is illustrated by an example.