A multivariate empirical characteristic function test of independence with normal marginals

• Published in: Journal of multivariate analysis Vol. 95; no. 2; pp. 345 - 369
• Main Authors: Bilodeau, M., Lafaye de Micheaux, P.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 2005; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: Characteristic function; Characteristic function Independence Multivariate analysis Serial independence Stochastic processes; Convergence; Distribution theory; Exact sciences and technology; Hypothesis testing; Independence; Inference from stochastic processes; time series analysis; Limit theorems; Mathematics; Multivariate analysis; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Serial independence; Statistics; Stochastic processes; Studies; 62H15; Characteristic function; 62M99; Serial independence; Stochastic processes; 62E20; Independence; Multivariate analysis; 60F05; Empirical process; Gaussian distribution; Eigenvalue; Stochastic process; Von Mises statistic; Marginal distribution; Distribution function; Random effect; 60F05 Characteristic function; Test statistic; 1991 subject classification: 62H15: 62M99; Empirical distribution function; Cornish Fisher expansion; Cramer von Mises test; Weak convergence; Statistical method; N variable function; Independence test; Parametric test
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2004.08.011

This paper proposes a semi-parametric test of independence (or serial independence) between marginal vectors each of which is normally distributed but without assuming the joint normality of these marginal vectors. The test statistic is a Cramér–von Mises functional of a process defined from the empirical characteristic function. This process is defined similarly as the process of Ghoudi et al. [J. Multivariate Anal. 79 (2001) 191] built from the empirical distribution function and used to test for independence between univariate marginal variables. The test statistic can be represented as a V-statistic. It is consistent to detect any form of dependence. The weak convergence of the process is derived. The asymptotic distribution of the Cramér–von Mises functionals is approximated by the Cornish–Fisher expansion using a recursive formula for cumulants and inversion of the characteristic function with numerical evaluation of the eigenvalues. The test statistic is finally compared with Wilks statistic for testing the parametric hypothesis of independence in the one-way MANOVA model with random effects.