Multivariate tests of independence

Abstract

If X_n=(X_n(1),...,X_n(p)), n=1,2,... is an i.i.d. sequence of p-variate random vectors, we consider here the problem of distribution free hypothesis testing on the probability law of X_n, that is, to test hypothesis invariant under continuous non decreasing transforms of the coordinates. Our main interest will be for tests of independence, typical among the preceding.

A fundamental tool for the study of these problems appears to be the dependence functions of which we summarize the main definitions and basic results in §1. Doing this, we introduce the new dependogram statistic which has proved itself to be an efficient way to visualize the information contained in a sample on its intern dependence structure.

We assume afterwords that the hypothesis of independence is true, and state in §2 some important results about the weak functional convergence of the normalized empirical dependence function toward a limiting Gaussian process.

As an application of the preceding, we derive in §3 the main original result of this paper, obtaining a general family of statistical tests for independence extending results proved for the bivariate case (p=2) in [2] and [4].

We give in §4 an explicit example of such a statistic for the trivariate (p=3) case. We obtain a (4) as a limiting distribution, and give a partition of this test in four asymptotically independent and (1) components.

In §5, we give an extension of a result proved in [5] about Kolmogorov-Smirnov tests of independence in the bivariate case (p=2).