Nonlinear canonical analysis and independence tests

´´ Measures of association between two random variables r.v. that are symmetric nondecreasing functions of the canonical coefficients provided by the nonlinear canonical analysis of the r.v.’s are studied. These measures can be used to characterize independence. Their estimators are obtained by estimating a suitable approximation to nonlinear canonical analysis. When some conditions ae satisfied, asymptotic distributions of the estimators, both under the independence hypothesis and under dependence, are given. A class of tests of independence with asymptotic level of significance can be investigated. 1. Introduction. Several measures of association between two random Ž. variables r.v. based on the canonical analysis of these r.v.’s have been introduced in the literature, and a review of such measures can be found in Ž. Ž . Cramer and Nicewander 1979 and Lazraq and Cleroux 1988 . A general ´ Ž. enough study of these measures is proposed in Lin 1987 and, more recently, Ž. in Dauxois and Nkiet 1997 , where a global study of the induced tests is developed. In these works the measures considered are constructed using Ž. linear canonical analysis LCA of the r.v.’s and, thus, characterize only lack of a linear relationship. It can be interesting to look for another class of measures of association which characterize independence. Ž. The properties of nonlinear canonical analysis NLCA suggest that such a class can be derived from this analysis, but there is little work in this direction. The canonical coefficients from NLCA have already been used for testing independence, but uniquely when the related random variables are Ž. categorical see, for instance, Tsai and Sen 1990 . That is restrictive because one knows that, in this particular case, NLCA is a LCA of suitable random vectors, and thus it suffices to use classical measures of association. Consequently, it appears that the use of NLCA is more interesting when the related random variables are not categorical. In this paper, we propose a class of measures of association between random variables with values in any measurable spaces, constructed using symmetric nondecreasing functions and the canonical coefficients derived from NLCA. The properties of these measures show that they are appropriate to characterize independence without any assumption on the distribution of