Nonparametric Tests of Dependence using the Empirical Characteristic Function

Abstract:

We were interested in the construction of multidimensionnal Independent Component Analysis (ICA) methods to be applied to fMRI data. ICA usually relies on the minimization of some measure of dependence. In this context, a nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is constructed, using empirical characteristic functions, as a Cramér-von Mises functional of a random process having an asymptotic complex Gaussian distribution. A closed-form expression of the test statistic is obtained. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed using an approximation of the limiting distribution of the test statistic, which is an infinite sum of weighted chi-squared distributions. The weights are estimated by solving a discretized version of an integral operator whose kernel is an estimator of the covariance function of the limiting Gaussian process mentioned above. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure.