Multivariate tests of independence based on a new class of measures of independence in Reproducing Kernel Hilbert Space

Abstract

In this paper, we propose a new class of independence measures based on the maximum mean discrepancy (MMD) in Reproducing Kernel Hilbert Space (RKHS). We use a novel way to build the independence measure by creating a metric on the space of all Borel probability measures. The proposed approach has several attractive properties including (i) no required assumptions about the data structure; (ii) insensitive to the dimension of the data; (iii) being more flexible than the Hilbert–Schmidt independence criterion (HSIC), which is the most popular independence measure based on RKHS. We show that the empirical estimator of the proposed independence measure possesses some desirable large sample properties regardless of the dimension of data. Based on the proposed independence measure, we develop two tests of independence in which the test statistics have simple forms and are easy to compute. The performance of the proposed tests of independence for high-dimensional data is evaluated through an extensive Monte Carlo simulation study.