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Over the last couple of decades, several copula based methods have beenproposed in the literature to test for the independence among several randomvariables. But these existing tests are not invariant under monotonetransformations of the variables, and they often perform poorly if thedependence among the variables is highly non-monotone in nature. In thisarticle, we propose a copula based measure of dependency and use it toconstruct some new distribution-free tests of independence. The proposedmeasure and the resulting tests, all are invariant under permutations andmonotone transformations of the variables. Our dependency measure involves akernel function, and we use the Gaussian kernel for that purpose. We adopt amulti-scale approach, where we look at the results obtained for several choicesof the bandwidth parameter associated with the Gaussian kernel and aggregatethem judiciously. Large sample properties of the dependency measure and theresulting tests are derived under appropriate regularity conditions. Severalsimulated and real data sets are analyzed to compare the performance of theproposed tests with some popular tests available in the literature.

Some New Copula Based Distribution-free Tests of Independence among Several Random Variables