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The purpose of this paper is twofold. First, we provide a novelcharacterization of independence of random vectors based on the checkerboardapproximation to a multivariate copula. Using this result, we then propose anew family of tests of multivariate independence for continuous random vectors.The tests rely on estimating the checkerboard approximation by means of thesample copula recently introduced in [9] and improved in [11]. Such estimatorshave nice properties, including a Glivenko-Cantelli-type theorem thatguarantees almost-sure uniform convergence to the checkerboard approximation.Each of our test statistics is defined in terms of one of a number of differentmetrics, including the supremum, total variation and Hellinger distances, aswell as the Kullback-Leibler divergence. All of these tests can be easilyimplemented since the corresponding test statistics can be efficientlysimulated under any alternative hypothesis, even for moderate and large samplesizes in relatively large dimensions. Finally, we assess the performance of ourtests by means of a simulation study and provide one real data example.

A novel characterization and new simple tests of multivariate independence using copulas