  The purpose of this paper is twofold. First, we provide a novel
characterization of independence of random vectors based on the checkerboard
approximation to a multivariate copula. Using this result, we then propose a
new family of tests of multivariate independence for continuous random vectors.
The tests rely on estimating the checkerboard approximation by means of the
sample copula recently introduced in [9] and improved in [11]. Such estimators
have nice properties, including a Glivenko-Cantelli-type theorem that
guarantees almost-sure uniform convergence to the checkerboard approximation.
Each of our test statistics is defined in terms of one of a number of different
metrics, including the supremum, total variation and Hellinger distances, as
well as the Kullback-Leibler divergence. All of these tests can be easily
implemented since the corresponding test statistics can be efficiently
simulated under any alternative hypothesis, even for moderate and large sample
sizes in relatively large dimensions. Finally, we assess the performance of our
tests by means of a simulation study and provide one real data example.
