Tests of independence among continuous random vectors based on Cramr-von Mises functionals of the empirical copula process

A decomposition of the independence empirical copula process into a finite number of asymptotically independent sub-processes was studied by Deheuvels. Starting from this decomposition, Genest and Rmillard recently investigated tests of independence among random variables based on Cramr-von Mises statistics derived from the sub-processes. A generalization of Deheuvels' decomposition to the case where independence is to be tested among continuous random vectors is presented. The asymptotic behavior of the resulting collection of Cramr-von Mises statistics is derived. It is shown that they are not distribution-free. One way of carrying out the resulting tests of independence then involves using the bootstrap or the permutation methodology. The former is shown to behave consistently, while the latter is employed in practice. Finally, simulations are used to study the finite-sample behavior of the tests.