Copula-based dependence measures for arbitrary data

In this article, we define extensions of copula-based dependence measures for data with arbitrary distributions, in the non-serial case, i.e., for independent and identically distributed random vectors, as well as in serial case, i.e., for time series. These dependence measures are covariances with respect to a multilinear copula associated with the data. We also consider multivariate extensions based on M\"obius transforms. We find the asymptotic distributions of the statistics under the hypothesis of independence or randomness and under contiguous alternatives. This enables us to find out locally most powerful test statistics for some alternatives, whatever the margins. Numerical experiments are performed for combinations of these statistics to assess the finite sample performance.