Semiparametric Estimation of Mutual Information and Related Criteria: Optimal Test of Independence

We derive independence tests by means of dependence measures thresholding in a semiparametric context. Precisely, the estimates of phi-mutual informations, associated to phi-divergences between a joint distribution and the product distribution of its marginals, are derived through the dual representation of phi-divergences. The asymptotic properties of the proposed estimates are established, including consistency, asymptotic distributions, and large deviations principle. The obtained tests of independence are compared via their relative asymptotic Bahadur efficiency and numerical simulations. It follows that the proposed semiparametric mutual information test is the optimal one. On the other hand, the proposed approach provides a new method for estimating the mutual information in a semiparametric setting, as well as a model selection procedure in a large class of dependence models, including semiparametric copulas.