  We derive independence tests by means of dependence measures thresholding in
a semiparametric context. Precisely, estimates of phi-mutual informations,
associated to phi-divergences between a joint distribution and the product
distribution of its margins, 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 Kullback-Leibler Mutual information test is
the optimal one. On the other hand, the proposed approach provides a new method
for estimating the Kullback-Leibler mutual information in a semiparametric
setting, as well as a model selection procedure in large class of dependency
models including semiparametric copulas.
