A goodness-of-fit test for copula densities

We consider the problem of testing hypotheses on the copula density from a two-dimensional random sample. We test the null hypothesis of a parametric class against a composite nonparametric alternative. Each density under the alternative is separated in the L2-norm from any density lying in the null hypothesis. The copula densities under consideration are assumed to belong to a range of Besov balls. According to the minimax approach, the testing problem is solved in an adaptive framework: it leads to a log log  loss term in the minimax rate of testing in comparison with the non-adaptive case. A smoothness-free test statistic that achieves the minimax rate is proposed. The lower bound is also proved.