A test of goodness-of-fit for the copula densities

We consider the problem of testing hypotheses on the copula density from n bidimensional observations. We wish to test the null hypothesis characterized by 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 supposed 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 term loss 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. Besides, the empirical performance of the test procedure is demonstrated with both simulated and real data. Index Terms — Adaptation, Copula Density, Minimax Theory of Test, Goodness Test of Fit. AMS Subject Classification — 62G10, 62G20, 62G30.