Non-parametric kk-sample tests: Density functions vs distribution functions

Tests for the comparison of kk samples based on kernel density estimators (KDE) are introduced. The Double Minimum method as a new and useful procedure for the crucial problem of bandwidth selection is developed. The statistical power of the proposed tests, as well as the impact of the smoothing degree and the performance of the Double Minimum algorithm, are studied via   Monte Carlo simulations. Finally, the results of the tests based on the KDE are compared to those of the traditional kk-sample tests based on empirical distribution functions (EDF), and to other tests based on the likelihood ratio introduced in the recent literature. Two main conclusions are obtained. First, the proposed bandwidth selection method attains quasi-optimal results. Second, the simulations suggest that KDE-based tests are the most powerful when the underlying populations are different in shape, and that the L1L1 distance among densities leads to optimal results in the considered situations.