Adaptive minimax testing in the discrete regression scheme

We consider the problem of testing hypotheses on the regression function from n observations on the regular grid on [0, 1]. We wish to test the null hypothesis that the regression function belongs to a given functional class (parametric or even nonparametric) against a composite nonparametric alternative. The functions under the alternative are separated in the L2-norm from any function in the null hypothesis. We assume that the regression function belongs to a wide range of Hölder classes but as the smoothness parameter of the regression function is unknown, an adaptive approach is considered. It leads to an optimal and unavoidable loss of order √ log(log n) in the minimax rate of testing compared with the non-adaptive setting. We propose a smoothness-free test that achieves the optimal rate, and finally we prove the lower bound showing that no test can be consistent if in the distance between the functions under the null hypothesis and those in the alternative, the loss is of order smaller than the optimal loss.