Minimax nonparametric hypothesis testing: the case of an inhomogeneous alternative

Abstract

We study the problem of testing a simple hypothesis for a nonparametric ''signal + white-noise'' model. It is assumed under the null hypothesis that the ''signal'' is completely specified, e.g., that no signal is present. This hypothesis is tested against a composite alternative of the following form: the underlying function (the signal) is separated away from the null in the L₂ norm and, in addition, it possesses some smoothness properties. We focus on the case of an inhomogeneous alternative when the smoothness properties of the signal are measured in an L_p norm with p<2. We consider tests whose errors have probabilities which do not exceed prescribed values and we measure the quality of testing by the minimal distance between the null and the alternative set for which such testing is still possible. We evaluate the optimal rate of decay for this distance to zero as the noise level tends to zero. Then a rate-optimal test is proposed which essentially uses a pointwise-adaptive estimation procedure.