Adaptive and spatially adaptive testing of a nonparametric hypothesis

The present paper continues studying the problem of nonparametric hypothesis testing started in Lepski and Spokoiny, 1995 and Spokoiny, 1995. Let a function ƒ be observed with noise. A null simple hypothesis ƒ ≡ ƒ0 is tested against a composite alternative of the form ƒ- ƒ0 r ≥ ᵨ. Additionally it is assumed that the underlying function ƒ possesses some smoothness properties, namely, that ƒ belongs to some Besov (or Sobolev) ball Bs,p,q(M) = {ƒ : ƒ Bs,p,q ≤ M}. The aim is to evaluate the fastest rate of decay of the radius g to zero as the noise level tends to zero (or, equivalently, as the number of observations tends to infinity) for which testing with prescribed error probabilities is still possible. The ealier results show that the answer depends heavily on the smoothness parameters s,p, q, M. Below we consider the problem of adaptive (assumption free) testing if these parameters are unknown. A test Φ* is proposed which is near minimax and adaptive at the same time. Compared with the optimal (minimax) rate, this test has a performance which is worse within a log log-factor that is inessential but unavoidable payment for adaptation