ADAPTIVE HYPOTHESIS-TESTING USING WAVELETS

Let a function f be observed with a noise. We wish to test the null hy pothesis that the function is identically zero, against a composite no nparametric alternative: functions from the alternative set are separa ted away from zero in an integral (e.g., L(2)) norm and also possess s ome smoothness properties. The minimax rate of testing for this proble m was evaluated in earlier papers by Ingster and by Lepski and Spokoin y under different kinds of smoothness assumptions. It was shown that b oth the optimal rate of testing and the structure of optimal (in rate) tests depend on smoothness parameters which are usually unknown in pr actical applications. In this paper the problem of adaptive (assumptio n free) testing is considered. It is shown that adaptive testing witho ut loss of efficiency is impossible, An extra log log-factor is inesse ntial but unavoidable payment for the adaptation. A simple adaptive te st based on wavelet technique is constructed which is nearly minimax f or a wide range of Besov classes.