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Description:
The present paper continues studying the problem of minimax nonparametric hypothesis testing started in Lepski and Spokoiny (1995). The null hypothesis assumes that the function observed with a noise is identically zero i.e. no signal is present. The alternative is composite and minimax: the function is assumed to be separated away from zero in an integral (L2-) norm and also to possess some smoothness properties. The minimax rate of testing for this problem was evaluated by Ingster for the case of Sobolev smoothness classes. Then this problem was studied by Lepski and Spokoiny in the sutiation of an alternative with inhomogeneous smoothness properties that leads to considering Besov smoothness classes. But for both cases the optimal rate and the structure of optimal (in rate) tests depends on smoothness parameters which are usually unknown in practical applications. In this paper the problem of adaptive (assumption free) testing is considered. It is shown that the adaptation without loss of efficiency is impossible. An extra (log log)-factor is nonsignificant but unavoidable payment for the adaptation. A simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of Besov classes.