For the problem of nonparametric detection of signal in Gaussian white noise we point out strong aasymptotically minimax tests. The sets of alternatives are a ball in Besov space B r 2∞ with "small" balls in L 2 removed.

Abstract. In the problem of signal detection in Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L2-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that the L2-norms of signal smoothed by the kernels exceed some constants ρ > 0. The constant ρ depends on the power of noise and ρ → 0 as → 0. Similar statements are proved also if an additional information on a signal smoothness is given. By theorems on asymptotic equivalence of statistical experiments these results are extended to the problems of testing nonparametric hypotheses on density and regression. The exact asymptotically minimax lower bounds of type II error probabilities are pointed out for all these settings. Similar results are also obtained for the problems of testing parametric hypotheses versus nonparametric sets of alternatives.Mathematics Subject Classification. 62G10, 62G20.

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For the problem of nonparametric estimation of signal in Gaussian noise we point out the strong asymptotically minimax estimators on maxisets for linear estimators (see [10,18]). It turns out that the order of rates of convergence of Pinsker estimator on this maxisets is worse than the order of rates of convergence for the class of linear estimators considered on this maxisets. We show that balls in Sobolev spaces are maxisets for Pinsker estimators.1991 Mathematics Subject Classification. 65M30, 65R30, 62G08, 62J07.