In the problem of signal detectionin Gaussian white noisewe show asymptotic minimaxity of kernel-based tests. The test statisticsequal L 2-norms of kernel estimates.The sets of alternatives are essentially nonparametric and are defined asthe sets of all signals such that the L 2-norms of signal smoothed by the kernels exceed some constants pε > 0.The constant pε depends on the power ϵof noise and pε → 0 as ε → 0.Similar statements are proved also if an additional informationon a signal smoothness is given.By theorems on asymptotic equivalence of statistical experimentsthese results are extended to the problems of testing nonparametrichypotheseson density and regression. The exact asymptotically minimaxlower bounds of type II error probabilities are pointed out forall these settings. Similar results are also obtained for the problemsof testing parametric hypotheses versus nonparametric sets of alternatives.
