Nonparametric signal detection with small type I and type II error probabilities

In the problem of signal detection in the heteroscedastic Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L ₂- norms of kernel estimators. The sets of alternatives are defined by the sets of all signals such that L ₂- norms of signals smoothed by the kernel exceed some constants ({rho_epsilon}) . The constants ({rho_epsilon}) depend on the power ({epsilon}) of noise and ({rho_epsilon to 0}) as ({epsilon to 0}) . The setup is considered in the zone of moderate deviation probabilities. We suppose that type I or type II error probabilities of tests tend to zero as ({epsilon to 0}).