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

Abstract

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 $${\epsilon \to 0}$$ .