On Asymptotic Minimaxity of Kernel-based Tests

Russian Academy of Sciences, Mechanical Engineering Problem Institute, Bolshoy Pr. V.O. 61, 199178 St. Petersburg, Russia; ermakov@random.ipme.ru..

Received: 21 March 2002
Revised: 15 February 2003

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 p_ε > 0. The constant p_ε depends on the power ϵ of noise and p_ε → 0 as ε → 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.