On Asymptotic Minimaxity of Kernel-based Tests
Russian Academy of Sciences, Mechanical
Engineering Problem Institute, Bolshoy Pr. V.O. 61, 199178 St.
Petersburg, Russia; email@example.com..
Revised: 15 February 2003
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 pε > 0. The constant pε depends on the power ϵ of noise and pε → 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.
Mathematics Subject Classification: 62G10 / 62G20
Key words: Nonparametric hypothesis testing / kernel-based tests / goodness-of-fit tests / efficiency / asymptotic minimaxity / kernel estimator.
© EDP Sciences, SMAI, 2003