On Asymptotic Minimaxity of Kernel-based Tests

Free access article

Issue		ESAIM: PS Volume 7, 2003


Page(s)		279 - 312
DOI		10.1051/ps:2003013

ESAIM: P&S, May 2003, Vol. 7, pp. 279-312
DOI: 10.1051/ps:2003013

On Asymptotic Minimaxity of Kernel-based Tests

Michael Ermakov

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

(Received March 21, 2002. Revised February 15, 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 $\rho_\epsilon > 0$ . The constant $\rho_\epsilon$ depends on the power $\epsilon$ of noise and $\rho_\epsilon \to 0$ as $\epsilon \to 0$ . 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

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