TY - JOUR

T1 - On asymptotic minimaxity of kernel–based tests

AU - Ermakov, Michael

N1 - Copyright: Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2003/5

Y1 - 2003/5

N2 - 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 ρε > 0. The constant ρε depends on the power ε of noise and ρε → 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.

AB - 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 ρε > 0. The constant ρε depends on the power ε of noise and ρε → 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.

KW - Asymptotic minimaxity

KW - Efficiency

KW - Goodness-of-fit tests

KW - Kernel estimator

KW - Kernel-based tests

KW - Nonparametric hypothesis testing

UR - http://www.scopus.com/inward/record.url?scp=84996132729&partnerID=8YFLogxK

U2 - 10.1051/ps:2003013

DO - 10.1051/ps:2003013

M3 - Article

AN - SCOPUS:84996132729

VL - 7

SP - 279

EP - 312

JO - ESAIM - Probability and Statistics

JF - ESAIM - Probability and Statistics

SN - 1292-8100

ER -