On asymptotic minimaxity of kernel-based tests

Michael Ermakov

ESAIM: Probability and Statistics (2003)

  • Volume: 7, page 279-312
  • ISSN: 1292-8100

Abstract

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In the problem of signal detection in gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L 2 -norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that the L 2 -norms of signal smoothed by the kernels exceed some constants ρ ϵ > 0 . The constant ρ ϵ depends on the power ϵ of noise and ρ 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.

How to cite

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Ermakov, Michael. "On asymptotic minimaxity of kernel-based tests." ESAIM: Probability and Statistics 7 (2003): 279-312. <http://eudml.org/doc/245872>.

@article{Ermakov2003,
abstract = {In the problem of signal detection in gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal $L_2$-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that the $L_2$-norms of signal smoothed by the kernels exceed some constants $\rho _\epsilon &gt; 0$. The constant $\rho _\epsilon $ depends on the power $\epsilon $ of noise and $\rho _\epsilon \rightarrow 0$ as $\epsilon \rightarrow 0$. 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.},
author = {Ermakov, Michael},
journal = {ESAIM: Probability and Statistics},
keywords = {nonparametric hypothesis testing; kernel-based tests; goodness-of-fit tests; efficiency; asymptotic minimaxity; kernel estimator; kernel estimators},
language = {eng},
pages = {279-312},
publisher = {EDP-Sciences},
title = {On asymptotic minimaxity of kernel-based tests},
url = {http://eudml.org/doc/245872},
volume = {7},
year = {2003},
}

TY - JOUR
AU - Ermakov, Michael
TI - On asymptotic minimaxity of kernel-based tests
JO - ESAIM: Probability and Statistics
PY - 2003
PB - EDP-Sciences
VL - 7
SP - 279
EP - 312
AB - In the problem of signal detection in gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal $L_2$-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that the $L_2$-norms of signal smoothed by the kernels exceed some constants $\rho _\epsilon &gt; 0$. The constant $\rho _\epsilon $ depends on the power $\epsilon $ of noise and $\rho _\epsilon \rightarrow 0$ as $\epsilon \rightarrow 0$. 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.
LA - eng
KW - nonparametric hypothesis testing; kernel-based tests; goodness-of-fit tests; efficiency; asymptotic minimaxity; kernel estimator; kernel estimators
UR - http://eudml.org/doc/245872
ER -

References

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