On Asymptotic Minimaxity of Kernel-based Tests

Michael Ermakov

ESAIM: Probability and Statistics (2010)

  • 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 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.

How to cite

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Ermakov, Michael. "On Asymptotic Minimaxity of Kernel-based Tests." ESAIM: Probability and Statistics 7 (2010): 279-312. <http://eudml.org/doc/104309>.

@article{Ermakov2010,
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 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. },
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-based tests; kernel estimators},
language = {eng},
month = {3},
pages = {279-312},
publisher = {EDP Sciences},
title = {On Asymptotic Minimaxity of Kernel-based Tests},
url = {http://eudml.org/doc/104309},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Ermakov, Michael
TI - On Asymptotic Minimaxity of Kernel-based Tests
JO - ESAIM: Probability and Statistics
DA - 2010/3//
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 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.
LA - eng
KW - Nonparametric hypothesis testing; kernel-based tests; goodness-of-fit tests; efficiency; asymptotic minimaxity; kernel estimator.; kernel-based tests; kernel estimators
UR - http://eudml.org/doc/104309
ER -

References

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  1. P.J. Bickel and M. Rosenblatt, On Some Global Measures of Deviation of Density Function Estimates. Ann. Statist.1 (1973) 1071-1095.  Zbl0275.62033
  2. P. Bickel, C. Klaassen, Y. Ritov and J. Wellner, Efficient and Adaptive Estimation for the Semiparametric Models. John Hopkins University Press, Baltimore (1993).  Zbl0786.62001
  3. B.M. Brown, Martingale Central Limit Theorems. Ann. Math. Statist.42 (1971) 59-66.  Zbl0218.60048
  4. L.D. Brown and M. Low, Asymptotic Equivalence of Nonparametric Regression and White Noise. Ann. Statist.24 (1996) 2384-2398.  Zbl0867.62022
  5. M.V. Burnashev, On the Minimax Detection of an Inaccurately Known Signal in a White Gaussian Noise. Theory Probab. Appl.24 (1979) 107-119.  Zbl0433.60043
  6. N.N. Chentsov, Statistical Decision Rules and Optimal Inference. Moskow, Nauka (1972).  
  7. M.S. Ermakov, Minimax Detection of a Signal in Gaussian White Noise. Theory Probab. Appl.35 (1990) 667-679.  Zbl0744.62117
  8. M.S. Ermakov, On Asymptotic Minimaxity of Rank Tests. Statist. Probab. Lett.15 (1992) 191-196.  Zbl0769.62035
  9. M.S. Ermakov, Minimax Nonparametric Testing Hypotheses on a Density Function. Theory Probab. Appl.39 (1994) 396-416.  
  10. M.S. Ermakov, Asymptotic Minimaxity of Tests of Kolmogorov and Omega-squared Types. Theory Probab. Appl.40 (1995) 54-67.  Zbl0898.62058
  11. M.S. Ermakov, Asymptotic Minimaxity of Chi-squared Tests. Theory Probab. Appl.42 (1997) 668-695.  Zbl0911.62039
  12. M.S. Ermakov, On Distinquishability of Two Nonparametric Sets of Hypotheses. Statist. Probab. Lett.48 (2000) 275-282.  Zbl0959.62039
  13. Y. Fan, Testing Goodness of Fit of a Parametric Density Function by Kernel Method. Econometric Theory10 (1994) 316-356.  
  14. E. Guerre and P. Lavergne, Minimax Rates for Nonparametric Specification Testing in Regression Models, Working Paper. Toulouse University of Social Sciences, Toulouse, France (1999).  Zbl1033.62042
  15. B.K. Ghosh and Wei-Mion Huang, The Power and Optimal Kernel of the Bickel-Rosenblatt Test for Goodness of Fit. Ann. Statist.19 (1991) 999-1009.  Zbl0741.62044
  16. P. Hall, Integrated Square Error Properties of Kernel Estimators of Regression Function. Ann. Statist.12 (1984) 241-260.  Zbl0544.62036
  17. P. Hall, Central Limit Theorem for Integrated Square Error of Multivariate Nonparametric Density Estimators. J. Multivar. Anal.14 (1984) 1-16.  Zbl0528.62028
  18. W. Hardle, Applied Nonparametric Regression. Cambridge University Press, Cambridge (1989).  Zbl0875.62159
  19. J.D. Hart, Nonparametric Smoothing and Lack-of-fit Tests. Springer-Verlag, New York (1997).  Zbl0886.62043
  20. J.L. Horowitz and V.G. Spokoiny, Adaptive, Rate-optimal Test of Parametric Model against a Nonparametric Alternative, Vol. 542, Preprint. Weierstrass-Institute of Applied Analysis and Stochastic, Berlin (1999).  Zbl1017.62012
  21. Yu.I. Ingster, Minimax Detection of Signal in lp-metrics. Z. Nauchn. Sem. (POMI)184 (1990) 152-168.  Zbl0738.94005
  22. Yu.I. Ingster and I.A. Suslina, Minimax Detection of Signals for Besov Balls and Bodies. Probl. Inform. Transm.34 (1998) 56-68.  Zbl1113.94303
  23. Yu.I. Ingster and I.A. Suslina, Nonparametric Goodness-of-Fit Testing under Gaussian Model. Springer-Verlag, New York, Lecture Notes in Statist. 169.  Zbl1013.62049
  24. V.D. Konakov, On a Global Measure of Deviation for an Estimate of the Regression Line. Theor. Probab. Appl.22 (1977) 858-868.  Zbl0391.62030
  25. O.V. Lepski and V.G. Spokoiny, Minimax Nonparametric Hypothesis Testing: The Case of an Inhomogeneous Alternative. Bernoulli5 (1999) 333-358.  Zbl0946.62050
  26. M.A. Lifshits, Gaussian Random Functions. TViMS Kiev (1995).  Zbl0832.60002
  27. M. Nussbaum, Asymptotic Equivalence of Density Estimation and Gaussian White Noise. Ann. Statist.24 (1996) 2399-2430.  Zbl0867.62035
  28. V.I. Piterbarg, Asymptotic Methods in Theory of Gaussian Proceses and Fields. Moskow University, Moskow (1988).  Zbl0652.60045
  29. J.C.W. Rayner and D.J. Best, Smooth Tests of Goodness of Fit. Oxford University Press, New York (1989).  Zbl0731.62064
  30. D. Slepian, The One-sided Barrier Problem for Gaussian Noise. Bell System Tech. J.41 (1962) 463-501.  
  31. V.G. Spokoiny, Adaptive Hypothesis Testing using Wavelets. Ann. Statist.24 (1996) 2477-2498.  Zbl0898.62056
  32. Ch. Stein, Efficient Nonparametric Testing and Estimation, in Third Berkeley Symp. Math. Statist. and Probab, Vol. 1. Univ. California Press, Berkeley (1956) 187-195.  
  33. W. Stute, Nonparametric Model Checks for Regression. Ann. Statist.25 (1997) 613-641.  Zbl0926.62035

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