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On Asymptotic Minimaxity of Kernel-based Tests

Published online by Cambridge University Press:  15 May 2003

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

Keywords

Nonparametric hypothesis testingkernel-based testsgoodness-of-fit testsefficiencyasymptotic minimaxitykernel estimator.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 279 - 312
Copyright
© EDP Sciences, SMAI, 2003

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References

Bickel, P.J. and Rosenblatt, M., Some Global Measures, On of Deviation of Density Function Estimates. Ann. Statist. 1 (1973) 1071-1095. CrossRef
P. Bickel, C. Klaassen, Y. Ritov and J. Wellner, Efficient and Adaptive Estimation for the Semiparametric Models. John Hopkins University Press, Baltimore (1993).
Brown, B.M., Martingale Central Limit Theorems. Ann. Math. Statist. 42 (1971) 59-66. CrossRef
Brown, L.D. and Low, M., Asymptotic Equivalence of Nonparametric Regression and White Noise. Ann. Statist. 24 (1996) 2384-2398.
Burnashev, M.V., On the Minimax Detection of an Inaccurately Known Signal in a White Gaussian Noise. Theory Probab. Appl. 24 (1979) 107-119. CrossRef
N.N. Chentsov, Statistical Decision Rules and Optimal Inference. Moskow, Nauka (1972).
Ermakov, M.S., Minimax Detection of a Signal in Gaussian White Noise. Theory Probab. Appl. 35 (1990) 667-679. CrossRef
Ermakov, M.S., Asymptotic Minimaxity, On of Rank Tests. Statist. Probab. Lett. 15 (1992) 191-196. CrossRef
Ermakov, M.S., Minimax Nonparametric Testing Hypotheses on a Density Function. Theory Probab. Appl. 39 (1994) 396-416. CrossRef
Ermakov, M.S., Asymptotic Minimaxity of Tests of Kolmogorov and Omega-squared Types. Theory Probab. Appl. 40 (1995) 54-67.
Ermakov, M.S., Asymptotic Minimaxity of Chi-squared Tests. Theory Probab. Appl. 42 (1997) 668-695.
Ermakov, M.S., Distinquishability, On of Two Nonparametric Sets of Hypotheses. Statist. Probab. Lett. 48 (2000) 275-282. CrossRef
Fan, Y., Testing Goodness of Fit of a Parametric Density Function by Kernel Method. Econometric Theory 10 (1994) 316-356. CrossRef
E. Guerre and P. Lavergne, Minimax Rates for Nonparametric Specification Testing in Regression Models, Working Paper. Toulouse University of Social Sciences, Toulouse, France (1999).
Ghosh, B.K. and Wei-Mion Huang, The Power and Optimal Kernel of the Bickel-Rosenblatt Test for Goodness of Fit. Ann. Statist. 19 (1991) 999-1009. CrossRef
Hall, P., Integrated Square Error Properties of Kernel Estimators of Regression Function. Ann. Statist. 12 (1984) 241-260. CrossRef
Hall, P., Central Limit Theorem for Integrated Square Error of Multivariate Nonparametric Density Estimators. J. Multivar. Anal. 14 (1984) 1-16. CrossRef
W. Hardle, Applied Nonparametric Regression. Cambridge University Press, Cambridge (1989).
J.D. Hart, Nonparametric Smoothing and Lack-of-fit Tests. Springer-Verlag, New York (1997).
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).
Yu.I. Ingster, Minimax Detection of Signal in l p -metrics. Z. Nauchn. Sem. (POMI) 184 (1990) 152-168.
Yu.I. Ingster, I.A. Suslina, Minimax Detection of Signals for Besov Balls and Bodies. Probl. Inform. Transm. 34 (1998) 56-68.
Yu.I. Ingster and I.A. Suslina, Nonparametric Goodness-of-Fit Testing under Gaussian Model. Springer-Verlag, New York, Lecture Notes in Statist. 169 .
Konakov, V.D., On a Global Measure of Deviation for an Estimate of the Regression Line. Theor. Probab. Appl. 22 (1977) 858-868. CrossRef
Lepski, O.V. and Spokoiny, V.G., Minimax Nonparametric Hypothesis Testing: The Case of an Inhomogeneous Alternative. Bernoulli 5 (1999) 333-358. CrossRef
M.A. Lifshits, Gaussian Random Functions. TViMS Kiev (1995).
Nussbaum, M., Asymptotic Equivalence of Density Estimation and Gaussian White Noise. Ann. Statist. 24 (1996) 2399-2430.
V.I. Piterbarg, Asymptotic Methods in Theory of Gaussian Proceses and Fields. Moskow University, Moskow (1988).
J.C.W. Rayner and D.J. Best, Smooth Tests of Goodness of Fit. Oxford University Press, New York (1989).
Slepian, D., The One-sided Barrier Problem for Gaussian Noise. Bell System Tech. J. 41 (1962) 463-501. CrossRef
Spokoiny, V.G., Adaptive Hypothesis Testing using Wavelets. Ann. Statist. 24 (1996) 2477-2498.
Ch. Stein, Efficient Nonparametric Testing and Estimation, in Third Berkeley Symp. Math. Statist. and Probab, Vol. 1. Univ. California Press, Berkeley (1956) 187-195.
Stute, W., Nonparametric Model Checks for Regression. Ann. Statist. 25 (1997) 613-641.