# On asymptotic minimaxity of kernel-based tests

ESAIM: Probability and Statistics (2003)

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

## Access Full Article

top## Abstract

top## How to cite

topErmakov, 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 $\rho _\epsilon > 0$. 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 $\rho _\epsilon > 0$. 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

top- [1] P.J. Bickel and M. Rosenblatt, On Some Global Measures of Deviation of Density Function Estimates. Ann. Statist. 1 (1973) 1071-1095. Zbl0275.62033MR348906
- [2] P. Bickel, C. Klaassen, Y. Ritov and J. Wellner, Efficient and Adaptive Estimation for the Semiparametric Models. John Hopkins University Press, Baltimore (1993). Zbl0786.62001MR1245941
- [3] B.M. Brown, Martingale Central Limit Theorems. Ann. Math. Statist. 42 (1971) 59-66. Zbl0218.60048MR290428
- [4] L.D. Brown and M. Low, Asymptotic Equivalence of Nonparametric Regression and White Noise. Ann. Statist. 24 (1996) 2384-2398. Zbl0867.62022MR1425958
- [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. Zbl0433.60043MR522240
- [6] N.N. Chentsov, Statistical Decision Rules and Optimal Inference. Moskow, Nauka (1972). MR343398
- [7] M.S. Ermakov, Minimax Detection of a Signal in Gaussian White Noise. Theory Probab. Appl. 35 (1990) 667-679. Zbl0744.62117MR1090496
- [8] M.S. Ermakov, On Asymptotic Minimaxity of Rank Tests. Statist. Probab. Lett. 15 (1992) 191-196. Zbl0769.62035MR1190253
- [9] M.S. Ermakov, Minimax Nonparametric Testing Hypotheses on a Density Function. Theory Probab. Appl. 39 (1994) 396-416. Zbl0855.62032MR1347182
- [10] M.S. Ermakov, Asymptotic Minimaxity of Tests of Kolmogorov and Omega-squared Types. Theory Probab. Appl. 40 (1995) 54-67. Zbl0898.62058MR1346730
- [11] M.S. Ermakov, Asymptotic Minimaxity of Chi-squared Tests. Theory Probab. Appl. 42 (1997) 668-695. Zbl0911.62039MR1618797
- [12] M.S. Ermakov, On Distinquishability of Two Nonparametric Sets of Hypotheses. Statist. Probab. Lett. 48 (2000) 275-282. Zbl0959.62039MR1765752
- [13] Y. Fan, Testing Goodness of Fit of a Parametric Density Function by Kernel Method. Econometric Theory 10 (1994) 316-356. MR1293204
- [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. Zbl0544.62036MR733511
- [17] P. Hall, Central Limit Theorem for Integrated Square Error of Multivariate Nonparametric Density Estimators. J. Multivar. Anal. 14 (1984) 1-16. Zbl0528.62028MR734096
- [18] W. Hardle, Applied Nonparametric Regression. Cambridge University Press, Cambridge (1989). Zbl0714.62030MR1161622
- [19] J.D. Hart, Nonparametric Smoothing and Lack-of-fit Tests. Springer-Verlag, New York (1997). Zbl0886.62043MR1461272
- [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 ${l}_{p}$-metrics. Z. Nauchn. Sem. (POMI) 184 (1990) 152-168. Zbl0738.94005MR1098696
- [22] Yu.I. Ingster and I.A. Suslina, Minimax Detection of Signals for Besov Balls and Bodies. Probl. Inform. Transm. 34 (1998) 56-68. Zbl1113.94303MR1654822
- [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. Zbl1013.62049MR1991446
- [24] V.D. Konakov, On a Global Measure of Deviation for an Estimate of the Regression Line. Theor. Probab. Appl. 22 (1977) 858-868. Zbl0391.62030MR483159
- [25] O.V. Lepski and V.G. Spokoiny, Minimax Nonparametric Hypothesis Testing: The Case of an Inhomogeneous Alternative. Bernoulli 5 (1999) 333-358. Zbl0946.62050MR1681702
- [26] M.A. Lifshits, Gaussian Random Functions. TViMS Kiev (1995). Zbl0998.60501MR1472736
- [27] M. Nussbaum, Asymptotic Equivalence of Density Estimation and Gaussian White Noise. Ann. Statist. 24 (1996) 2399-2430. Zbl0867.62035MR1425959
- [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). Zbl0731.62064MR1029526
- [30] D. Slepian, The One-sided Barrier Problem for Gaussian Noise. Bell System Tech. J. 41 (1962) 463-501. MR133183
- [31] V.G. Spokoiny, Adaptive Hypothesis Testing using Wavelets. Ann. Statist. 24 (1996) 2477-2498. Zbl0898.62056MR1425962
- [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. Zbl0074.34801MR84921
- [33] W. Stute, Nonparametric Model Checks for Regression. Ann. Statist. 25 (1997) 613-641. Zbl0926.62035MR1439316

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.