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October 2002 Concentration inequalities, large and moderate deviations for self-normalized empirical processes
Bernard Bercu, Elisabeth Gassiat, Emmanuel Rio
Ann. Probab. 30(4): 1576-1604 (October 2002). DOI: 10.1214/aop/1039548367

Abstract

We consider the supremum $\mathcal{W}_n$ of self-normalized empirical processes indexed by unbounded classes of functions $\mathcal{F}$. Such variables are of interest in various statistical applications, for example, the likelihood ratio tests of contamination. Using the Herbst method, we prove an exponential concentration inequality for $\mathcal{W}_n$ under a second moment assumption on the envelope function of $\mathcal{F}$. This inequality is applied to obtain moderate deviations for $\mathcal{W}_n$. We also provide large deviations results for some unbounded parametric classes $\mathcal{F}$.

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Bernard Bercu. Elisabeth Gassiat. Emmanuel Rio. "Concentration inequalities, large and moderate deviations for self-normalized empirical processes." Ann. Probab. 30 (4) 1576 - 1604, October 2002. https://doi.org/10.1214/aop/1039548367

Information

Published: October 2002
First available in Project Euclid: 10 December 2002

zbMATH: 1021.60013
MathSciNet: MR1944001
Digital Object Identifier: 10.1214/aop/1039548367

Subjects:
Primary: 60E15 , 60F10
Secondary: 62E20 , 62F05

Keywords: Concentration inequalities , Empirical processes , large deviations , logarithmic Sobolev inequalities , Maximal inequalities , Moderate deviations , self-normalized sums

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 4 • October 2002
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