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 $\mathcal{F}$.
Citation
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
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