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 $\mathcal{F}$.
@article{1039548367,
author = {Bercu, Bernard and Gassiat, Elisabeth and Rio, Emmanuel},
title = {Concentration inequalities, large and moderate deviations for self-normalized empirical processes},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 1576-1604},
language = {en},
url = {http://dml.mathdoc.fr/item/1039548367}
}
Bercu, Bernard; Gassiat, Elisabeth; Rio, Emmanuel. Concentration inequalities, large and moderate deviations for self-normalized empirical processes. Ann. Probab., Tome 30 (2002) no. 1, pp. 1576-1604. http://gdmltest.u-ga.fr/item/1039548367/