Concentration inequalities, large and moderate deviations for self-normalized empirical processes

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 $\mathcal{F}$.