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

We consider the supremum W-n of self-normalized empirical processes indexed by unbounded classes of functions 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 W-n under a second moment assumption on the envelope function of F. This inequality is applied to obtain moderate deviations for W-n. We also provide large deviations results for some unbounded parametric classes F.