We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant δ. The bound is expressed in the uniform entropy integral of the class at δ. The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of the contrast functions.
"A local maximal inequality under uniform entropy." Electron. J. Statist. 5 192 - 203, 2011. https://doi.org/10.1214/11-EJS605