A Local Maximual Inequality under Uniform Entropy

  Abstract: 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. AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35. Keywords and phrases: Empirical process, modulus of continuity, minimum contrast estimator, rate of convergence.