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A local maximal inequality under uniform entropy. (English) Zbl 1268.60027
Summary: 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 \(\delta \). The bound is expressed in the uniform entropy integral of the class at \(\delta \). The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of contrast functions.

MSC:
60E15 Inequalities; stochastic orderings
60F17 Functional limit theorems; invariance principles
60G05 Foundations of stochastic processes
62G20 Asymptotic properties of nonparametric inference
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