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.


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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[1] Birgé, L., and Massart, P. Rates of convergence for minimum contrast estimators., Probab. Theory Related Fields 97 , 1-2 (1993), 113-150. · Zbl 0805.62037
[2] Boyd, S., and Vandenberghe, L., Convex optimization . Cambridge University Press, Cambridge, 2004. · Zbl 1058.90049
[3] Dudley, R. M. Central limit theorems for empirical measures., Ann. Probab. 6 , 6 (1978), 899-929 (1979). · Zbl 0404.60016
[4] Giné, E., and Koltchinskii, V. Concentration inequalities and asymptotic results for ratio type empirical processes., Ann. Probab. 34 , 3 (2006), 1143-1216. · Zbl 1152.60021
[5] Kolchins’kiĭ, V. Ī. On the central limit theorem for empirical measures., Teor. Veroyatnost. i Mat. Statist. 24 (1981), 63-75, 152. · Zbl 0478.60039
[6] Ledoux, M., and Talagrand, M., Probability in Banach spaces , vol. 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, 1991. Isoperimetry and processes. · Zbl 0748.60004
[7] Massart, P., and Nédélec, É. Risk bounds for statistical learning., Ann. Statist. 34 , 5 (2006), 2326-2366. · Zbl 1108.62007
[8] Ossiander, M. A central limit theorem under metric entropy with, L 2 bracketing. Ann. Probab. 15 , 3 (1987), 897-919. · Zbl 0665.60036
[9] Pollard, D. A central limit theorem for empirical processes., J. Austral. Math. Soc. Ser. A 33 , 2 (1982), 235-248. · Zbl 0504.60023
[10] Pollard, D., Empirical processes: theory and applications . NSF-CBMS Regional Conference Series in Probability and Statistics, 2. Institute of Mathematical Statistics, Hayward, CA, 1990. · Zbl 0741.60001
[11] van de Geer, S. The method of sieves and minimum contrast estimators., Math. Methods Statist. 4 , 1 (1995), 20-38. · Zbl 0831.62029
[12] van der Vaart, A. W., and Wellner, J. A., Weak convergence and empirical processes . Springer Series in Statistics. Springer-Verlag, New York, 1996. With applications to statistics. · Zbl 0862.60002
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