Electronic Journal of Statistics

A local maximal inequality under uniform entropy

Aad van der Vaart and Jon A. Wellner

Full-text: Open access


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.

Article information

Electron. J. Statist., Volume 5 (2011), 192-203.

First available in Project Euclid: 14 April 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Empirical process modulus of continuity minimum contrast estimator rate of convergence


van der Vaart, Aad; Wellner, Jon A. A local maximal inequality under uniform entropy. Electron. J. Statist. 5 (2011), 192--203. doi:10.1214/11-EJS605. https://projecteuclid.org/euclid.ejs/1302784853

Export citation


  • [1] Birgé, L., and Massart, P. Rates of convergence for minimum contrast estimators., Probab. Theory Related Fields 97, 1-2 (1993), 113–150.
  • [2] Boyd, S., and Vandenberghe, L., Convex optimization. Cambridge University Press, Cambridge, 2004.
  • [3] Dudley, R. M. Central limit theorems for empirical measures., Ann. Probab. 6, 6 (1978), 899–929 (1979).
  • [4] Giné, E., and Koltchinskii, V. Concentration inequalities and asymptotic results for ratio type empirical processes., Ann. Probab. 34, 3 (2006), 1143–1216.
  • [5] Kolchins’kiĭ, V. Ī. On the central limit theorem for empirical measures., Teor. Veroyatnost. i Mat. Statist. 24 (1981), 63–75, 152.
  • [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.
  • [7] Massart, P., and Nédélec, É. Risk bounds for statistical learning., Ann. Statist. 34, 5 (2006), 2326–2366.
  • [8] Ossiander, M. A central limit theorem under metric entropy with, L2 bracketing. Ann. Probab. 15, 3 (1987), 897–919.
  • [9] Pollard, D. A central limit theorem for empirical processes., J. Austral. Math. Soc. Ser. A 33, 2 (1982), 235–248.
  • [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.
  • [11] van de Geer, S. The method of sieves and minimum contrast estimators., Math. Methods Statist. 4, 1 (1995), 20–38.
  • [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.