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.
References
[2] Boyd, S., and Vandenberghe, L., Convex optimization. Cambridge University Press, Cambridge, 2004. 1058.90049[2] Boyd, S., and Vandenberghe, L., Convex optimization. Cambridge University Press, Cambridge, 2004. 1058.90049
[3] Dudley, R. M. Central limit theorems for empirical measures., Ann. Probab. 6, 6 (1978), 899–929 (1979). 0404.60016 10.1214/aop/1176995384 euclid.aop/1176995384[3] Dudley, R. M. Central limit theorems for empirical measures., Ann. Probab. 6, 6 (1978), 899–929 (1979). 0404.60016 10.1214/aop/1176995384 euclid.aop/1176995384
[4] Giné, E., and Koltchinskii, V. Concentration inequalities and asymptotic results for ratio type empirical processes., Ann. Probab. 34, 3 (2006), 1143–1216. 1152.60021 10.1214/009117906000000070 euclid.aop/1151418495[4] Giné, E., and Koltchinskii, V. Concentration inequalities and asymptotic results for ratio type empirical processes., Ann. Probab. 34, 3 (2006), 1143–1216. 1152.60021 10.1214/009117906000000070 euclid.aop/1151418495
[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. 0748.60004[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. 0748.60004
[7] Massart, P., and Nédélec, É. Risk bounds for statistical learning., Ann. Statist. 34, 5 (2006), 2326–2366. 1108.62007 10.1214/009053606000000786 euclid.aos/1169571799[7] Massart, P., and Nédélec, É. Risk bounds for statistical learning., Ann. Statist. 34, 5 (2006), 2326–2366. 1108.62007 10.1214/009053606000000786 euclid.aos/1169571799
[8] Ossiander, M. A central limit theorem under metric entropy with, L2 bracketing. Ann. Probab. 15, 3 (1987), 897–919. 0665.60036 10.1214/aop/1176992072 euclid.aop/1176992072[8] Ossiander, M. A central limit theorem under metric entropy with, L2 bracketing. Ann. Probab. 15, 3 (1987), 897–919. 0665.60036 10.1214/aop/1176992072 euclid.aop/1176992072
[9] Pollard, D. A central limit theorem for empirical processes., J. Austral. Math. Soc. Ser. A 33, 2 (1982), 235–248. 0504.60023 10.1017/S1446788700018371[9] Pollard, D. A central limit theorem for empirical processes., J. Austral. Math. Soc. Ser. A 33, 2 (1982), 235–248. 0504.60023 10.1017/S1446788700018371
[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. 0741.60001[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. 0741.60001
[11] van de Geer, S. The method of sieves and minimum contrast estimators., Math. Methods Statist. 4, 1 (1995), 20–38. 0831.62029[11] van de Geer, S. The method of sieves and minimum contrast estimators., Math. Methods Statist. 4, 1 (1995), 20–38. 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. MR1385671 0862.60002[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. MR1385671 0862.60002
