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May 2006 Concentration inequalities and asymptotic results for ratio type empirical processes
Evarist Giné, Vladimir Koltchinskii
Ann. Probab. 34(3): 1143-1216 (May 2006). DOI: 10.1214/009117906000000070


Let ℱ be a class of measurable functions on a measurable space $(S,\mathscr{S})$ with values in [0,1] and let $$P_n=n^{−1}\sum_{i=1}^nδ{X_i}$$ be the empirical measure based on an i.i.d. sample (X1,…,Xn) from a probability distribution P on $(S,\mathscr{S})$. We study the behavior of suprema of the following type: $$\sup_{r_{n}<\sigma_{P}f\leq \delta_{n}}\frac{|P_{n}f-Pf|}{\phi(\sigma_{P}f)},$$ where σPf≥Var1/2Pf and ϕ is a continuous, strictly increasing function with ϕ(0)=0. Using Talagrand’s concentration inequality for empirical processes, we establish concentration inequalities for such suprema and use them to derive several results about their asymptotic behavior, expressing the conditions in terms of expectations of localized suprema of empirical processes. We also prove new bounds for expected values of sup-norms of empirical processes in terms of the largest σPf and the L2(P) norm of the envelope of the function class, which are especially suited for estimating localized suprema. With this technique, we extend to function classes most of the known results on ratio type suprema of empirical processes, including some of Alexander’s results for VC classes of sets. We also consider applications of these results to several important problems in nonparametric statistics and in learning theory (including general excess risk bounds in empirical risk minimization and their versions for L2-regression and classification and ratio type bounds for margin distributions in classification).


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Evarist Giné. Vladimir Koltchinskii. "Concentration inequalities and asymptotic results for ratio type empirical processes." Ann. Probab. 34 (3) 1143 - 1216, May 2006.


Published: May 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1152.60021
MathSciNet: MR2243881
Digital Object Identifier: 10.1214/009117906000000070

Primary: 60E15
Secondary: 60F15 , 60F17 , 62G08 , 68T10

Keywords: ‎classification‎ , Concentration inequalities , localized sup-norms , moment bounds for empirical processes , Nonparametric regression , ratio limit theorems , Ratio type empirical processes , VC type classes , weighted central limit theorems

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 3 • May 2006
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