Concentration around the mean for maxima of empirical processes

Abstract

In this paper we give optimal constants in Talagrand's concentration inequalities for maxima of empirical processes associated to independent and eventually nonidentically distributed random variables. Our approach is based on the entropy method introduced by Ledoux.Comment: Published at http://dx.doi.org/10.1214/009117905000000044 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Risk bounds for new M-estimation problems

“…Another point is missing before we apply this result in our context, it is the fact that the result is valid for countable classes of functions, and so, we need to extend the Theorem 1.1 in [11]. We prove the following proposition.…”

“…For this, we propose to use the work of T. Klein and E. Rio [11], in particular Theorem 1.1, that deal with right hand side deviations of the empirical process. They show that for an empirical process W p indexed by a countable class of functions G with values in…”

“…In our purpose, we don't really work with sup g∈G W p (g) but rather with sup g∈G |W p (g)| = W p G corresponding to a two-side control. Hence, according to the work of T. Klein and E. Rio [11], it exists some function ϕ G : R + → [0, 1] decreasing to zero such that for all positive t…”

“…Empirical processes give power tools for evaluating statistical estimation and inference problems. In particular, we use concentration inequalities to derive risk bounds following the work of [20], [15] and [11] among others. Estimation based on minimizing a function was introduced by Huber in 1964 [8] where he proposed generalizing maximum likelihood estimation.…”

“…Inequality (11) can not be applied to our framework because the induced procedure θ n involves the quantity ρ h (θ) intractable for complex models.…”