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比率型经验过程的集中不等式及其渐近结果

Concentration inequalities and asymptotic results for ratio type empirical processes

Evarist Giné, Vladimir Koltchinskii

备注：
Published at http://dx.doi.org/10.1214/009117906000000070 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：
Annals of Probability 2006, Vol. 34, No. 3, 1143-1216
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摘要：

设$\mathcal{F}$是可测空间$(S,\mathcal{S})$上的一类可测函数，其值在$[0,1]$中，设\[P_n=n^{-1}\sum_{i=1}^nδ_{X_i}\]是基于i.i.d.从$(S,\mathcal{S})$上的概率分布$P$采样$(X_1,...,X_n)$。我们研究了以下类型的上确界的行为：\[\sup_{r_n<σ_Pf\leq δ_n}\frac{|P_nf-Pf|}{ϕ(σ_Pf)},\]其中$σ_Pf\ge\operatorname {Var}^{1/2}_Pf$和$ϕ$是连续的，严格递增的函数，具有$ϕ(0)=0$。使用Talagrand的浓度不等式的经验过程，我们建立这样的上确界的浓度不等式，并使用它们来推导出几个结果，其渐近行为，表达的条件在期望的局部上确界的经验过程。我们还证明了新的期望值的上限规范的经验过程中最大的$σ_Pf$和$L_2(P)$规范的包络的函数类，这是特别适合于估计本地化的上确界。利用这种技术，我们扩展到函数类的比例型上确界的经验过程，包括一些亚历山大的VC类集的结果的已知结果。我们还考虑应用这些结果的几个重要问题，在非参数统计和学习理论（包括一般的超额风险界限的经验风险最小化和他们的版本$L_2$ -回归和分类和比率类型界限的利润率分布分类）。

英文摘要：

Let $\mathcal{F}$ be a class of measurable functions on a measurable space $(S,\mathcal{S})$ 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 $(X_1,...,X_n)$ from a probability distribution $P$ on $(S,\mathcal{S})$. We study the behavior of suprema of the following type: \[\sup_{r_n<σ_Pf\leq δ_n}\frac{|P_nf-Pf|}{ϕ(σ_Pf)},\] where $σ_Pf\ge\operatorname {Var}^{1/2}_Pf$ 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 $σ_Pf$ and the $L_2(P)$ 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 $L_2$-regression and classification and ratio type bounds for margin distributions in classification).

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