Local invariance principles and their application to density estimation
Local invariance principles and their application to density estimation
复制标题
局部不变性原理及其在密度估计中的应用
DOI：
10.1007/bf01311347
复制
发表时间：
1994
影响因子：
2
通讯作者：
E. Rio
中科院分区：
文献类型：
--
作者：
E. Rio
摘要Letx 1，.，xn是在[0，1]d上具有均匀分布的独立随机变量，X（n）是与tox 1，.相关的中心化和归一化经验过程，xn。给定一个从[0，1]d到R的有界变差的有界函数的Vapnik-Chervonenkis类L，利用Komlós，Major和Tusnády的一维并矢格式，得到经验过程{E（n）（h）：h∈L}的达德利一致中心极限定理中的最佳可能速率.当L满足一定的条件时，我们证明了存在以L为指标的布朗桥序列Bn，使得其中K（L）表示L中元素的最大变差.然后，这一结果适用于最大偏差分布的核密度估计的最小假设下的序列的带宽参数。我们还得到了一些结果，强逼近的经验过程的集合类一致小周长索引。例如，从Beck的论文中可以得出，上述结果是最优的，直到一个可能的因子
$$\sqrt {\log n}$$
当L是半径小于r的欧氏球类时，本文给出了一个新的欧氏球类.
SummaryLetx1,...,xn be independent random variables with uniform distribution over [0, 1]d, andX(n) be the centered and normalized empirical process associated tox1,...,xn. Given a Vapnik-Chervonenkis classL of bounded functions from [0, 1]d intoR of bounded variation, we apply the one-dimensional dyadic scheme of Komlós, Major and Tusnády to get the best possible rate in Dudley's uniform central limit theorem for the empirical process {E(n)(h):h∈L}. WhenL fulfills some extra condition, we prove there exists some sequenceBn of Brownian bridges indexed byL such that whereK (L) denotes the maximal variation of the elements ofL. This result is then applied to maximal deviations distributions for kernel density estimators under minimal assumptions on the sequence of bandwith parameters. We also derive some results concerning strong approximations for empirical processes indexed by classes of sets with uniformly small perimeter. For example, it follows from Beck's paper that the above result is optimal, up to a possible factor
$$\sqrt {\log n}$$
, whenL is the class of Euclidean balls with radius less thanr.