GAUSSIAN APPROXIMATION OF SUPREMA OF EMPIRICAL PROCESSES
GAUSSIAN APPROXIMATION OF SUPREMA OF EMPIRICAL PROCESSES
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DOI：
10.1214/14-aos1230
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发表时间：
2014-08-01
影响因子：
4.5
通讯作者：
Kato, Kengo
中科院分区：
文献类型：
--
作者：
Chernozhukov, Victor;Chetverikov, Denis;Kato, Kengo
本文提出了用高斯过程的上界序列来逼近一般经验过程的上界的一种新的直接方法，而不采用以超范数逼近整个经验过程的方法。我们证明了一个适用于各种统计问题的抽象逼近定理，例如构造函数的一致置信带。值得注意的是，主逼近定理中的界是非渐近的，并且该定理允许指示经验过程的函数是无界的，并且具有随样本大小发散的熵。逼近定理的证明建立在关于随机向量和的极大值的一个新的耦合不等式的基础上，它的证明依赖于有效地使用Stein的正态逼近方法和一些新的经验处理技巧。我们研究了这一逼近定理在非参数估计中的局部和级数经验过程中的应用，其中函数类随样本量变化且非Donsker。重要的是，在这些例子中，我们的新技术能够证明在弱正则性条件下上确定型统计量的高斯近似，特别是关于带宽和级数函数的数目。
This paper develops a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the sup-norm. We prove an abstract approximation theorem applicable to a wide variety of statistical problems, such as construction of uniform confidence bands for functions. Notably, the bound in the main approximation theorem is nonasymptotic and the theorem allows for functions that index the empirical process to be unbounded and have entropy divergent with the sample size. The proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors, the proof of which depends on an effective use of Stein's method for normal approximation, and some new empirical process techniques. We study applications of this approximation theorem to local and series empirical processes arising in nonparametric estimation via kernel and series methods, where the classes of functions change with the sample size and are non-Donsker. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples.