GAUSSIAN APPROXIMATION OF SUPREMA OF EMPIRICAL PROCESSES

被引:144
作者
Chernozhukov, Victor [1 ,2 ]
Chetverikov, Denis [3 ]
Kato, Kengo [4 ]
机构
[1] MIT, Dept Econ, Cambridge, MA 02142 USA
[2] MIT, Ctr Operat Res, Cambridge, MA 02142 USA
[3] Univ Calif Los Angeles, Dept Econ, Los Angeles, CA 90095 USA
[4] Univ Tokyo, Grad Sch Econ, Bunkyo Ku, Tokyo 1130033, Japan
基金
美国国家科学基金会; 日本学术振兴会;
关键词
Coupling; empirical process; Gaussian approximation; kernel estimation; local empirical process; series estimation; supremum; INVARIANCE-PRINCIPLES; UNIFORM CONSISTENCY; DENSITY ESTIMATORS; LIMIT-THEOREMS; RATES; SUMS; INEQUALITY; REGRESSION; LOGARITHM; BOUNDS;
D O I
10.1214/14-AOS1230
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
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.
引用
收藏
页码:1564 / 1597
页数:34
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