Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Chernozhukov, Victor^1,2  Chetverikov, Denis³  Kato, Kengo⁴ 

[1] MIT, Dept Econ, 50 Mem Dr, Cambridge, MA 02142 USA

[2] MIT, Ctr Stat, 50 Mem Dr, Cambridge, MA 02142 USA

[3] Univ Calif Los Angeles, Dept Econ, Bunche Hall,315 Portola Plaza, Los Angeles, CA 90095 USA

[4] Univ Tokyo, Grad Sch Econ, Bunkyo Ku, 7-3-1 Hongo, Tokyo 1130033, Japan

关键词: Coupling;    Empirical process;    Multiplier bootstrap process;    Empirical bootstrap process;    Gaussian approximation;    Supremum;   

DOI  :  10.1016/j.spa.2016.04.009

来源: Elsevier

【 摘 要 】

We derive strong approximations to the supremum of the non-centered empirical process indexed by a possibly unbounded VC-type class of functions by the suprema of the Gaussian and bootstrap processes. The bounds of these approximations are non-asymptotic, which allows us to work with classes of functions whose complexity increases with the sample size. The construction of couplings is not of the Hungarian type and is instead based on the Slepian-Stein methods and Gaussian comparison inequalities. The increasing complexity of classes of functions and non-centrality of the processes make the results useful for applications in modern nonparametric statistics (Gine and Nickl 2015), in particular allowing us to study the power properties of nonparametric tests using Gaussian and bootstrap approximations. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free