centre for microdata methods and practice

ESRC centre

cemmap is an ESRC research centre

ESRC

Keep in touch

Subscribe to cemmap news

Gaussian approximation of suprema of empirical processes

Authors: Victor Chernozhukov , Denis Chetverikov and Kengo Kato
Date: 17 December 2012
Type: cemmap Working Paper, CWP44/12
DOI: 10.1920/wp.cem.2012.4412

Abstract

We develop 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 empirical processes themselves in the sup-norm. We prove an abstract approximation theorem that is applicable to a wide variety of problems, primarily in statistics. Especially, the bound in the main approximation theorem is non-asymptotic and the theorem does not require uniform boundedness of the class of functions. 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 processes techniques. We study applications of this approximation theorem to local empirical processes and series estimation in nonparametric regression where the classes of functions change with the sample size and are not Donsker-type. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under considerably weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples.

Download full version
New version:
Victor Chernozhukov, Denis Chetverikov and Kengo Kato August 2016, Gaussian approximation of suprema of empirical processes, cemmap Working Paper, CWP41/16, The IFS
Victor Chernozhukov, Denis Chetverikov and Kengo Kato December 2013, Gaussian approximation of suprema of empirical processes, cemmap Working Paper, CWP75/13, The IFS

Publications feeds

Subscribe to cemmap working papers via RSS

Search cemmap

Search by title, topic or name.

Contact cemmap

Centre for Microdata Methods and Practice

How to find us

Tel: +44 (0)20 7291 4800

E-mail us