Gaussian approximation of suprema of empirical processes

Victor Chernozhukov, Denis Chetverikov () and Kengo Kato
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Denis Chetverikov: Institute for Fiscal Studies and UCLA
Kengo Kato: Institute for Fiscal Studies

No CWP41/16, CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies

Abstract: 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 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 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.

Keywords: coupling; empirical process; Gaussian approximation; kernel estimation; local empirical process; series estimation; supremum (search for similar items in EconPapers)
Date: 2016-08-26
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Related works:
Working Paper: Gaussian approximation of suprema of empirical processes (2016)
Working Paper: Gaussian approximation of suprema of empirical processes (2013)
Working Paper: Gaussian approximation of suprema of empirical processes (2013)
Working Paper: Gaussian approximation of suprema of empirical processes (2012)
Working Paper: Gaussian approximation of suprema of empirical processes (2012)
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