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 CWP44/12, CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies

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.

Date: 2012-12-17
New Economics Papers: this item is included in nep-ecm
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Working Paper: Gaussian approximation of suprema of empirical processes (2012)
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