Institute of Mathematical Statistics Lecture Notes - Monograph Series

Empirical processes indexed by estimated functions

Jon A. Wellner Wellner and Aad W. van der Vaart

Full-text: Open access


We consider the convergence of empirical processes indexed by functions that depend on an estimated parameter $\eta$ and give several alternative conditions under which the ``estimated parameter'' $\eta_n$ can be replaced by its natural limit $\eta_0$ uniformly in some other indexing set $\Theta$. In particular we reconsider some examples treated by Ghoudi and Remillard. We recast their examples in terms of empirical process theory, and provide an alternative general view which should be of wide applicability.

Chapter information

Eric A. Cator, Geurt Jongbloed, Cor Kraaikamp, Hendrik P. Lopuhaä, Jon A. Wellner, eds., Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2007), 234-252

First available in Project Euclid: 4 December 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G08: Nonparametric regression 62G20: Asymptotic properties 62F05: Asymptotic properties of tests 62F15: Bayesian inference

delta-method Donsker class entropy integral pseudo observation

Copyright © 2007, Institute of Mathematical Statistics


van der Vaart, Aad W.; Wellner, Jon A. Wellner. Empirical processes indexed by estimated functions. Asymptotics: Particles, Processes and Inverse Problems, 234--252, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000382.

Export citation