Empirical processes indexed by estimated functions

Aad W. van der Vaart; Jon A. Wellner

doi:10.1214/074921707000000382

Empirical processes indexed by estimated functions

Statistics Theory 2007-09-12 v1 Statistics Theory

Authors: Aad W. van der Vaart , Jon A. Wellner

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Abstract

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 [Asymptotic Methods in Probability and Statistics (1998) 171--197, Fields Inst. Commun. 44 (2004) 381--406]. We recast their examples in terms of empirical process theory, and provide an alternative general view which should be of wide applicability.

Keywords

point processes stochastic processes ruin theory

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@article{arxiv.0709.1013,
  title  = {Empirical processes indexed by estimated functions},
  author = {Aad W. van der Vaart and Jon A. Wellner},
  journal= {arXiv preprint arXiv:0709.1013},
  year   = {2007}
}

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Published at http://dx.doi.org/10.1214/074921707000000382 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

Empirical processes indexed by estimated functions

Abstract

Keywords

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