Empirical processes indexed by estimated functions

TL;DR

This paper studies the convergence behavior of empirical processes indexed by functions depending on estimated parameters, providing conditions for replacing estimated parameters with their limits uniformly across index sets.

Contribution

It offers new conditions for empirical process convergence involving estimated parameters and recasts existing examples within a broader empirical process framework.

Findings

Established alternative conditions for replacing estimated parameters with limits

Reinterpreted previous examples using empirical process theory

Provided a general, widely applicable framework for empirical process convergence

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.