Empirical processes indexed by estimated functions
Aad van der Vaart, Jon A. Wellner · Institute of Mathematical Statistics eBooks · 2007
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.