Projection Estimates of Constrained Functional Parameters

Abstract

Curve estimation problems can often be formulated in terms of a closed and convex parameter set embedded in a real Hilbert space. This is the case, for instance, if the curve of interest is a monotone or convex density or regression function, the support function of a convex set, or the Pickands dependence function of an extreme-value copula. The topic of this paper is the estimator that results when an arbitrary initial estimator possibly falling outside the parameter set is projected onto this parameter set. If direct computation of the projection is infeasible, the full parameter set can be replaced by an approximating sequence of finite-dimensional subsets. Asymptotic properties of the initial estimator sequence in the Hilbert space topology transfer easily to those of the projected sequence and its approximating sequence.