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. 1 INTRODUCTION Suppose we wish to estimate a function or a vector of functions subject to shape constraints. The functions could for instance be regression functions, probability density functions, hazard rates, and so on, and the shape constraint could for instance be that the functions are monotone, convex, non-negative, or a combination thereof. We have at our disposal an estimator, but unfortunately this estimator is not guaranteed to satisfy the constraints. Then how to modify this estimator so that the constraints are met? If all relevant information in the sample is already contained in the initial estimator, then the modified estimator should depend on the data only through this initial estimator. Moreover, the modification should be as small as possible, to be measured along some metric on the appropriate function class. Consider for instance the problem of estimating a regression function that is known to be non-decreasing. Mammen [18] proposes a two-step procedure whereby an initial Nadaraya-Watson kernel estimator is isotonized

A projection approach for multiple monotone regression

“…Proof of Proposition 2.1. Our proposition falls as a special case of Lemma 2.3 of [17], we give another proof here which gives more insights onto the projection algorithm.…”

“…The following proposition concerns the asymptotic distribution of the projection, which follows from the general results of Theorem 3.4 in [17].…”

“…Such estimates are intuitive, and can result in reduction of the estimation error compared to that of a naïve initial estimator. Note that a general projection framework for constrained functional parameters, in particular for constrained functions forming a closed convex cone in a Hilbert space, is proposed in [17]. Our approach falls into this general framework; however, our projection algorithm makes use of the fact that the convex cone of a multivariate monotone function is an intersection of a collection of convex cones of univariate monotone functions.…”