Functional estimation under shape constraints

Abstract

In the problem of nonparametric regression for a fixed design model, we may want to use additional information about the shape of the regression function, when available, to improve the estimation. The regression function may, for example, be convex or monotone or more generally belong to a cone in some functional space. We devise a method for improving any ordinary consistent estimate by projecting it onto a discretized version of the cone, using the theory of reproducing kernel Hilbert spaces and convex optimization techniques. The initial estimate can be chosen as a smoothing spline or a convolution type kernel estimate. The latter is shown to be mean square consistent in a Sobolev norm sense. The consistency (in the same sense) of the constrained estimate follows.