Smoothing Splines and Shape Restrictions

Constrained smoothing splines are discussed under order restrictions on the shape of the function m. We consider shape constraints of the type$m^{(r)}\geq 0$, i.e. positivity, monotonicity, convexity,... (Here for an integer$r\geq 0m^{(r)}$denotes the rth derivative of m.) The paper contains three results: (1) constrained smoothing splines achieve optimal rates in shape restricted Sobolev classes; (2) they are equivalent to two step procedures of the following type: (a) in a first step the unconstrained smoothing spline is calculated; (b) in a second step the unconstrained smoothing spline is "projected" onto the constrained set. The projection is calculated with respect to a Sobolev-type norm; this result can be used for two purposes, it may motivate new algorithmic approaches and it helps to understand the form of the estimator and its asymptotic properties; (3) the infinite number of constraints can be replaced by a finite number with only a small loss of accuracy, this is discussed for estimation of a convex function.