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)) greater than or equal to 0, i.e. positivity, monotonicity, convexity,.... (Here for an integer r greater than or equal to 0, m((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.