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 de rivative of m.) The paper contains three results: (1) constrained smoothing splines achieve optimal rates in shape restricted Sobolev classes; (2) the y are equivalent to two step procedures of the following type: (a) in a fir st step the unconstrained smoothing spline is calculated; (b) in a second s tep 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 appro aches and it helps to understand the form of the estimator and its asymptot ic 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 est imation of a convex function.