Shape constrained smoothing using smoothing splines

In some regression settings one would like to combine the flexibility of non-parametric smoothing with some prior knowledge about the regression curve. Such prior knowledge may come from a physical or economic theory, leading to shape constraints such as the underlying regression curve being positive, monotone, convex or concave. We propose a new method for calculating smoothing splines that fulfill these kinds of constraints. Our approach leads to a quadratic programming problem and the infinite number of constraints are replaced by a finite number of constraints that are chosen adaptively. We show that the resulting problem can be solved using the algorithm of Goldfarb and Idnani (1982, 1983) and illustrate our method on several real data sets.