Isotonic smoothing spline regression

This article considers the monotone smoothing spline regression problem. We add, as the smoothness penalty term, the integral of the squared second derivative of the regression function. The objective function is minimized over the space of natural cubic splines with knots at the design points. We give a necessary and sufficient condition for a cubic function to be nondecreasing over an interval. Estimation of the unknown parameters is formulated into a second-order cone programming problem. The resulting estimated regression function is nondecreasing in the whole domain and also has enough smoothness. Simulation results suggests the method performs well and we illustrate the method using the ASA car data.