Constrained Penalized Splines

The penalized spline is a popular method for function estimation when the assumption of “smoothness” is valid. In this paper, methods for estimation and inference are proposed using penalized splines under the additional constraints of shape, such as monotonicity or convexity. The constrained penalized spline estimator is shown to have the same convergence rates as the corresponding unconstrained penalized spline, although in practice the squared error loss is typically smaller for the constrained versions. The penalty parameter may be chosen with generalized cross-validation, which also provides a method for determining if the shape restrictions hold. The method is not a formal hypothesis test, but is shown to have nice large-sample properties, and simulations show that is compares well with existing tests for monotonicity. Extensions to the partial linear model, the generalized regression model, and the varying coefficient model are given, and several examples demonstrate the utility of the methods.