Summary

In this paper we study the class of penalized regression spline estimators, which enjoy similarities to both regression splines (without penalty and with less knots than data points) and smoothing splines (with knots equal to the data points and a penalty controlling the roughness of the fit). Depending on an assumption on the number of knots, sample size and penalty, we show that the theoretical properties of penalized regression spline estimators are either similar to those of regression splines or to those of smoothing splines, with a clear breakpoint distinguishing the cases. We prove that using less knots results in better asymptotic rates than when using a large number of knots. We obtain expressions for bias and variance and asymptotic rates for the number of knots and penalty parameter.