Asymptotic theory of penalized splines

Abstract

The paper gives a unified study of the large sample asymptotic theory of penalized splines including the O-splines using B-splines and an integrated squared derivative penalty [22], the P-splines which use B-splines and a discrete difference penalty [13], and the T-splines which use truncated polynomials and a ridge penalty [24]. Extending existing results for O-splines [7], it is shown that, depending on the number of knots and appropriate smoothing parameters, the $L_{2}$ risk bounds of penalized spline estimators are rate-wise similar to either those of regression splines or to those of smoothing splines and could each attain the optimal minimax rate of convergence [32]. In addition, convergence rate of the $L_{\infty }$ risk bound, and local asymptotic bias and variance are derived for all three types of penalized splines.