Local Asymptotics of P-Spline Smoothing

TL;DR

This paper investigates the asymptotic behavior of penalized spline estimators, showing their approximation by kernel smoothing and establishing their normality, with results independent of spline degree and knot count.

Contribution

It provides a comprehensive analysis of the asymptotic properties of penalized splines, including kernel approximation and boundary behavior, for arbitrary spline degrees and difference penalties.

Findings

Penalized splines asymptotically behave like kernel smoothers.

Asymptotic normality is established at interior points.

Convergence rate is unaffected by spline degree and knot number, given sufficient growth.

Abstract

This paper addresses asymptotic properties of general penalized spline estimators with an arbitrary B-spline degree and an arbitrary order difference penalty. The estimator is approximated by a solution of a linear differential equation subject to suitable boundary conditions. It is shown that, in certain sense, the penalized smoothing corresponds approximately to smoothing by the kernel method. The equivalent kernels for both inner points and boundary points are obtained with the help of Green's functions of the differential equation. Further, the asymptotic normality is established for the estimator at interior points. It is shown that the convergence rate is independent of the degree of the splines, and the number of knots does not affect the asymptotic distribution, provided that it tends to infinity fast enough.