Asymptotic properties of penalized splines for functional data

Abstract

Penalized spline methods are popular for functional data analysis but their asymptotic properties have not been established. We present a theoretic study of the $L_{2}$ and uniform convergence of penalized splines for estimating the mean and covariance functions of functional data under general settings. The established convergence rates for the mean function estimation are mini-max rate optimal and the rates for the covariance function estimation are comparable to those using other smoothing methods.