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 L2 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.