Free-knot spline smoothing for functional data

This article introduces free-knot regression spline estimators for the mean and the variance components of a sample of curves. The asymptotic distribution of the mean estimator is derived, and asymptotic confidence bands are constructed. A comparative simulation study shows that free-knot splines estimate salient features of the functions (such as sharp peaks) more accurately than smoothing splines. This adaptive behavior is also illustrated by an analysis of weather data. Key words and phrases: Functional data analysis; Karhunen—Loève decomposi-tion; Longitudinal data analysis; Variance components