Abstract: In this paper, the asymptotic behavior of penalized spline estimators is studied using bivariate splines over triangulations and an energy functional as the penalty. A convergence rate for the penalized spline estimators is derived that achieves the optimal nonparametric convergence rate established by [#!25!#]. The asymptotic normality of the proposed estimators is established and shown to hold uniformly over the points where the regression function is estimated. The size of the asymptotic conditional variance is evaluated, and a simple expression for the asymptotic variance is given. Simulation experiments have provided strong evidence that corroborates the asymptotic theory. A comparison with thin-plate splines is provided to illustrate some advantages of this spline smoothing approach.
Key words and phrases: Asymptotic normality, least squares, penalty, spline, triangulation.