A reproducing kernel perspective of smoothing spline estimators

Spline functions have a long history as smoothers of noisy time series data, and several equivalent kernel representations have been proposed in terms of the Green's function solving the related boundary value problem. In this study we make use of the reproducing kernel property of the Green's function to obtain an hierarchy of time-invariant spline kernels of different order. The reproducing kernels give a good representation of smoothing splines for medium and long length filters, with a better performance of the asymmetric weights in terms of signal passing, noise suppression and revisions. Empirical comparisons of time-invariant filters are made with the classical non linear ones. The former are shown to loose part of their optimal properties when we fixed the length of the filter according to the noise to signal ratio as done in nonparametric seasonal adjustment procedures.