SPLINES AS LOCAL SMOOTHERS

A smoothing spline is a nonparametric curve estimate that is defined a s the solution to a minimization problem. One problem with this repres entation is that it obscures the fact that a spline, like most other n onparametric estimates, is a local, weighted average of the observed d ata. This property has been used extensively to study the limiting pro perties of kernel estimates and it is advantageous to apply similar te chniques to spline estimates. Although equivalent kernels have been id entified for a smoothing spline, these functions are either not accura te enough for asymptotic approximations or are restricted to equally s paced points. This paper extends this previous work to understand a sp line estimate's local properties. It is shown that the absolute value of the spline weight function decreases exponentially away from its ce nter. This result is not asymptotic. The only requirement is that the empirical distribution of the observation points be sufficiently close to a continuous distribution with a strictly positive density functio n. These bounds are used to derive the asymptotic form for the bias an d variance of a first order smoothing spline estimate. The arguments l eading to this result can be easily extended to higher order splines.