A smoothing spline is a nonparametric curve estimate that is defined as the solution to a minimization problem. One problem with this representation is that it obscures the fact that a spline, like most other nonparametric estimates, is a local, weighted average of the observed data. This property has been used extensively to study the limiting properties of kernel estimates and it is advantageous to apply similar techniques to spline estimates. Although equivalent kernels have been identified for a smoothing spline, these functions are either not accurate enough for asymptotic approximations or are restricted to equally spaced points. This paper extends this previous work to understand a spline estimate's local properties. It is shown that the absolute value of the spline weight function decreases exponentially away from its center. 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 function. These bounds are used to derive the asymptotic form for the bias and variance of a first order smoothing spline estimate. The arguments leading to this result can be easily extended to higher order splines.
"Splines as Local Smoothers." Ann. Statist. 23 (4) 1175 - 1197, August, 1995. https://doi.org/10.1214/aos/1176324704