Spline Smoothing

In the kernel methods of Chapter 2, the estimator of a function is defined first and then a measure of precision is invoked to assess how close the estimator is to the true function. In this chapter, this is reversed. The starting point is a precision criterion, and the spline smoother is the result of minimizing it. Before presenting and developing the machinery of smoothing splines, it is worthwhile to introduce interpolating splines. This parallels the discussion of Early smoothers and the considerations leading to local smoothers, thereby giving insight into the formulation of smoothing splines. Like the kernel-based methods of the last chapter, splines suffer from the Curse of Dimensionality. Nevertheless, there is a parallel theory for multivariate splines. Eubank (1988), Chapter 6.2.3 touches on it with some references. Such Laplacian smoothing splines are neglected here, as are partial splines, which generalize splines to include an extra nonparametric component.