ABSTRACT
In this chapter we will study smoothing spline estimators for the regression curve. To motivate the concept, suppose that we wish to fit a data set using a function that reflects the key features of the data but retains some degree of smoothness. A natural measure of smoothness associated with a function f E wr(O, 1] is J01 f(m)(t)2dt while a standard measure of goodness-of-fit to the data is the (average) residual sum-of-squares n-1 E~=l (Yi-f(ti)) 2 • Thus, an overall assessment of the quality of a candidate estimator f is provided by the convex sum
for some 0 < q < 1. An "optimal" estimator could then be obtained by minimizing this functional over wr(o, 1]. Upon setting..\= q/(1-q) this becomes equivalent to estimating J.t by the function J.t>. which minimizes
over f E wr(O, 1]. The result is the smoothing spline estimator of the regression function to be studied in this chapter.
