Selection criteria for scatterplot smoothers

Scatterplot smoothers estimate a regression function y = f(x) by local aver aging of the observed data points (x(i), y(i)). In using a smoother, the st atistician must choose a "window width," a crucial smoothing parameter that says just how locally the averaging is done. This paper concerns the data- based choice of a smoothing parameter for splinelike smoothers, focusing on the comparison of two popular methods, C-p and generalized maximum likelih ood. The latter is the MLE within a normal-theory empirical Bayes model. We show that C-p is also maximum likelihood within a closely related nonnorma l family, both methods being examples of a class of selection criteria, Eac h member of the class is the MLE within its own one-parameter curved expone ntial family. Exponential family theory facilitates a finite-sample nonasym ptotic comparison of the criteria. In particular it explains the eccentric behavior of C-p, which even in favorable circumstances can easily select sm all window widths and wiggly estimates of f(x). The theory leads to simple geometric pictures of both C-p and MLE that are valid whether or not one be lieves in the probability models.