Selection criteria for scatterplot smoothers

Abstract

Scatterplot smoothers estimate a regression function y = f(x) by local averaging of the observed data points (x_i, y_i). In using a smoother, the statistician must choose a “window width,” a crucial smoothing parameter that says just how locally the averaging is done. This paper concerns the data­based choice of a smoothing parameter for splinelike smoothers, focusing on the comparison of two popular methods, C_p and generalized maximum likelihood. The latter is the MLE within a normal­theory empirical Bayes model. We show that C_p is also maximum likelihood within a closely related nonnormal family, both methods being examples of a class of selection criteria. Each member of the class is the MLE within its own one­parameter curved exponential family. Exponential family theory facilitates a finite­sample nonasymptotic comparison of the criteria. In particular it explains the eccentric behavior of C_p, which even in favorable circumstances can easily select small 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 believes in the probability models.