Scatterplot smoothers estimate a regression function y =
f(x) by local averaging of the observed data points
(xi, yi). 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,
Cp and generalized maximum likelihood. The latter is
the MLE within a normaltheory empirical Bayes model. We show that
Cp 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 Cp, 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 Cp and MLE that are valid whether or not one
believes in the probability models.