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April 2001 Selection criteria for scatterplot smoothers
Bradley Efron
Ann. Statist. 29(2): 470-505 (April 2001). DOI: 10.1214/aos/1009210549


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.


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Bradley Efron. "Selection criteria for scatterplot smoothers." Ann. Statist. 29 (2) 470 - 505, April 2001.


Published: April 2001
First available in Project Euclid: 24 December 2001

zbMATH: 1012.62040
MathSciNet: MR1863966
Digital Object Identifier: 10.1214/aos/1009210549

Primary: 62F10 , 62L08

Keywords: C p , choice of smoothing parameter , curved exponential families , Empirical Bayes , GML

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 2 • April 2001
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