Selection criteria for scatterplot smoothers

• 成果类型: Article
• 署名作者: Efron, B
• 署名单位: Stanford University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1009210549
• 发表日期: 2001
• 页码: 470-504
• 关键词: generalized cross-validation; maximum-likelihood; regression; parameter

摘要

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 data-based 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 normal-theory 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 one-parameter curved exponential family. Exponential family theory facilitates a finite-sample 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.