Topics in U-Statistics and Risk Estimation

• Activity Number: 333
• Type: Topic Contributed
• Date/Time: Tuesday, August 2, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Qing Wang*+ and Bruce George Lindsay
• Companies: Penn State University and Penn State University
• Keywords: U-statistics ; best unbiased estimator ; L2 distance ; Kullback-Leibler distance ; two-stage bandwidth selector

Abstract

A major concern with cross-validation estimators is their large variance. This has led to considerable research in other "plug-in" type methodologies. In this work, we consider how cross-validation can be improved. Our key example will be the use of cross-validation bandwidth selection in nonparametric kernel density estimation. Cross-validation estimators can be considered as U-statistic form estimators for the risk that arises from L2 and Kullback-Leibler loss functions. These cross-validation estimators can then be used to select the bandwidth in the kernel density estimator by choosing the bandwidth that has the smallest risk estimate. Our first objective is to better estimate the variance of a U-statistic when, as often occurs in cross validation, the kernel size (subsample size) is large relative to the sample size. We consider a new method to estimate the variance. The proposed variance estimator is the best unbiased estimator and is applicable even when the asymptotic assumption does not hold. Our second objective is to introduce a two-stage bandwidth selector that can help to reduce the variability of the traditional bandwidth selector dramatically.