LOWER BOUNDS FOR BANDWIDTH SELECTION IN DENSITY-ESTIMATION

• 成果类型: Article
• 署名作者: HALL, P; MARRON, JS
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01192160
• 发表日期: 1991
• 页码: 149-173
• 关键词: cross-validation; CONVERGENCE; derivatives; regression; error

摘要

This paper establishes asymptotic lower bounds which specify, in a variety of contexts, how well (in terms of relative rate of convergence) one may select the bandwidth of a kernel density estimator. These results provide important new insights concerning how the bandwidth selection problem should be considered. In particular it is shown that if the error criterion is Integrated Squared Error (ISE) then, even under very strong assumptions on the underlying density, relative error of the selected bandwidth cannot be reduced below order n-1/10 (as the sample size grows). This very large error indicates that any technique which aims specifically to minimize ISE will be subject to serious practical difficulties arising from sampling fluctuations. Cross-validation exhibits this very slow convergence rate, and does suffer from unacceptably large sampling variation. On the other hand, if the error criterion is Mean Integrated Squared Error (MISE) then relative error of bandwidth selection can be reduced to order n-1/2, when enough smoothness is assumed. Therefore bandwidth selection techniques which aim to minimize MISE can be much more stable, and less sensitive to small sampling fluctuations, than those which try to minimize ISE. We feel this indicates that performance in minimizing MISE, rather than ISE, should become the benchmark for measuring performance of bandwidth selection methods.