Tensor product space ANOVA models

• 成果类型: Article
• 署名作者: Lin, Y
• 署名单位: University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1015951996
• 发表日期: 2000
• 页码: 734-755
• 关键词: penalized likelihood; Nonparametric Regression; asymptotic equivalence; DENSITY-ESTIMATION; white-noise; CONVERGENCE; rates

摘要

To deal with the curse of dimensionality in high-dimensional nonparametric problems, we consider using tensor product space ANOVA models, which extend the popular additive models and are able to capture interactions of any order. The multivariate function is given an ANOVA decomposition, that is, it is expressed as a constant plus the sum of functions of one variable (main effects), plus the sum of functions of two variables (two-factor interactions) and so on. We assume the interactions to be in tensor product spaces. We show in both regression and white noise settings, the optimal rate of convergence for the TPS;ANOVA model is within a fog factor of the one-dimensional optimal rate, and that the penalized likelihood estimator in TPS-ANOVA achieves this rate of convergence. The quick optimal rate of the TPS-ANOVA model makes it very preferable in high-dimensional function estimation. Many properties of the tensor product space of Sobolev-Hilbert spaces are also given.