Multiscale generalised linear models for nonparametric function estimation

• 成果类型: Article
• 署名作者: Kolaczyk, ED; Nowak, RD
• 署名单位: Boston University; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/92.1.119
• 发表日期: 2005
• 页码: 119133
• 关键词: wavelet shrinkage; regression

摘要

We present a method for extracting information about both the scale and trend of local components of an inhomogeneous function in a nonparametric generalised linear model. Our multiscale framework combines recursive partitions, which allow for the incorporation of scale in a natural manner, with systems of piecewise polynomials supported on the partition intervals, which serve to summarise the smooth trend within each interval. Our estimators are formulated as solutions of complexity-penalised likelihood optimisations, where the penalty seeks to limit the number of intervals used to model the data. The actual calculation of the estimators may be accomplished using standard software routines for generalised linear models, within the context of efficient, tree-based, polynomial-time algorithms. A risk analysis shows that these estimators achieve the same asymptotic rates in the nonparametric generalised linear model as the classical wavelet-based estimators in the Gaussian 'function plus noise' model, for suitably defined ranges of Besov spaces. Numerical simulations show that the method tends to perform at least as well as, and often better than, alternative wavelet-based methodologies in the context of finite samples, while applications to gamma-ray burst data in astronomy and packet loss data in computer network traffic analysis confirm its practical relevance.