CONSISTENCIES AND RATES OF CONVERGENCE OF JUMP-PENALIZED LEAST SQUARES ESTIMATORS

• 成果类型: Article
• 署名作者: Boysen, Leif; Kempe, Angela; Liebscher, Volkmar; Munk, Axel; Wittich, Olaf
• 署名单位: University of Gottingen; Helmholtz Association; Helmholtz-Center Munich - German Research Center for Environmental Health; Universitat Greifswald; Eindhoven University of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/07-AOS558
• 发表日期: 2009
• 页码: 157-183
• 关键词: large underdetermined systems; bayesian restoration; change-points; regression; shrinkage; smoothers; EQUATIONS; sequence

摘要

We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in L-2([0, 1)) our results cover other metrics like Skorokhod metric on the space of cadlag functions and uniform metrics on C([0, 1]). We will show that these estimators are in an adaptive sense rate optimal over certain classes of approximation spaces. Special cases are the class of functions of bounded variation (piecewise) Holder continuous functions of order 0 < alpha <= 1 and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.