Open Access
February 2009 Consistencies and rates of convergence of jump-penalized least squares estimators
Leif Boysen, Angela Kempe, Volkmar Liebscher, Axel Munk, Olaf Wittich
Ann. Statist. 37(1): 157-183 (February 2009). DOI: 10.1214/07-AOS558


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 L2([0, 1)) our results cover other metrics like Skorokhod metric on the space of càdlàg 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) Hölder continuous functions of order 0<α≤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.


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Leif Boysen. Angela Kempe. Volkmar Liebscher. Axel Munk. Olaf Wittich. "Consistencies and rates of convergence of jump-penalized least squares estimators." Ann. Statist. 37 (1) 157 - 183, February 2009.


Published: February 2009
First available in Project Euclid: 16 January 2009

zbMATH: 1155.62034
MathSciNet: MR2488348
Digital Object Identifier: 10.1214/07-AOS558

Primary: 62G05 , 62G20
Secondary: 41A10 , 41A25

Keywords: adaptive estimation , approximation spaces , change-point analysis , jump detection , multiscale resolution analysis , Nonparametric regression , penalized maximum likelihood , Potts functional , regressogram , Skorokhod topology , Variable selection

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 1 • February 2009
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