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Boysen, L.* ; Kempe, A. ; Liebscher, V.* ; Munk, A.* ; Wittich, O.*

Consistencies and rates of convergence of jump-penalized least squares estimators.

Ann. Stat. 37, 157-183 (2009)
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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.
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Publication type Article: Journal article
Document type Scientific Article
Keywords Jump detection; adaptive estimation; penalized maximum likelihood; approximation spaces; change-point analysis; multiscale resolution analysis; Potts functional; nonparametric regression; regressogram; Skorokhod topology; variable selection; large underdetermined systems; nonparametric regression; bayesian restoration; smooth regression; wavelet shrinkage; change-points; selection; equations; sequence
ISSN (print) / ISBN 0090-5364
Quellenangaben Volume: 37, Issue: 1, Pages: 157-183 Article Number: , Supplement: ,
Publisher Institute of Mathematical Statistics (IMS)
Reviewing status Peer reviewed