The Annals of Statistics

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

Leif Boysen, Angela Kempe, Volkmar Liebscher, Axel Munk, and Olaf Wittich

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Abstract

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.

Article information

Source
Ann. Statist., Volume 37, Number 1 (2009), 157-183.

Dates
First available in Project Euclid: 16 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1232115931

Digital Object Identifier
doi:10.1214/07-AOS558

Mathematical Reviews number (MathSciNet)
MR2488348

Zentralblatt MATH identifier
1155.62034

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 41A25: Rate of convergence, degree of approximation

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

Citation

Boysen, Leif; Kempe, Angela; Liebscher, Volkmar; Munk, Axel; Wittich, Olaf. Consistencies and rates of convergence of jump-penalized least squares estimators. Ann. Statist. 37 (2009), no. 1, 157--183. doi:10.1214/07-AOS558. https://projecteuclid.org/euclid.aos/1232115931


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