Institute of Mathematical Statistics Lecture Notes - Monograph Series

Scale space consistency of piecewise constant least squares estimators – another look at the regressogram

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

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Abstract

We study the asymptotic behavior of piecewise constant least squares regression estimates, when the number of partitions of the estimate is penalized. We show that the estimator is consistent in the relevant metric if the signal is in $L^2([0,1])$, the space of càdlàg functions equipped with the Skorokhod metric or $C([0,1])$ equipped with the supremum metric. Moreover, we consider the family of estimates under a varying smoothing parameter, also called scale space. We prove convergence of the empirical scale space towards its deterministic target.

Chapter information

Source
Eric A. Cator, Geurt Jongbloed, Cor Kraaikamp, Hendrik P. Lopuhaä, Jon A. Wellner, eds., Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2007), 65-84

Dates
First available in Project Euclid: 4 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.lnms/1196797068

Digital Object Identifier
doi:10.1214/074921707000000274

Mathematical Reviews number (MathSciNet)
MR2459931

Zentralblatt MATH identifier
1176.62033

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
Hard thresholding nonparametric regression penalized maximum likelihood regressogram scale spaces Skorokhod topology

Rights
Copyright © 2007, Institute of Mathematical Statistics

Citation

Boysen, Leif; Liebscher, Volkmar; Munk, Axel; Wittich, Olaf. Scale space consistency of piecewise constant least squares estimators – another look at the regressogram. Asymptotics: Particles, Processes and Inverse Problems, 65--84, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000274. https://projecteuclid.org/euclid.lnms/1196797068


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