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VOL. 55 | 2007 Scale space consistency of piecewise constant least squares estimators – another look at the regressogram
Leif Boysen, Volkmar Liebscher, Axel Munk, Olaf Wittich

Editor(s) Eric A. Cator, Geurt Jongbloed, Cor Kraaikamp, Hendrik P. Lopuhaä, Jon A. Wellner

## 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.

## Information

Published: 1 January 2007
First available in Project Euclid: 4 December 2007

zbMATH: 1176.62033
MathSciNet: MR2459931

Digital Object Identifier: 10.1214/074921707000000274

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

Rights: Copyright © 2007, Institute of Mathematical Statistics

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Vol. 55 • 1 January 2007