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

Boysen, Leif, Universit¨at Göttingen; Liebscher, Volkmar, Universit¨at Greifswald; Munk, Axel, Universit¨at G¨ottingen; Wittich, Olaf, Technische Universiteit Eindhoven

doi:10.1214/074921707000000274

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

IMS Lecture Notes Monogr. Ser., 2007: 65-84 (2007) DOI: 10.1214/074921707000000274

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.