Scale space consistency of piecewise constant least squares estimators – another look at the regressogram
Leif Boysen, Volkmar Liebscher, Axel Munk, Olaf Wittich
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.
Full-text: Open access
Permanent link to this document: http://projecteuclid.org/euclid.lnms/1196797068
Digital Object Identifier: doi:10.1214/074921707000000274
Institute of Mathematical Statistics Lecture Notes - Monograph Series