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Scale space consistency of piecewise constant least squares estimators – another look at the regressogram. (English) Zbl 1176.62033

Cator, Eric A. (ed.) et al., Asymptotics: particles, processes and inverse problems. Festschrift for Piet Groeneboom. Papers based on the presentations at the workshop, Delft, The Netherland, July 10–14, 2006. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-71-3/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 55, 65-84 (2007).
Summary: 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.
For the entire collection see [Zbl 1165.60002].

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
41A10 Approximation by polynomials

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