Consistencies and rates of convergence of jump-penalized least squares estimators

Details Consistencies and rates of convergence of jump-penalized least squares estimators Consistencies and rates of convergence of jump-penalized least squares estimators

2009-02-27

arxiv.org/abs/0902.4838v1

Annals of Statistics 2009, Vol. 37, No. 1, 157-183

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/07-AOS558 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/07-AOS558

We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in $L^2([0,1))$ our results cover other metrics like Skorokhod metric on the space of c\`{a}dl\`{a}g functions and uniform metrics on $C([0,1])$. We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) H\"{o}lder continuous functions of order $0<\alpha\le1$ and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.

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