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

TL;DR

This paper analyzes the asymptotic behavior and convergence rates of jump-penalized least squares estimators for piecewise constant functions, demonstrating their optimality and addressing automatic smoothing parameter selection.

Contribution

It provides new asymptotic results and convergence rates for jump-penalized estimators across various metrics and function classes, including adaptive optimality and automatic parameter choice.

Findings

Establishes consistency and convergence rates in multiple metrics.

Shows adaptive rate optimality over classes like bounded variation and Hölder functions.

Addresses automatic selection of the smoothing parameter.

Abstract

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.