Asymptotics for change-point models under varying degrees of mis-specification

TL;DR

This paper investigates the asymptotic behavior of change-point estimators under different levels of model mis-specification, bridging the gap between correctly specified models and completely mis-specified smooth curves.

Contribution

It introduces a unified framework to analyze change-point estimators' limits under varying degrees of mis-specification, revealing a family of intermediate asymptotic regimes.

Findings

Identifies how the convergence rate varies with mis-specification degree

Derives a family of intermediate limit distributions

Provides insights into the transition between extreme asymptotic behaviors

Abstract

Change-point models are widely used by statisticians to model drastic changes in the pattern of observed data. Least squares/maximum likelihood based estimation of change-points leads to curious asymptotic phenomena. When the change-point model is correctly specified, such estimates generally converge at a fast rate ($n$) and are asymptotically described by minimizers of jump process. Under complete mis-specification by a smooth curve, i.e. when a change-point model is fitted to data described by a smooth curve, the rate of convergence slows down to $n^{1/3}$ and the limit distribution changes to that of the minimizer of a continuous Gaussian process. In this paper we provide a bridge between these two extreme scenarios by studying the limit behavior of change-point estimates under varying degrees of model mis-specification by smooth curves, which can be viewed as local alternatives. We find that the limiting regime depends on how quickly the alternatives approach a change-point model. We unravel a family of `intermediate' limits that can transition, at least qualitatively, to the limits in the two extreme scenarios.