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February 2016 Asymptotics for change-point models under varying degrees of mis-specification
Rui Song, Moulinath Banerjee, Michael R. Kosorok
Ann. Statist. 44(1): 153-182 (February 2016). DOI: 10.1214/15-AOS1362

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 a jump process. Under complete mis-specification by a smooth curve, that is, 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. The theoretical results are illustrated via a set of carefully designed simulations. We also demonstrate how inference for the change-point parameter can be performed in absence of knowledge of the underlying scenario by resorting to sub-sampling techniques that involve estimation of the convergence rate.

Citation

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Rui Song. Moulinath Banerjee. Michael R. Kosorok. "Asymptotics for change-point models under varying degrees of mis-specification." Ann. Statist. 44 (1) 153 - 182, February 2016. https://doi.org/10.1214/15-AOS1362

Information

Received: 1 May 2014; Revised: 1 July 2015; Published: February 2016
First available in Project Euclid: 10 December 2015

zbMATH: 1331.62251
MathSciNet: MR3449765
Digital Object Identifier: 10.1214/15-AOS1362

Subjects:
Primary: 62G05, 62G20
Secondary: 62E20

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 1 • February 2016
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