ASYMPTOTICS FOR CHANGE-POINT MODELS UNDER VARYING DEGREES OF MIS-SPECIFICATION

• 成果类型: Article
• 署名作者: Song, Rui; Banerjee, Moulinath; Kosorok, Michael R.
• 署名单位: North Carolina State University; University of Michigan System; University of Michigan; University of North Carolina; University of North Carolina Chapel Hill; University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1362
• 发表日期: 2016
• 页码: 153-182
• 关键词: Nonparametric regression; gradual changes; threshold; estimators; covariate; CURVES

摘要

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.