×

Asymptotics for change-point models under varying degrees of mis-specification. (English) Zbl 1331.62251

Summary: 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.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics

Software:

unbalhaar; seqCBS
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47-78. · Zbl 1056.62523 · doi:10.2307/2998540
[2] Bai, J. and Perron, P. (2003). Computation and analysis of multiple structural change models. J. Appl. Econometrics 18 1-22.
[3] Banerjee, M. and McKeague, I. W. (2007). Confidence sets for split points in decision trees. Ann. Statist. 35 543-574. · Zbl 1117.62037 · doi:10.1214/009053606000001415
[4] Basseville, M. and Nikiforov, I. V. (1993). Detection of Abrupt Changes : Theory and Application . Prentice Hall, Englewood Cliffs, NJ.
[5] Bertail, P., Politis, D. N. and Romano, J. P. (1999). On subsampling estimators with unknown rate of convergence. J. Amer. Statist. Assoc. 94 569-579. · Zbl 1072.62551 · doi:10.2307/2670177
[6] Bhattacharya, P. K. and Brockwell, P. J. (1976). The minimum of an additive process with applications to signal estimation and storage theory. Z. Wahrsch. Verw. Gebiete 37 51-75. · Zbl 0326.60053 · doi:10.1007/BF00536298
[7] Bühlmann, P. and Yu, B. (2002). Analyzing bagging. Ann. Statist. 30 927-961. · Zbl 1029.62037 · doi:10.1214/aos/1031689014
[8] Csörgő, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analysis . Wiley, Chichester. · Zbl 0884.62023
[9] Fryzlewicz, P. (2007). Unbalanced Haar technique for nonparametric function estimation. J. Amer. Statist. Assoc. 102 1318-1327. · Zbl 1333.62014 · doi:10.1198/016214507000000860
[10] Gijbels, I., Hall, P. and Kneip, A. (1999). On the estimation of jump points in smooth curves. Ann. Inst. Statist. Math. 51 231-251. · Zbl 0934.62035 · doi:10.1023/A:1003802007064
[11] Hall, P. and Molchanov, I. (2003). Sequential methods for design-adaptive estimation of discontinuities in regression curves and surfaces. Ann. Statist. 31 921-941. · Zbl 1028.62069 · doi:10.1214/aos/1056562467
[12] Horowitz, J. L. (1992). A smoothed maximum score estimator for the binary response model. Econometrica 60 505-531. · Zbl 0761.62166 · doi:10.2307/2951582
[13] Hušková, M. (1999). Gradual changes versus abrupt changes. J. Statist. Plann. Inference 76 109-125. · Zbl 1054.62520 · doi:10.1016/S0378-3758(98)00173-6
[14] Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference . Springer, New York. · Zbl 1180.62137
[15] Kosorok, M. R. and Song, R. (2007). Inference under right censoring for transformation models with a change-point based on a covariate threshold. Ann. Statist. 35 957-989. · Zbl 1136.62376 · doi:10.1214/009053606000001244
[16] Lai, T. L. (1995). Sequential changepoint detection in quality control and dynamical systems. J. Roy. Stat. Soc. Ser. B. Stat. Methodol. 57 613-658. · Zbl 0832.62072
[17] Lai, T. L. (2001). Sequential analysis: Some classical problems and new challenges. Statist. Sinica 11 303-408. · Zbl 1037.62081
[18] Lan, Y., Banerjee, M. and Michailidis, G. (2009). Change-point estimation under adaptive sampling. Ann. Statist. 37 1752-1791. · Zbl 1168.62018 · doi:10.1214/08-AOS602
[19] Loader, C. R. (1996). Change point estimation using nonparametric regression. Ann. Statist. 24 1667-1678. · Zbl 0867.62033 · doi:10.1214/aos/1032298290
[20] Lund, R. and Reeves, J. (2002). Detection of undocumented changepoints: A revision of the two-phase regression model. Journal of Climate 15 2547-2554.
[21] Müller, H.-G. (1992). Change-points in nonparametric regression analysis. Ann. Statist. 20 737-761. · Zbl 0783.62032 · doi:10.1214/aos/1176348654
[22] Müller, H.-G. and Song, K.-S. (1997). Two-stage change-point estimators in smooth regression models. Statist. Probab. Lett. 34 323-335. · Zbl 0874.62035 · doi:10.1016/S0167-7152(96)00197-6
[23] Pons, O. (2003). Estimation in a Cox regression model with a change-point according to a threshold in a covariate. Ann. Statist. 31 442-463. · Zbl 1040.62090 · doi:10.1214/aos/1051027876
[24] Ritov, Y. (1990). Asymptotic efficient estimation of the change point with unknown distributions. Ann. Statist. 18 1829-1839. · Zbl 0714.62027 · doi:10.1214/aos/1176347881
[25] Seijo, E. and Sen, B. (2011). A continuous mapping theorem for the smallest argmax functional. Electron. J. Stat. 5 421-439. · Zbl 1329.60090 · doi:10.1214/11-EJS613
[26] Seo, M. H. (2012). Forecasting with a regime-switching model. Unpublished manuscript.
[27] Seo, M. H. and Linton, O. (2007). A smoothed least squares estimator for threshold regression models. J. Econometrics 141 704-735. · Zbl 1418.62355 · doi:10.1016/j.jeconom.2006.11.002
[28] Shen, J. J. and Zhang, N. R. (2012). Change-point model on nonhomogeneous Poisson processes with application in copy number profiling by next-generation DNA sequencing. Ann. Appl. Stat. 6 476-496. · Zbl 1243.62112 · doi:10.1214/11-AOAS517
[29] Song, R., Banerjee, M. and Kosorok, M. (2015). Supplement to “Asymptotics for change-point models under varying degrees of mis-specification.” . · Zbl 1331.62251 · doi:10.1214/15-AOS1362
[30] Thomson, R. E. and Fine, I. V. (2003). Estimating mixed layer depth from oceanic profile data. Journal of Atmospheric and Oceanic Technology 20 319-329.
[31] Vogt, M. and Dette, H. (2015). Detecting gradual changes in locally stationary processes. Ann. Statist. 43 713-740. · Zbl 1312.62045 · doi:10.1214/14-AOS1297
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.