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**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 |

### References:

[1] | Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47-78. · Zbl 1056.62523 |

[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 |

[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 |

[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 |

[7] | Bühlmann, P. and Yu, B. (2002). Analyzing bagging. Ann. Statist. 30 927-961. · Zbl 1029.62037 |

[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 |

[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 |

[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 |

[12] | Horowitz, J. L. (1992). A smoothed maximum score estimator for the binary response model. Econometrica 60 505-531. · Zbl 0761.62166 |

[13] | Hušková, M. (1999). Gradual changes versus abrupt changes. J. Statist. Plann. Inference 76 109-125. · Zbl 1054.62520 |

[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 |

[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 |

[19] | Loader, C. R. (1996). Change point estimation using nonparametric regression. Ann. Statist. 24 1667-1678. · Zbl 0867.62033 |

[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 |

[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 |

[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 |

[24] | Ritov, Y. (1990). Asymptotic efficient estimation of the change point with unknown distributions. Ann. Statist. 18 1829-1839. · Zbl 0714.62027 |

[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 |

[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 |

[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 |

[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 |

[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 |

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