A MOSUM procedure for the estimation of multiple random change points

Birte Eichinger; Claudia Kirch

doi:10.3150/16-BEJ887

Bernoulli

Bernoulli
Volume 24, Number 1 (2018), 526-564.

A MOSUM procedure for the estimation of multiple random change points

Birte Eichinger and Claudia Kirch

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Abstract

In this work, we investigate statistical properties of change point estimators based on moving sum statistics. We extend results for testing in a classical situation with multiple deterministic change points by allowing for random exogenous change points that arise in Hidden Markov or regime switching models among others. To this end, we consider a multiple mean change model with possible time series errors and prove that the number and location of change points are estimated consistently by this procedure. Additionally, we derive rates of convergence for the estimation of the location of the change points and show that these rates are strict by deriving the limit distribution of properly scaled estimators. Because the small sample behavior depends crucially on how the asymptotic (long-run) variance of the error sequence is estimated, we propose to use moving sum type estimators for the (long-run) variance and derive their asymptotic properties. While they do not estimate the variance consistently at every point in time, they can still be used to consistently estimate the number and location of the changes. In fact, this inconsistency can even lead to more precise estimators for the change points. Finally, some simulations illustrate the behavior of the estimators in small samples showing that its performance is very good compared to existing methods.

Article information

Source
Bernoulli, Volume 24, Number 1 (2018), 526-564.

Dates
Received: August 2015
Revised: July 2016
First available in Project Euclid: 27 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1501142454

Digital Object Identifier
doi:10.3150/16-BEJ887

Mathematical Reviews number (MathSciNet)
MR3706768

Zentralblatt MATH identifier
06778339

Keywords
binary segmentation change point hidden Markov model moving sum statistics regime switching model

Citation

Eichinger, Birte; Kirch, Claudia. A MOSUM procedure for the estimation of multiple random change points. Bernoulli 24 (2018), no. 1, 526--564. doi:10.3150/16-BEJ887. https://projecteuclid.org/euclid.bj/1501142454

References

[1] Antoch, J. and Hušková, M. (1999). Estimators of changes. In Asymptotics, Nonparametrics, and Time Series. Statist. Textbooks Monogr. 158 533–577. New York: Dekker.
[2] Antoch, J., Hušková, M. and Jarušková, D. (2000). Change Point Detection. 5th ERS IASC Summer School.
[3] Bai, J. (1997). Estimating multiple breaks one at a time. Econometric Theory 13 315–352.
[4] Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47–78.
[5] Bauer, P. and Hackl, P. (1980). An extension of the MOSUM technique for quality control. Technometrics 22 1–7.
[6] Beveridge, S. and Nelson, C.R. (1981). A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle. J. Monet. Econ. 7 151–174.
[7] Braun, J.V., Braun, R.K. and Müller, H.-G. (2000). Multiple changepoint fitting via quasilikelihood, with application to DNA sequence segmentation. Biometrika 87 301–314.
[8] Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer Series in Statistics. New York: Springer.
[9] Chen, J., Gupta, A.K. and Pan, J. (2006). Information criterion and change point problem for regular models. Sankhyā 68 252–282.
[10] Chu, C.-S.J., Hornik, K. and Kuan, C.-M. (1995). MOSUM tests for parameter constancy. Biometrika 82 603–617.
[11] Csörgő, M. and Horváth, L. (1997). Limit Theorems in Change-point Analysis. Chichester: Wiley.
[12] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
[13] Davis, R.A., Lee, T.C.M. and Rodriguez-Yam, G.A. (2006). Structural break estimation for nonstationary time series models. J. Amer. Statist. Assoc. 101 223–239.
[14] Dümbgen, L. (1991). The asymptotic behavior of some nonparametric change-point estimators. Ann. Statist. 19 1471–1495.
[15] Dümbgen, L. and Spokoiny, V.G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124–152.
[16] Eichinger, B. and Kirch, C. (2016). Supplement to “A MOSUM procedure for the estimation of multiple random change points.” DOI:10.3150/16-BEJ887SUPP.
[17] Franke, J. (2011). Markov switching time series models. In Handbook of Statistics. Time Series – Methods and Applications 30 (C.R. Rao and T. Subba Rao, eds.). Amsterdam: Elsevier.
[18] Frick, K., Munk, A. and Sieling, H. (2014). Multiscale change point inference. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 495–580.
[19] Fryzlewicz, P. (2014). Wild binary segmentation for multiple change-point detection. Ann. Statist. 42 2243–2281.
[20] Harchaoui, Z. and Lévy-Leduc, C. (2010). Multiple change-point estimation with a total variation penalty. J. Amer. Statist. Assoc. 105 1480–1493.
[21] Hušková, M. (1990). Asymptotics for robust MOSUM. Comment. Math. Univ. Carolin. 31 345–356.
[22] Hušková, M. and Slabý, A. (2001). Permutation tests for multiple changes. Kybernetika (Prague) 37 605–622.
[23] Killick, R., Fearnhead, P. and Eckley, I.A. (2012). Optimal detection of changepoints with a linear computational cost. J. Amer. Statist. Assoc. 107 1590–1598.
[24] Kirch, C. (2006). Resampling methods for the change analysis of dependent data. Ph.D. Thesis, University of Cologne.
[25] Kirch, C. and Tajduidje Kamgaing, J. (2016). Detection of change points in discrete valued time series. In Handbook of Discrete Valued Time Series (R.A. Davis, S.A. Holan, R.B. Lund and N. Ravishanker, eds.). CRC Press, Boca Raton.
[26] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent $\mathrm{RV}$’s and the sample $\mathrm{DF}$. I. Probab. Theory Related Fields 32 111–131.
[27] Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent RV’s, and the sample DF. II. Probab. Theory Related Fields 34 33–58.
[28] Kuelbs, J. and Philipp, W. (1980). Almost sure invariance principles for partial sums of mixing $B$-valued random variables. Ann. Probab. 8 1003–1036.
[29] Lange, T. and Rahbek, A. (2009). An introduction to regime switching time series models. In Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.-P. Kreiß and T. Mikosch, eds.) Springer, Berlin.
[30] Ling, S. (2007). Testing for change points in time series models and limiting theorems for NED sequences. Ann. Statist. 35 1213–1237.
[31] Liu, J., Wu, S. and Zidek, J.V. (1997). On segmented multivariate regression. Statist. Sinica 7 497–525.
[32] Marušiaková, M. (2009). Tests for multiple changes in linear regression models. Ph.D. Thesis, Charles University, Prague.
[33] Messer, M., Kirchner, M., Schiemann, J., Roeper, J., Neininger, R. and Schneider, G. (2014). A multiple filter test for the detection of rate changes in renewal processes with varying variance. Ann. Appl. Stat. 8 2027–2067.
[34] Muggeo, V.M. and Adelfio, G. (2010). Efficient change point detection for genomic sequences of continuous measurements. Bioinformatics btq647.
[35] Muhsal, B. (2013). Change-point methods for multivariate autoregressive models and multiple structural breaks in the mean. Ph.D. Thesis, Karlsruhe Institute of Technology, Karlsruhe.
[36] Pan, J. and Chen, J. (2006). Application of modified information criterion to multiple change point problems. J. Multivariate Anal. 97 2221–2241.
[37] Phillips, P.C.B. and Solo, V. (1992). Asymptotics for linear processes. Ann. Statist. 20 971–1001.
[38] Politis, D.N. and Romano, J.P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16 67–103.
[39] Preuss, P., Puchstein, R. and Dette, H. (2015). Detection of multiple structural breaks in multivariate time series. J. Amer. Statist. Assoc. 110 654–668.
[40] Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464.
[41] Stout, W.F. (1974). Almost Sure Convergence. New York: Academic Press.
[42] Tong, H. (1990). Non-linear Time Series: A Dynamical System Approach. London: Oxford Univ. Press.
[43] Vostrikova, L.J. (1981). Detecting disorder in multidimensional random processes. Sov. Math. Dokl. 24 55–59.
[44] Yao, Y.-C. (1988). Estimating the number of change-points via Schwarz’ criterion. Statist. Probab. Lett. 6 181–189.
[45] Yao, Y.-C. and Au, S.T. (1989). Least-squares estimation of a step function. Sankhyā Ser. A 51 370–381.
[46] Yau, C.Y. and Zhao, Z. (2016). Inference for multiple change-points in time series via likelihood ratio scan statistics. J. Roy. Statist. Soc. Ser. A 78 895–916.
[47] Yokoyama, R. (1980). Moment bounds for stationary mixing sequences. Probab. Theory Related Fields 52 45–57.

Supplemental materials

Additional simulation results. In this supplement, we give some additional simulations illustrating the performance of the above change point estimators in small samples with an emphasis on how the variance estimator and the bandwidth influence the performance.
Digital Object Identifier: doi:10.3150/16-BEJ887SUPP
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