• Bernoulli
  • Volume 23, Number 2 (2017), 1408-1447.

A robust approach for estimating change-points in the mean of an $\operatorname{AR}(1)$ process

S. Chakar, E. Lebarbier, C. Lévy-Leduc, and S. Robin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider the problem of multiple change-point estimation in the mean of an $\operatorname{AR}(1)$ process. Taking into account the dependence structure does not allow us to use the dynamic programming algorithm, which is the only algorithm giving the optimal solution in the independent case. We propose a robust estimator of the autocorrelation parameter, which is consistent and satisfies a central limit theorem in the Gaussian case. Then, we propose to follow the classical inference approach, by plugging this estimator in the criteria used for change-points estimation. We show that the asymptotic properties of these estimators are the same as those of the classical estimators in the independent framework. The same plug-in approach is then used to approximate the modified BIC and choose the number of segments. This method is implemented in the R package AR1seg and is available from the Comprehensive R Archive Network (CRAN). This package is used in the simulation section in which we show that for finite sample sizes taking into account the dependence structure improves the statistical performance of the change-point estimators and of the selection criterion.

Article information

Bernoulli, Volume 23, Number 2 (2017), 1408-1447.

Received: February 2015
Revised: October 2015
First available in Project Euclid: 4 February 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

auto-regressive model change-points model selection robust estimation of the $\operatorname{AR}(1)$ parameter time series


Chakar, S.; Lebarbier, E.; Lévy-Leduc, C.; Robin, S. A robust approach for estimating change-points in the mean of an $\operatorname{AR}(1)$ process. Bernoulli 23 (2017), no. 2, 1408--1447. doi:10.3150/15-BEJ782.

Export citation


  • [1] Arcones, M.A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242–2274.
  • [2] Auger, I.E. and Lawrence, C.E. (1989). Algorithms for the optimal identification of segment neighborhoods. Bull. Math. Biol. 51 39–54.
  • [3] Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47–78.
  • [4] Bai, J. and Perron, P. (2003). Computation and analysis of multiple structural change models. J. Appl. Econometrics 18 1–22.
  • [5] Bardet, J.-M., Kengne, W.C. and Wintenberger, O. (2012). Multiple breaks detection in general causal time series using penalized quasi-likelihood. Electron. J. Stat. 6 435–477.
  • [6] Basseville, M. and Nikiforov, I.V. (1993). Detection of Abrupt Changes: Theory and Application. Prentice Hall Information and System Sciences Series. Englewood Cliffs, NJ: Prentice Hall.
  • [7] Boysen, L., Kempe, A., Liebscher, V., Munk, A. and Wittich, O. (2009). Consistencies and rates of convergence of jump-penalized least squares estimators. Ann. Statist. 37 157–183.
  • [8] 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.
  • [9] Braun, J.V. and Müller, H.-G. (1998). Statistical methods for DNA sequence segmentation. Statist. Sci. 13 142–162.
  • [10] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [11] Csörgő, S. and Mielniczuk, J. (1996). The empirical process of a short-range dependent stationary sequence under Gaussian subordination. Probab. Theory Related Fields 104 15–25.
  • [12] 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.
  • [13] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge Univ. Press.
  • [14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed. New York: Wiley.
  • [15] Gazeaux, J., Williams, S., King, M., Bos, M., Dach, R., Deo, M., Moore, A.W., Ostini, L., Petrie, E., Roggero, M., Teferle, F.N., Olivares, G. and Webb, F.H. (2013). Detecting offsets in GPS time series: First results from the detection of offsets in GPS experiment. J. Geophys. Res. (Solid Earth) 118 2397–2407.
  • [16] Harchaoui, Z. and Lévy-Leduc, C. (2010). Multiple change-point estimation with a total variation penalty. J. Amer. Statist. Assoc. 105 1480–1493.
  • [17] Lai, T.L., Liu, H. and Xing, H. (2005). Autoregressive models with piecewise constant volatility and regression parameters. Statist. Sinica 15 279–301.
  • [18] Lai, W.R., Johnson, M.D., Kucherlapati, R. and Park, P.J. (2005). Comparative analysis of algorithms for identifying amplifications and deletions in array CGH data. Bioinformatics 21 3763–3770.
  • [19] Lavielle, M. (1999). Detection of multiple changes in a sequence of dependent variables. Stochastic Process. Appl. 83 79–102.
  • [20] Lavielle, M. (2005). Using penalized contrasts for the change-point problem. Signal Process. 85 1501–1510.
  • [21] Lavielle, M. and Moulines, E. (2000). Least-squares estimation of an unknown number of shifts in a time series. J. Time Ser. Anal. 21 33–59.
  • [22] Lebarbier, É. (2005). Detecting multiple change-points in the mean of Gaussian process by model selection. Signal Processing 85 717–736.
  • [23] Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M.S. and Reisen, V.A. (2011). Robust estimation of the scale and of the autocovariance function of Gaussian short- and long-range dependent processes. J. Time Series Anal. 32 135–156.
  • [24] Li, S. and Lund, R. (2012). Multiple changepoint detection via genetic algorithms. J. Climate 25 674–686.
  • [25] Lu, Q., Lund, R. and Lee, T.C.M. (2010). An MDL approach to the climate segmentation problem. Ann. Appl. Stat. 4 299–319.
  • [26] Ma, Y. and Genton, M.G. (2000). Highly robust estimation of the autocovariance function. J. Time Ser. Anal. 21 663–684.
  • [27] Maidstone, R., Hocking, T., Rigaill, G. and Fearnhead, P. (2014). On optimal multiple changepoint algorithms for large data. Available at arXiv:1409.1842.
  • [28] Mestre, O. (2000). Méthodes statistiques pour l’homogénéisation de longues séries climatiques. Ph.D. thesis, Univ. Toulouse 3.
  • [29] Olshen, A.B., Venkatraman, E.S., Lucito, R. and Wigler, M. (2004). Circular binary segmentation for the analysis of array-based DNA copy number data. Biostatistics 5 557–572.
  • [30] Picard, F., Lebarbier, E., Hoebeke, M., Rigaill, G., Thiam, B. and Robin, S. (2011). Joint segmentation, calling, and normalization of multiple CGH profiles. Biostatistics 12 413–428.
  • [31] Picard, F., Robin, S., Lavielle, M., Vaisse, C. and Daudin, J.-J. (2005). A statistical approach for array CGH data analysis. BMC Bioinformatics 6 27.
  • [32] Rigaill, G., Lebarbier, E. and Robin, S. (2012). Exact posterior distributions and model selection criteria for multiple change-point detection problems. Stat. Comput. 22 917–929.
  • [33] Rousseeuw, P.J. and Croux, C. (1993). Alternatives to the median absolute deviation. J. Amer. Statist. Assoc. 88 1273–1283.
  • [34] Van der Vaart, A.W. (2000). Asymptotic Statistics 3. Cambridge: Cambridge Univ. Press.
  • [35] Venkatraman, E.S. (1992). Consistency results in multiple change-point problems. Ph.D. thesis, Stanford Univ.
  • [36] Williams, S. (2003). Offsets in global positioning system time series. J. Geophys. Res. (Solid Earth) 108.
  • [37] Yao, Y.-C. (1988). Estimating the number of change-points via Schwarz’ criterion. Statist. Probab. Lett. 6 181–189.
  • [38] Yao, Y.-C. and Au, S.T. (1989). Least-squares estimation of a step function. Sankhyā Ser. A 51 370–381.
  • [39] Zhang, N.R. and Siegmund, D.O. (2007). A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data. Biometrics 63 22–32, 309.