Numéro spécial : Special Issue on Change-Point Detection
Sequential change-point detection in Poisson autoregressive models
[Détection séquentielle de ruptures dans les models autorégressifs de Poisson]
Journal de la société française de statistique, Tome 156 (2015) no. 4, pp. 98-112.

Nous considérons la détection séquentielle de ruptures dans une classe assez générale de modèles de Poisson autorégressifs de séries temporelles à valeurs entières. La moyenne conditionnelle du processus dépend d’un paramètre θ 0 * Θ I R d susceptible de changer dans le temps au fur et à mesure que les données sont observées. Nous proposons une procédure séquentielle dont le temps de suivi peut être fini ou infini basée sur l’estimateur du maximum de vraisemblance du paramètre. Sous l’hypothèse nulle selon laquelle aucun changement n’intervient dans le paramètre, la statistique de test converge vers une distribution connue. Des résultats de simulations nous permettent d’évaluer la puissance (empirique) ainsi que l’efficacité en terme du délai de détection et un exemple d’application aux données réelles est fourni.

We consider the sequential change-point detection in a general class of Poisson autoregressive models. The conditional mean of the process depends on a parameter θ 0 * Θ I R d which may change over time as and when data are observed. We propose a closed and open-end procedure based on the maximum likelihood estimator of the parameter. Under the null hypothesis of no change, it is shown that the detector converges to a well know distribution. The (empirical) power and the efficiency in terms of the detection delay are assessed through a simulation study and a real data example is provided.

Keywords: Sequential detection, change-point, time series of counts, Poisson autoregression, likelihood estimation
Mot clés : Détection séquentielle, rupture, séries temporelles à valeurs entières, autorégression de Poisson, estimation par vraisemblance
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     title = {Sequential change-point detection in {Poisson} autoregressive models},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {98--112},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {156},
     number = {4},
     year = {2015},
     zbl = {1343.62062},
     language = {en},
     url = {http://archive.numdam.org/item/JSFS_2015__156_4_98_0/}
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Kengne, William. Sequential change-point detection in Poisson autoregressive models. Journal de la société française de statistique, Tome 156 (2015) no. 4, pp. 98-112. http://archive.numdam.org/item/JSFS_2015__156_4_98_0/

[1] Aue, Alexander; Hörmann, Siegfried; Horváth, Lajos; Reimherr, Matthew Break detection in the covariance structure of multivariate time series models, The Annals of Statistics, Volume 37 (2009) no. 6B, pp. 4046-4087 | Zbl

[2] Aue, Alexander; Horváth, Lajos; Hušková, Marie; Kokoszka, Piotr Change-point monitoring in linear models, The Econometrics Journal, Volume 9 (2006) no. 3, pp. 373-403 | Zbl

[3] Aue, Alexander; Horváth, Lajos; Reimherr, Matthew L Delay times of sequential procedures for multiple time series regression models, Journal of Econometrics, Volume 149 (2009) no. 2, pp. 174-190 | Zbl

[4] Berkes, István; Gombay, Edit; Horváth, Lajos; Kokoszka, Piotr Sequential change-point detection in GARCH (p, q) models, Econometric Theory, Volume 20 (2004) no. 06, pp. 1140-1167 | Zbl

[5] Billingsley, P. Convergence of probability Measures (1968) | Zbl

[6] Bardet, Jean-Marc; Kengne, William Monitoring procedure for parameter change in causal time series, Journal of Multivariate Analysis, Volume 125 (2014), pp. 204-221 | Zbl

[7] Bardet, Jean-Marc; Kengne, William; Wintenberger, Olivier Multiple breaks detection in general causal time series using penalized quasi-likelihood, Electronic Journal of Statistics, Volume 6 (2012), pp. 435-477 | Zbl

[8] Basseville, Michèle; Nikiforov, Igor V Detection of abrupt changes: theory and application, 104, Prentice Hall Englewood Cliffs, 1993

[9] Csörgö, Miklós; Horváth, Lajos Limit theorems in change-point analysis, Wiley New York, 1997 | Zbl

[10] Chu, Chia-Shang James; Stinchcombe, Maxwell; White, Halbert Monitoring structural change, Econometrica: Journal of the Econometric Society (1996), pp. 1045-1065 | Zbl

[11] Doukhan, Paul; Fokianos, Konstantinos; Tjøstheim, Dag On weak dependence conditions for Poisson autoregressions, Statistics & Probability Letters, Volume 82 (2012) no. 5, pp. 942-948 | Zbl

[12] Doukhan, Paul; Kengne, William Inference and testing for structural change in general Poisson autoregressive models, Electronic Journal of Statistics, Volume 9 (2015), pp. 1267-1314 | Zbl

[13] Doukhan, Paul; Wintenberger, Olivier Weakly dependent chains with infinite memory, Stochastic Processes and their Applications, Volume 118 (2008) no. 11, pp. 1997-2013 | Zbl

[14] Fokianos, Konstantinos; Fried, Roland Interventions in INGARCH processes, Journal of Time Series Analysis, Volume 31 (2010) no. 3, pp. 210-225 | Zbl

[15] Fokianos, Konstantinos; Fried, Roland Interventions in log-linear Poisson autoregression, Statistical Modelling, Volume 12 (2012) no. 4, pp. 299-322 | Zbl

[16] Fokianos, Konstantinos; Gombay, Edit; Hussein, Abdulkadir Retrospective change detection for binary time series models, Journal of Statistical Planning and Inference, Volume 145 (2014), pp. 102-112 | Zbl

[17] Ferland, René; Latour, Alain; Oraichi, Driss Integer-Valued GARCH Process, Journal of Time Series Analysis, Volume 27 (2006) no. 6, pp. 923-942 | Zbl

[18] Fokianos, Konstantinos; Neumann, Michael H A goodness-of-fit test for Poisson count processes, Electronic Journal of Statistics, Volume 7 (2013), pp. 793-819 | Zbl

[19] Fokianos, Konstantinos; Rahbek, Anders; Tjøstheim, Dag Poisson autoregression, Journal of the American Statistical Association, Volume 104 (2009) no. 488, pp. 1430-1439 | Zbl

[20] Fryzlewicz, Piotr; Subba Rao, Suhasini Multiple-change-point detection for auto-regressive conditional heteroscedastic processes, Journal of the Royal Statistical Society: series B (statistical methodology) (2013) | Zbl

[21] Fokianos, Konstantinos; Tjøstheim, Dag Nonlinear poisson autoregression, Annals of the Institute of Statistical Mathematics, Volume 64 (2012) no. 6, pp. 1205-1225 | Zbl

[22] Gombay, Edit; Serban, Daniel Monitoring parameter change in time series models, Journal of Multivariate Analysis, Volume 100 (2009) no. 4, pp. 715-725 | Zbl

[23] Horváth, Lajos; Hušková, Marie; Kokoszka, Piotr; Steinebach, Josef Monitoring changes in linear models, Journal of Statistical Planning and Inference, Volume 126 (2004) no. 1, pp. 225-251 | Zbl

[24] Horvath, Lajos; Kokoszka, Piotr; Steinebach, Josef On sequential detection of parameter changes in linear regression, Statistics & probability letters, Volume 77 (2007) no. 9, pp. 885-895 | Zbl

[25] Kengne, William Charky Testing for parameter constancy in general causal time-series models, Journal of Time Series Analysis, Volume 33 (2012) no. 3, pp. 503-518 | Zbl

[26] Kang, Jiwon; Lee, Sangyeol Parameter change test for random coefficient integer-valued autoregressive processes with application to polio data analysis, Journal of Time Series Analysis, Volume 30 (2009) no. 2, pp. 239-258 | Zbl

[27] Kang, Jiwon; Lee, Sangyeol Parameter change test for Poisson autoregressive models, Scandinavian Journal of Statistics, Volume 41 (2014) no. 4, pp. 1136-1152 | Zbl

[28] Leisch, Friedrich; Hornik, Kurt; Kuan, Chung-Ming Monitoring structural changes with the generalized fluctuation test, Econometric Theory, Volume 16 (2000) no. 06, pp. 835-854 | Zbl

[29] Neumann, Michael H Absolute regularity and ergodicity of Poisson count processes, Bernoulli, Volume 17 (2011) no. 4, pp. 1268-1284 | Zbl

[30] Na, Okyoung; Lee, Youngmi; Lee, Sangyeol Monitoring parameter change in time series models, Statistical Methods & Applications, Volume 20 (2011) no. 2, pp. 171-199 | Zbl

[31] Weiß, Christian H Modelling time series of counts with overdispersion, Statistical Methods and Applications, Volume 18 (2009) no. 4, pp. 507-519 | Zbl

[32] Zeileis, Achim; Leisch, Friedrich; Kleiber, Christian; Hornik, Kurt Monitoring structural change in dynamic econometric models, Journal of Applied Econometrics, Volume 20 (2005) no. 1, pp. 99-121

[33] Zhu, Fukang; Wang, Dehui Diagnostic checking integer-valued ARCH (p) models using conditional residual autocorrelations, Computational Statistics & Data Analysis, Volume 54 (2010) no. 2, pp. 496-508 | Zbl