Doukhan, Paul; Kengne, William Inference and testing for structural change in general Poisson autoregressive models. (English) Zbl 1349.62397 Electron. J. Stat. 9, No. 1, 1267-1314 (2015). Summary: We consider here together the inference questions and the change-point problem in a large class of Poisson autoregressive models (see [D. Tjøstheim, Test 21, No. 3, 413–438 (2012; Zbl 1362.62174)]). The conditional mean (or intensity) of the process is involved as a non-linear function of its past values and the past observations. Under Lipschitz-type conditions, it can be written as a function of lagged observations. For the latter model, assume that the link function depends on an unknown parameter \(\theta_{0}\). The consistency and the asymptotic normality of the maximum likelihood estimator of the parameter are proved. These results are used to study change-point problem in the parameter \(\theta_{0}\). From the likelihood of the observations, two tests are proposed. Under the null hypothesis (i.e. no change), each of these tests statistics converges to an explicit distribution. Consistencies under alternatives are proved for both tests. Simulation results show how those procedures work in practice, and applications to real data are also processed. Cited in 14 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F12 Asymptotic properties of parametric estimators 62F10 Point estimation 62F03 Parametric hypothesis testing 62F05 Asymptotic properties of parametric tests 65C60 Computational problems in statistics (MSC2010) 60F05 Central limit and other weak theorems Keywords:Poisson autoregression; maximum likelihood estimator; consistency; asymptotic normality; limit distribution; time series; change-point; semiparametric test Citations:Zbl 1362.62174 PDFBibTeX XMLCite \textit{P. Doukhan} and \textit{W. Kengne}, Electron. J. Stat. 9, No. 1, 1267--1314 (2015; Zbl 1349.62397) Full Text: DOI arXiv Euclid References: [1] Bardet, J.-M. and Wintenberger, O., Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes., Ann. Statist. 37 (2009), 2730-2759. · Zbl 1173.62063 [2] Bardet, J.-M., Kengne, W. and Wintenberger, O., Detecting multiple change-points in general causal time series using penalized quasi-likelihood., Electronic Journal of Statistics 6 (2012), 435-477. · Zbl 1337.62210 [3] Berkes, I., Horváth, L., Kokoszka, P.S. and Shao, Q.-M., On discriminating between long-range dependence and changes in mean., Ann. Statist. 34 (2006), 1140-1165. · Zbl 1112.62085 [4] Billingsley, P., Convergence of Probability Measures . John Wiley & Sons Inc., New York (1968). · Zbl 0172.21201 [5] Brännäs, K. and Shahiduzzaman Quoreshi, A.M.M., Integer-valued moving average modelling of the number of transactions in stocks., Applied Financial Economics 20 (2010), 1429-1440. [6] Csörgo, M., Csörgo, S., Horváth, L. and Mason, D.M., Weighted empirical and quantile processes., The Annals of Probability 14 (1986), 31-85. · Zbl 0589.60029 [7] Csörgo, M. and Horváth, L., Weighted Approximations in Probability and Statistics . Wiley Chichester (1993). · Zbl 0770.60038 [8] Davis, R.A., Dunsmuir, W. and Streett, S., Maximum likelihood estimation for an observation driven model for Poisson counts., Methodol. Comput. Appl. Probab. 7 (2005), 149-159. · Zbl 1078.62091 [9] Davis, R.A. and Liu, H., Theory and Inference for a Class of Observation-Driven Models with Application to Time Series of Counts., Preprint, arXiv :1204.3915 . · Zbl 1356.62137 [10] Davis, R.A. and Wu, R., A negative binomial model for time series of counts., Biometrika 96 (2009), 735-749. · Zbl 1170.62062 [11] Douc, R., Doukhan, P. and Moulines, E., Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator., Stochastic Process. Appl. 123 (2013), 2620-2647. · Zbl 1285.62104 [12] Doukhan, P. and Wintenberger, O., Weakly dependent chains with infinite memory., Stochastic Process. Appl. 118 (2008), 1997-2013. · Zbl 1166.60031 [13] Doukhan, P., Fokianos, K. and Tjøstheim, D., On weak dependence conditions for Poisson autoregressions., Statist. and Probab. Letters 82 (2012), 942-948. · Zbl 1241.62109 [14] Doukhan, P., Fokianos, K. and Tjøstheim, D., Correction to “On weak dependence conditions for Poisson autoregressions” [Statist. Probab. Lett. 82 (2012) 942-948]., Statist. and Probab. Letters 83 (2013), 1926-1927. · Zbl 1241.62109 [15] Ferland, R., Latour, A. and Oraichi, D., Integer-valued GARCH process., J. Time Ser. Anal. 27 (2006), 923-942. · Zbl 1150.62046 [16] Fokianos, K. and Fried, R., Interventions in INGARCH processes., J. Time Ser. Anal. 31 (2010), 210-225. · Zbl 1173.62063 [17] Fokianos, K. and Fried, R., Interventions in log-linear Poisson autoregression., Statistical Modelling 12 (2012), 1-24. [18] Fokianos, K. and Neumann, M., A goodness-of-fit test for Poisson count processes., Electronic Journal of Statistics 7 (2013), 793-819. · Zbl 1327.62455 [19] Fokianos, K., Rahbek, A. and Tjøstheim, D., Poisson autoregression., Journal of the American Statistical Association 104 (2009), 1430-1439. · Zbl 1205.62130 [20] Fokianos, K. and Tjøstheim, D., Nonlinear Poisson autoregression., Ann. Inst. Stat. Math. 64 (2012), 1205-1225. · Zbl 1253.62058 [21] Franke, J., Kirch, C. and Tadjuidje Kamgaing, J., Changepoints in times series of counts., J. Time Ser. Anal. 33 (2012), 757-770. · Zbl 1281.62181 [22] Hairer, M. and Mattingly, J., Ergodicity of the 2d navier-stokes equations with degenerate stochastic forcings., Ann. Math. 164 (2006), 993-1032. · Zbl 1130.37038 [23] Held, L., Höhle, M. and Hofmann, M., A statistical framework for the analysis of multivariate infectious disease surveillance counts., Statistical Modelling 5 (2005), 187-199. · Zbl 1111.62105 [24] Inclán, C. and Tiao, G.C., Use of cumulative sums of squares for retrospective detection of changes of variance., Journal of the American Statistical Association 89 (1994), 913-923. · Zbl 0825.62678 [25] Kang, J. and Lee, S., Parameter change test for random coefficient integer-valued autoregressive processes with application to polio data analysis., Journal of Time Series Analysis 30(2) (2009), 239-258. · Zbl 1221.62126 [26] Kedem, B. and Fokianos, K., Regression Models for Time Series Analysis . Hoboken, Wiley, NJ (2002). · Zbl 1011.62089 [27] Kengne W., Testing for parameter constancy in general causal time-series models., J. Time Ser. Anal. 33 (2012), 503-518. · Zbl 1242.62095 [28] Kierfer, J., K-sample analogues of the Kolmogorov-Smirnov and Cramér-v.Mises tests., Ann. Math. Statist 30 (1959), 420-447. · Zbl 0134.36707 [29] Kounias, E.G. and Weng, T.-S., An inequality and almost sure convergence., Annals of Mathematical Statistics 40 (1969), 1091-1093. · Zbl 0176.48003 [30] Lambert, D., Zero-inflated Poisson regression, with an application to defects in manufacturing., Technometrics 34 (1992), 1-14. · Zbl 0850.62756 [31] Neumann, M., Absolute regularity and ergodicity of Poisson count processes., Bernoulli 17 (2011), 1268-1284. · Zbl 1277.60089 [32] Rabemananjara, R. and Zakoïan, J.M., Threshold ARCH models and asymmetries in volatility., Journal of Applied Econometrics 8 (1993), 31-49. [33] Straumann, D. and Mikosch, T., Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach., Ann. Statist. 34 (2006), 2449-2495. · Zbl 1108.62094 [34] Tjøstheim, D., Some recent theory for autoregressive count time series., TEST 21 (2012), 413-438. · Zbl 1362.62174 [35] Weiß, C.H., Modelling time series of counts with overdispersion., Stat. Methods Appl. 18 (2009), 507-519. · Zbl 1332.62348 [36] Zakoïan, J.-M., Threshold heteroskedastic models., Journal of Economic Dynamics and Control 18 (1994), 931-955. · Zbl 0875.90197 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.