## Electronic Journal of Statistics

### Inference and testing for structural change in general Poisson autoregressive models

#### Abstract

We consider here together the inference questions and the change-point problem in a large class of Poisson autoregressive models (see Tjøstheim, 2012 [34]). The conditional mean (or intensity) of the process is involved as a non-linear function of it 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.

#### Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 1267-1314.

Dates
First available in Project Euclid: 19 June 2015

https://projecteuclid.org/euclid.ejs/1434720539

Digital Object Identifier
doi:10.1214/15-EJS1038

Mathematical Reviews number (MathSciNet)
MR3358325

Zentralblatt MATH identifier
1349.62397

#### Citation

Doukhan, Paul; Kengne, William. Inference and testing for structural change in general Poisson autoregressive models. Electron. J. Statist. 9 (2015), no. 1, 1267--1314. doi:10.1214/15-EJS1038. https://projecteuclid.org/euclid.ejs/1434720539

#### 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.
• [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.
• [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.
• [4] Billingsley, P., Convergence of Probability Measures. John Wiley & Sons Inc., New York (1968).
• [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.
• [7] Csörgo, M. and Horváth, L., Weighted Approximations in Probability and Statistics. Wiley Chichester (1993).
• [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.
• [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.
• [10] Davis, R.A. and Wu, R., A negative binomial model for time series of counts., Biometrika 96 (2009), 735–749.
• [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.
• [12] Doukhan, P. and Wintenberger, O., Weakly dependent chains with infinite memory., Stochastic Process. Appl. 118 (2008), 1997–2013.
• [13] Doukhan, P., Fokianos, K. and Tjøstheim, D., On weak dependence conditions for Poisson autoregressions., Statist. and Probab. Letters 82 (2012), 942–948.
• [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.
• [15] Ferland, R., Latour, A. and Oraichi, D., Integer-valued GARCH process., J. Time Ser. Anal. 27 (2006), 923–942.
• [16] Fokianos, K. and Fried, R., Interventions in INGARCH processes., J. Time Ser. Anal. 31 (2010), 210–225.
• [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.
• [19] Fokianos, K., Rahbek, A. and Tjøstheim, D., Poisson autoregression., Journal of the American Statistical Association 104 (2009), 1430–1439.
• [20] Fokianos, K. and Tjøstheim, D., Nonlinear Poisson autoregression., Ann. Inst. Stat. Math. 64 (2012), 1205–1225.
• [21] Franke, J., Kirch, C. and Tadjuidje Kamgaing, J., Changepoints in times series of counts., J. Time Ser. Anal. 33 (2012), 757–770.
• [22] Hairer, M. and Mattingly, J., Ergodicity of the 2d navier-stokes equations with degenerate stochastic forcings., Ann. Math. 164 (2006), 993–1032.
• [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.
• [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.
• [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.
• [26] Kedem, B. and Fokianos, K., Regression Models for Time Series Analysis. Hoboken, Wiley, NJ (2002).
• [27] Kengne W., Testing for parameter constancy in general causal time-series models., J. Time Ser. Anal. 33 (2012), 503–518.
• [28] Kierfer, J., K-sample analogues of the Kolmogorov-Smirnov and Cramér-v.Mises tests., Ann. Math. Statist 30 (1959), 420–447.
• [29] Kounias, E.G. and Weng, T.-S., An inequality and almost sure convergence., Annals of Mathematical Statistics 40 (1969), 1091–1093.
• [30] Lambert, D., Zero-inflated Poisson regression, with an application to defects in manufacturing., Technometrics 34 (1992), 1–14.
• [31] Neumann, M., Absolute regularity and ergodicity of Poisson count processes., Bernoulli 17 (2011), 1268–1284.
• [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.
• [34] Tjøstheim, D., Some recent theory for autoregressive count time series., TEST 21 (2012), 413–438.
• [35] Weiß, C.H., Modelling time series of counts with overdispersion., Stat. Methods Appl. 18 (2009), 507–519.
• [36] Zakoïan, J.-M., Threshold heteroskedastic models., Journal of Economic Dynamics and Control 18 (1994), 931–955.