A seasonal Integer-Valued AR(1) model with delaporte marginal distribution

Document Type : Original Paper

Authors

Department of Statistics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

Abstract

Real-count data time series often show the phenomenon of over-dispersion. In this paper, we introduce the first-order integer-valued autoregressive process with seasonal structure. The univariate marginal distribution is derived from the Delaporte distribution and the innovations are convolution of Poisson with α-fold zero modified geometric distribution, based on binomial thinning operator, for modeling integer-valued time series with over-dispersion. Some properties of the model are derived. The methods of Yule-Walker, conditional least squares, and conditional maximum likelihood are used to estimate the parameters. The Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. At the end, this model is illustrated using a real data set and is compared to some INAR(1) models.

Keywords

Main Subjects


[1] Aghababaei Jazi, M., Jones, G. and Lai, C.D., 2012. First‐order integer valued AR processes
with zero inflated Poisson innovations. Journal of Time Series Analysis, 33(6), pp.954-963.
http://dx.doi.org/10.1111/j.1467-9892.2012.00809.x
[2] Aghababaei Jazi, M., Jones, G. and Lai, C.D., 2022. Integer valued AR (1) with geometric innovations. Journal of the Iranian Statistical Society, 11(2), pp.173-190.
[3] Al‐Osh, M.A. and Alzaid, A.A., 1987. First‐order integer‐valued autoregressive (INAR (1))
process. Journal of Time Series Analysis, 8(3), pp.261-275. http://dx.doi.org/10.1111/j.1467-
9574.1988.tb01521.x
[4] Barreto‐Souza, W., 2015. Zero‐modified geometric INAR (1) process for modelling count time
series with deflation or inflation of zeros. Journal of Time Series Analysis, 36(6), pp.839-852.
http://dx.doi.org/10.1111/jtsa.12131
[5] Barreto-Souza, W., 2019. Mixed poisson INAR (1) processes. Statistical papers, 60(6), pp.2119-2139. http://dx.doi.org/10.1007/s00362-017-0912-x
[6] Bourguignon, M., Rodrigues, J. and Santos-Neto, M., 2019. Extended Poisson INAR (1) processes with equidispersion, underdispersion and overdispersion. Journal of Applied Statistics, 46(1), pp.101-118. http://dx.doi.org/10.1080/02664763.2018.1458216
[7] Bourguignon, M., LP Vasconcellos, K., Reisen, V.A. and Ispány, M., 2016. A Poisson INAR (1)
process with a seasonal structure. Journal of Statistical Computation and Simulation, 86(2), pp.373-387. http://dx.doi.org/10.1080/00949655.2015.1015127
[8] Bourguignon, M. and Vasconcellos, K.L., 2015. First order non-negative integer valued autoregressive processes with power series innovations. Brazilian Journal of Probability and Statistics, 29(1) , pp.71–93. http://dx.doi.org/10.1214/13-bjps229
[9] Bourguignon, M. and Weiß, C.H., 2017. An INAR (1) process for modeling count
time series with equidispersion, underdispersion and overdispersion. Test, 26(4), pp.847-868.
http://dx.doi.org/10.1007/s11749-017-0536-4
[10] Brannas, K., 1995. Explanatory variables in the AR (1) count data model. Umea economic studies, 381. http://dx.doi.org/10.2139/ssrn.1313842
[11] Buteikis, A. and Leipus, R., 2020. An integer-valued autoregressive process for seasonality. Journal of Statistical Computation and Simulation, 90(3), pp.391-411.
http://dx.doi.org/10.1080/00949655.2019.1685995
[12] Delaporte, P., 1959. Quelques problèmes de statistique mathématique posés par l’assurance automobile et le bonus pour non sinistre. Bulletin Trimestriel de l’Institut des Actuaires Français, 227,pp.87-102.
[13] Ferland, R., Latour, A. and Oraichi, D., 2006. Integer‐valued GARCH process. Journal of time series analysis, 27(6), pp.923-942. http://dx.doi.org/10.1111/j.1467-9892.2006.00496.x
[14] Fernández-Fontelo, A., Fontdecaba, S., Alba, A. and Puig, P., 2017. Integer-valued
AR processes with Hermite innovations and time-varying parameters: An application to
bovine fallen stock surveillance at a local scale. Statistical modelling, 17(3), pp.172-195.
http://dx.doi.org/10.1177/1471082x16683113
[15] Freeland, R.K., 1998. Statistical analysis of discrete time series with application to the analysis of workers’ compensation claims data (Doctoral dissertation, University of British Columbia).
http://dx.doi.org/10.14288/1.0088709
[16] Freeland, R.K. and McCabe, B., 2005. Asymptotic properties of CLS estimators in the Poisson AR (1) model. Statistics & probability letters, 73(2), pp.147-153. http://dx.doi.org/10.1016/j.spl.2005.03.006
[17] Freeland, R.K. and McCabe, B.P., 2004. Analysis of low count time series data by Poisson autoregression. Journal of time series analysis, 25(5), pp.701-722. http://dx.doi.org/10.1111/j.1467- 9892.2004.01885.x
[18] Jose, K.K. and Mariyamma, K.D., 2016. A note on an integer valued time series model with Poisson–negative binomial marginal distribution. Communications in Statistics-Theory and Methods, 45(1), pp.123-131. http://dx.doi.org/10.1080/03610926.2013.826979
[19] Jung, R.C., Ronning, G. and Tremayne, A.R., 2005. Estimation in conditional first order autoregression with discrete support. Statistical papers, 46, pp.195-224. http://dx.doi.org/10.1007/bf02762968
[20] Kim, H.Y. and Park, Y., 2008. A non-stationary integer-valued autoregressive model. Statistical
papers, 49, pp.485-502. http://dx.doi.org/10.1007/s00362-006-0028-1
[21] Kim, H. and Lee, S., 2017. On first-order integer-valued autoregressive process with Katz
family innovations. Journal of Statistical Computation and Simulation, 87(3), pp.546-562.
http://dx.doi.org/10.1080/00949655.2016.1219356
[22] Klimko, L.A. and Nelson, P.I., 1978. On conditional least squares estimation for stochastic processes. The Annals of statistics, pp.629-642. http://dx.doi.org/10.1214/aos/1176344207
[23] McKenzie, E., 1985. Some simple models for discrete variate time series 1. JAWRA Journal of
the American Water Resources Association, 21(4), pp.645-650. http://dx.doi.org/10.1111/j.1752-
1688.1985.tb05379.x
[24] Quoreshi, S., Uddin, R. and Mamode Khan, N., 2020. A review of INMA integervalued model class, application and further development. Filomat, 34(1), pp.143-152.
http://dx.doi.org/10.2298/fil2001143q
[25] Scotto, M.G., Weiss, C.H. and Gouveia, S., 2015. Thinning-based models in the analysis of integer-valued time series: a review. Statistical Modelling, 15(6), pp.590-618.
http://dx.doi.org/10.1177/1471082x15584701
[26] Shalbaf, M., Parham, G. and Chinipardaz, R., 2022. Binomial Thinning Integer-Valued AR (1) with Poisson–ff Fold Zero Modified Geometric Innovations. Journal of Sciences, Islamic Republic of Iran,33(1), pp.55-63. https://doi.org/10.22059/jsciences.2021.320996.1007633
[27] Steutel, F.W. and van Harn, K., 1979. Discrete analogues of self-decomposability and stability. The Annals of Probability, pp.893-899. http://dx.doi.org/10.1214/aop/1176994950
[28] Silva, M.E. and Oliveira, V.L., 2005. Difference equations for the higher order moments
and cumulants of the INAR (p) model. Journal of Time Series Analysis, 26(1), pp.17-36.
http://dx.doi.org/10.1111/j.1467-9892.2005.00388.x
[29] Tian, S., Wang, D. and Cui, S., 2020. A seasonal geometric INAR process based on negative binomial thinning operator. Statistical Papers, 61, pp.2561-2581. http://dx.doi.org/10.1007/s00362-018-1060-7
[30] Weiß, C.H., 2008. Thinning operations for modeling time series of counts—a survey. AStA Advances in Statistical Analysis, 92, pp.319-341. http://dx.doi.org/10.1007/s10182-008-0072-3
[31] Weiß, C.H., 2009. Controlling jumps in correlated processes of Poisson counts. Applied Stochastic Models in Business and Industry, 25(5), pp.551-564. https://doi.org/10.1002/asmb.744
[32] Weiß, C.H., 2009. Modelling time series of counts with overdispersion. Statistical Methods and Applications, 18, pp.507-519. https://doi.org/10.1007/s10260-008-0108-6
[33] Willmot, G.E. and Sundt, B., 1989. On evaluation of the Delaporte distribution and related distributions. Scandinavian Actuarial Journal, 1989(2), pp.101-113.
https://doi.org/10.1080/03461238.1989.10413859
[34] Zheng, H., Basawa, I.V. and Datta, S., 2006. Inference for pth‐order random coefficient
integer‐valued autoregressive processes. Journal of Time Series Analysis, 27(3), pp.411-440.
https://doi.org/10.1111/j.1467-9892.2006.00472.x
[35] Zhu, R. and Joe, H., 2006. Modelling count data time series with Markov processes based on binomial thinning. Journal of Time Series Analysis, 27(5), pp.725-738. https://doi.org/10.1111/j.1467-9892.2006.00485.x