This paper considers a new parameterization of the generalized binomial thinning operator that is to be incorporated in a simple ordered integer-valued autoregressive process (INAR(1)) with the Poisson–Lindley innovations. The statistical properties of the resulting INAR(1) process are explored along with the estimation procedures. Monte Carlo simulation experiments are executed to assess the consistency of the estimates under the new INAR(1) process. Finally, the importance of the proposed INAR(1) model is confirmed through the analysis of a real data set.

References

• Al-Osh, M. A., & Aly, E. E. A. (1992). First order autoregressive time series with negative binomial and geometric marginals. Communications in Statistics-Theory and Methods, 21(9), 2483–2492. https://doi.org/10.1080/03610929208830925
• Al-Osh, M. A., & Alzaid, A. A. (1987). First-order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis, 8(3), 261–275. https://doi.org/10.1111/j.1467-9892.1987.tb00438.x
• Altun, E., Bhati, D., & Khan, N. M. (2021). A new approach to model the counts of earthquakes: INARPQX(1) process. SN Applied Sciences, 3(1), 1–17. https://doi.org/10.1007/s42452-020-04109-8
• Aly, E. E. A., & Bouzar, N. (1994a). Explicit stationary distributions for some Galton-Watson processes with immigration. Stochastic Models, 10(2), 499–517. https://doi.org/10.1080/15326349408807305
• Aly, E. E. A., & Bouzar, N. (1994b). On some integer-valued autoregressive moving average models. Journal of Multivariate Analysis, 50(1), 132–151. https://doi.org/10.1006/jmva.1994.1038
• Aly, E. E. A., & Bouzar, N. (2019). Expectation thinning operators based on linear fractional probability generating functions. Journal of the Indian Society for Probability and Statistics, 20(1), 89–107. https://doi.org/10.1007/s41096-018-0056-x
• Andersen, E. B. (1970). Asymptotic properties of conditional maximum-likelihood estimators. Journal of the Royal Statistical Society: Series B :Statistical Methodology, 32(2), 283–301. https://doi.org/10.1111/j.2517-6161.1970.tb00842.x
• Billingsley, P. (1965). Ergodic Theory and Information (Vol. 1). Wiley.
• Borges, P., Molinares, F. F., & Bourguignon, M. (2016). A geometric time series model with inflated-parameter Bernoulli counting series. Statistics & Probability Letters, 119, 264–272. https://doi.org/10.1016/j.spl.2016.08.012
• Bu, R., & McCabe, B. (2008). Model selection, estimation and forecasting in INAR(p) models: A likelihood-based Markov chain approach. International Journal of Forecasting, 24(1), 151–162. https://doi.org/10.1016/j.ijforecast.2007.11.002
• Bu, R., McCabe, B., & Hadri, K. (2008). Maximum likelihood estimation of higher-order integer-valued autoregressive processes. Journal of Time Series Analysis, 29(6), 973–994. https://doi.org/10.1111/j.1467-9892.2008.00590.x
• Cramér, H., & Wold, H. (1936). Some theorems on distribution functions. Journal of the London Mathematical Society, s1-11(4), 290–294. https://doi.org/10.1112/jlms/s1-11.4.290
• Eliwa, M. S., Altun, E., El-Dawoody, M., & El-Morshedy, M. (2020). A new three-parameter discrete distribution with associated INAR(1) process and applications. IEEE Access, 8, 91150–91162. https://doi.org/10.1109/ACCESS.2020.2993593
• Ghitany, M. E., & Al-Mutairi, D. K. (2009). Estimation methods for the discrete Poisson-Lindley distribution. Journal of Statistical Computation and Simulation, 79(1), 1–9. https://doi.org/10.1080/00949650701550259
• Harvey, A. C., & Fernandes, C. (1989). Time series models for count or qualitative observations. Journal of Business & Economic Statistics, 7(4), 407–417. https://doi.org/10.1080/07350015.1989.10509750
• Huang, J., & Zhu, F. (2021). A new first-order integer-valued autoregressive model with Bell innovations. Entropy, 23(6), 1-17. https://doi.org/10.3390/e23060713
• Irshad, M. R., Chesneau, C., D'cruz, V., & Maya, R. (2021). Discrete pseudo Lindley distribution: Properties, estimation and application on INAR(1) process. Mathematical and Computational Applications, 26(4), 1-22. https://doi.org/10.3390/mca26040076
• Jazi, M. A., Jones, G., & Lai, C. D. (2012). First-order integer valued AR processes with zero inflated Poisson innovations. Journal of Time Series Analysis, 33(6), 954–963. https://doi.org/10.1111/j.1467-9892.2012.00809.x
• Kang, Y., Wang, D., & Yang, K. (2020). Extended binomial AR(1) processes with generalized binomial thinning operator. Communications in Statistics-Theory and Methods, 49(14), 3498–3520. https://doi.org/10.1080/03610926.2019.1589519
• Karlsen, H., & Tjøstheim, D. (1988). Consistent estimates for the NEAR(2) and NLAR(2) time series models. Journal of the Royal Statistical Society: Series B (Methodological), 50(2), 313–320. https://doi.org/10.1111/j.2517-6161.1988.tb01730.x
• Khoo, W. C., Ong, S. H., & Biswas, A. (2017). Modeling time series of counts with a new class of INAR(1) model. Statistical Papers, 58(2), 393–416. https://doi.org/10.1007/s00362-015-0704-0
• Liu, Z., & Zhu, F. (2020). A new extension of thinning-based integer-valued autoregressive models for count data. Entropy, 23(1), 1-17. https://doi.org/10.3390/e23010062
• Lívio, T., Khan, N. M., Bourguignon, M., & Bakouch, H. S. (2018). An INAR(1) model with Poisson-Lindley innovations. Economics Bulletin, 38(3), 1505–1513.
• Maya, R., Jodrá, P, Aswathy, S, & Irshad, M. R. (2024) The discrete new XLindley distribution and the associated autoregressive process. International Journal of Data Science and Analytics, 20, 1767–1793. https://doi.org/10.1007/s41060-024-00563-4
• McKenzie, E. (1985) Some simple models for discrete variate time series. JAWRA Journal of the American Water Resources Association, 21(4), 645–650. https://doi.org/10.1111/j.1752-1688.1985.tb05379.x
• Ristić, M. M., Bakouch, H. S., & Nastić, A. S. (2009). A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. Journal of Statistical Planning and Inference, 139(7), 2218–2226. https://doi.org/10.1016/j.jspi.2008.10.007
• Rostami, A., Mahmoudi, E, & Roozegar, R (2018) A new integer valued AR(1) process with Poisson-Lindley innovations. arXiv: 1802.00994
• Sankaran, M. (1970). The discrete Poisson-Lindley distribution. Biometrics, 26(1), 145–149. https://doi.org/10.2307/2529053
• Schweer, S., & Weiß, C. H. (2014). Compound Poisson INAR(1) processes: Stochastic properties and testing for overdispersion. Computational Statistics & Data Analysis, 77, 267–284. https://doi.org/10.1016/j.csda.2014.03.005
• Steutel, F. W., & van Harn, K. (1979). Discrete analogues of self-decomposability and stability. The Annals of Probability, 7(5), 893–899. https://doi.org/10.1214/aop/1176994950
• Weiß, C. H. (2008) Thinning operations for modelling time series of counts – a survey. AStA Advances in Statistical Analysis, 92, 319–341. https://doi.org/10.1007/s10182-008-0072-3
• Yang, K., Kang, Y, Wang, D, Li, H, & Diao, Y. (2019) Modeling overdispersed or underdispersed count data with generalized Poisson integer-valued autoregressive processes. Metrika, 82, 863–889. https://doi.org/10.1007/s00184-019-00714-9
• Zhang, H., Wang, D., & Zhu, F. (2010). Inference for INAR(p) processes with signed generalized power series thinning operator. Journal of Statistical Planning and Inference, 140(3), 667–683. https://doi.org/10.1016/j.jspi.2009.08.012

To cite this article: Emad-Eldin Aly Ahmed Aly, Naushad Mamode Khan, Muhammed Rasheed Irshad, Veena D'cruz & Radhakumari Maya (25 Sep 2025): A new generalized binomial thinningbased INAR(1) process with Poisson–Lindley innovations, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2025.2557723

To link to this article: https://doi.org/10.1080/24754269.2025.2557723