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First‐Order Integer‐Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties

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  • A. Alzaid
  • M. Al‐Osh

Abstract

Some properties of a first‐order integer‐valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self‐decomposability and unimodality of the 1‐dimensional marginals of the process {Xn} generated according to the scheme Xn=α°Xn‐i +en, where α°Xn‐1 denotes a sum of Xn ‐ 1, independent 0 ‐ 1 random variables Y(n‐1), independent of Xn‐1 with Pr‐(y(n ‐ 1)= 1) = 1 ‐ Pr (y(n‐i)= 0) =α. The distribution of the innovation process (en) is obtained when the marginal distribution of the process (Xn) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.

Suggested Citation

  • A. Alzaid & M. Al‐Osh, 1988. "First‐Order Integer‐Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 42(1), pages 53-61, March.
    • Handle: RePEc:bla:stanee:v:42:y:1988:i:1:p:53-61
    DOI: 10.1111/j.1467-9574.1988.tb01521.x
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    Cited by:

    1. Shirozhan, M. & Bakouch, Hassan S. & Mohammadpour, M., 2023. "A flexible INAR(1) time series model with dependent zero-inflated count series and medical contagious cases," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 216-230.
    2. Shihang Yu & Dehui Wang & Xia Chen, 2018. "Large and Moderate Deviations for the Total Population Arising from a Sub-critical Galton–Watson Process with Immigration," Journal of Theoretical Probability, Springer, vol. 31(1), pages 41-67, March.
    3. Shengqi Tian & Dehui Wang & Shuai Cui, 2020. "A seasonal geometric INAR process based on negative binomial thinning operator," Statistical Papers, Springer, vol. 61(6), pages 2561-2581, December.
    4. Chen, Zezhun & Dassios, Angelos, 2022. "Cluster point processes and Poisson thinning INARMA," LSE Research Online Documents on Economics 113652, London School of Economics and Political Science, LSE Library.
    5. Johannes Ferreira & Ané van der Merwe, 2022. "A Noncentral Lindley Construction Illustrated in an INAR(1) Environment," Stats, MDPI, vol. 5(1), pages 1-19, January.
    6. Nisreen Shamma & Mehrnaz Mohammadpour & Masoumeh Shirozhan, 2020. "A time series model based on dependent zero inflated counting series," Computational Statistics, Springer, vol. 35(4), pages 1737-1757, December.
    7. Wagner Barreto‐Souza & Hernando Ombao, 2022. "The negative binomial process: A tractable model with composite likelihood‐based inference," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 568-592, June.
    8. Ayaz Aliev & Madina Magomadova & Anna Budkina & Mustafa Harputlu & Alagez Yusifova, 2023. "EU: The Effect of Energy Factors on Economic Growth," Energies, MDPI, vol. 16(6), pages 1-19, March.
    9. Manik Awale & N. Balakrishna & T. V. Ramanathan, 2019. "Testing the constancy of the thinning parameter in a random coefficient integer autoregressive model," Statistical Papers, Springer, vol. 60(5), pages 1515-1539, October.
    10. Drost, F.C. & van den Akker, R. & Werker, B.J.M., 2008. "Efficient Estimation of Autoregression Parameters and Innovation Distributions forSemiparametric Integer-Valued AR(p) Models (Revision of DP 2007-23)," Other publications TiSEM cef533d0-6b49-4ce9-8cd2-7, Tilburg University, School of Economics and Management.
    11. Emrah Altun & Naushad Mamode Khan, 2022. "Modelling with the Novel INAR(1)-PTE Process," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1735-1751, September.
    12. Feilong Lu & Dehui Wang, 2022. "A new estimation for INAR(1) process with Poisson distribution," Computational Statistics, Springer, vol. 37(3), pages 1185-1201, July.
    13. Ané van der Merwe & Johannes T. Ferreira, 2022. "An Adapted Discrete Lindley Model Emanating from Negative Binomial Mixtures for Autoregressive Counts," Mathematics, MDPI, vol. 10(21), pages 1-21, November.

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