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Stationary Discrete Autoregressive‐Moving Average Time Series Generated By Mixtures

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  • P. A. Jacobs
  • P. A. W. Lewis

Abstract

. Two simple stationary processes of discrete random variables with arbitrarily chosen first‐order marginal distributions, DARMA (p, N+ 1) and NDARMA (p, N), are given. The correlation structure of these processes mimics that of the usual linear ARMA (p, q) processes. The relationship of these processes to mover‐stayer models, and to models for discrete time series given separately by Lindqvist and Pegram is discussed. Ad hoc nonparametric estimators for the parameters in the DARMA (p, N+ 1) and NDARMA (p, N) are given. A simulation study shows them to be as good as maximum likelihood estimators for the first‐order autoregressive case, and to be much simpler to compute than the maximum likelihood estimators.

Suggested Citation

  • P. A. Jacobs & P. A. W. Lewis, 1983. "Stationary Discrete Autoregressive‐Moving Average Time Series Generated By Mixtures," Journal of Time Series Analysis, Wiley Blackwell, vol. 4(1), pages 19-36, January.
    • Handle: RePEc:bla:jtsera:v:4:y:1983:i:1:p:19-36
    DOI: 10.1111/j.1467-9892.1983.tb00354.x
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    Cited by:

    1. Kharin, Yuriy & Voloshko, Valeriy, 2021. "Robust estimation for Binomial conditionally nonlinear autoregressive time series based on multivariate conditional frequencies," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    2. Christian H. Weiß, 2018. "Goodness-of-fit testing of a count time series’ marginal distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(6), pages 619-651, August.
    3. Damian Eduardo Taranto & Giacomo Bormetti & Fabrizio Lillo, 2014. "The adaptive nature of liquidity taking in limit order books," Papers 1403.0842, arXiv.org, revised Apr 2014.
    4. Jungtaek Oh & Christian H. Weiß, 2020. "On the Individuals Chart with Supplementary Runs Rules under Serial Dependence," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1257-1273, September.
    5. Bondon, Pascal, 2009. "Estimation of autoregressive models with epsilon-skew-normal innovations," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1761-1776, September.
    6. Carsten Jentsch & Lena Reichmann, 2019. "Generalized Binary Time Series Models," Econometrics, MDPI, vol. 7(4), pages 1-26, December.
    7. 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.
    8. 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.
    9. Raju Maiti & Atanu Biswas, 2015. "Coherent forecasting for stationary time series of discrete data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 99(3), pages 337-365, July.
    10. Dehnert, M. & Helm, W.E. & Hütt, M.-Th., 2003. "A discrete autoregressive process as a model for short-range correlations in DNA sequences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 327(3), pages 535-553.
    11. Christian H. Weiß, 2019. "Measures of Dispersion and Serial Dependence in Categorical Time Series," Econometrics, MDPI, vol. 7(2), pages 1-23, April.
    12. Atanu Biswas & Maria Carmen Pardo & Apratim Guha, 2014. "Auto-association measures for stationary time series of categorical data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 487-514, September.
    13. Christopher S. Withers & Saralees Nadarajah & Shou Hsing Shih, 2015. "Moments and Cumulants of a Mixture," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 541-564, September.
    14. Weiß, Christian H., 2021. "On Edgeworth models for count time series," Statistics & Probability Letters, Elsevier, vol. 171(C).
    15. Kosmidis, Kosmas & Hütt, Marc-Thorsten, 2023. "DNA visibility graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 626(C).
    16. Guodong Pang & Ward Whitt, 2012. "The Impact of Dependent Service Times on Large-Scale Service Systems," Manufacturing & Service Operations Management, INFORMS, vol. 14(2), pages 262-278, April.
    17. Möller, Simon & Hameister, Heike & Hütt, Marc-Thorsten, 2014. "A genome signature derived from the interplay of word frequencies and symbol correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 414(C), pages 216-226.
    18. Biswas, Atanu & Song, Peter X.-K., 2009. "Discrete-valued ARMA processes," Statistics & Probability Letters, Elsevier, vol. 79(17), pages 1884-1889, September.
    19. Song‐Hee Kim & Ward Whitt, 2014. "Choosing arrival process models for service systems: Tests of a nonhomogeneous Poisson process," Naval Research Logistics (NRL), John Wiley & Sons, vol. 61(1), pages 66-90, February.
    20. Carsten Jentsch & Lena Reichmann, 2022. "Generalized binary vector autoregressive processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(2), pages 285-311, March.
    21. Maria Eduarda Da Silva & Vera Lúcia Oliveira, 2004. "Difference Equations for the Higher‐Order Moments and Cumulants of the INAR(1) Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(3), pages 317-333, May.
    22. A. M. M. Shahiduzzaman Quoreshi & Reaz Uddin & Naushad Mamode Khan, 2019. "Quasi-Maximum Likelihood Estimation for Long Memory Stock Transaction Data—Under Conditional Heteroskedasticity Framework," JRFM, MDPI, vol. 12(2), pages 1-13, April.

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