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Generalized Random Environment INAR Models of Higher Order

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Abstract

The random environment integer-valued autoregressive models of higher orders introduced by Nastić et al. (J Time Ser Anal 37: 267–287, 2016) are generalized. Generalizations are made by relaxing two assumptions: the assumption about the equality of negative binomial thinning operators and the assumption that the maximal orders are equal for all the process random states. Some statistical properties of the introduced models are derived and discussed. Their unknown parameters are estimated by the Yule–Walker method and the performances of the obtained estimators are checked using simulated data sets. A possible application of the models on the real-life data is discussed at the end of the paper.

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References

  1. Al-Osh, M.A., Aly, E.E.A.A.: First order autoregressive time series with negative binomial and geometric marginals. Commun. Stat. Theory Methods 21, 2483–2492 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Osh, M.A., Alzaid, A.A.: First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8, 261–275 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aly, E.E.A.A., Bouzar, N.: On some integer-valued autoregressive moving average models. J. Multivar. Anal. 50, 132–151 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alzaid, A.A., Al-Osh, M.A.: First-order integer-valued autoregressive (INAR(1)) process: distributional and regression properties. Stat. Neerl. 42, 53–61 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alzaid, A.A., Al-Osh, M.A.: Some autoregressive moving average processes with generalized Poisson marginal distributions. Ann. Inst. Stat. Math. 45, 223–232 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bakouch, H.S., Ristić, M.M.: Zero Truncated Poisson Integer Valued AR(1) Model. Metrika 72(2), 265–280 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jowaheer, V., Khan, N.M., Sunecker, R.: A BINAR(1) time series model with cross-correlated COM/Poisson innovations. Commun. Stat. Theory Methods. https://doi.org/10.1080/03610926.2017.1316400 (2017)

  8. Khan, N.M., Sunecher, R., Jowaheer, V.: Modelling a non-stationary BINAR(1) Poisson process. J. Stat. Comput. Simul. 86(15), 3106–3126 (2016)

    Article  MathSciNet  Google Scholar 

  9. Latour, A.: Existence and stochastic structure of a non-negative integer-valued autoregressive process. J. Time Ser. Anal. 19, 439–455 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. McKenzie, E.: Some simple models for discrete variate time series. Water Resour. Bull. 21, 645–650 (1985)

    Article  Google Scholar 

  11. McKenzie, E.: Autoregressive moving-average processes with negative binomial and geometric distributions. Adv. Appl. Prob. 18, 679–705 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nastić, A.S., Ristić, M.M., Bakouch, H.S.: A combined geometric INAR(\(p\)) model based on negative binomial thinning. Math. Comput. Modelling 55, 1665–1672 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nastić, A.S., Ristić, M.M.: Some geometric mixed integer-valued autoregressive (INAR) models. Stat. Probab. Lett. 82, 805–811 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nastić, A.S., Laketa, P.N., Ristić, M.M.: Random environment integer-valued autoregressive process. J. Time Ser. Anal. 37, 267–287 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nastić, A.S., Laketa, P.N., Ristić, M.M.: Random environment INAR models of higher order. Stat. J., To Appear, RevStat (2017)

  16. Ristić, M.M., Bakouch, H.S., Nastić, A.S.: A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. J. Stat. Plan. Inference 139, 2218–2226 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tang, M., Wang, Y.: Asymptotic behavior of random coefficient INAR model under random environment defined by difference equation. Adv. Differ. Equ. 2014(1), 19 (2014)

    Article  MATH  Google Scholar 

  18. Weiß, C.H.: The combined INAR\((p)\) models for time series of counts. Stat. Probab. Lett. 72, 1817–1822 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zheng, H., Basawa, I.V., Datta, S.: Inference for \(p\)th-order random coefficient integer-valued autoregressive processes. J. Time Ser. Anal. 27, 411–440 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zheng, H., Basawa, I.V., Datta, S.: First-order random coefficient integer-valued autoregressive processes. J. Stat. Plan. Inference 137, 212–229 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhu, R., Joe, H.: Modelling count data time series with Markov processes based on binomial thinning. J. Time Ser. Anal. 27, 727–738 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Zhu, R., Joe, H.: Negative binomial time series models based on expectation thinning operators. J. Stat. Plan. Inference 140, 1874–1888 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Niš, Faculty of Sciences and Mathematics, Niš, Serbia

    Petra N. Laketa, Aleksandar S. Nastić & Miroslav M. Ristić

Authors
  1. Petra N. Laketa
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  2. Aleksandar S. Nastić
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  3. Miroslav M. Ristić
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Correspondence to Miroslav M. Ristić.

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Laketa, P.N., Nastić, A.S. & Ristić, M.M. Generalized Random Environment INAR Models of Higher Order. Mediterr. J. Math. 15, 9 (2018). https://doi.org/10.1007/s00009-017-1054-z

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  • DOI: https://doi.org/10.1007/s00009-017-1054-z

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