Periodic INAR(1) Models with Skellam-Distributed Innovations

  • Cláudia SantosEmail author
  • Isabel Pereira
  • Manuel Scotto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11621)


In this paper, an integer-valued autoregressive model of order one (INAR(1)) with time-varying parameters and driven by a periodic sequence of innovations is introduced. The proposed INAR(1) model is based on the signed thinning operator defined by Kachour and Truquet (2011) and conveniently adapted to the periodic case. Basic notations and definitions concerning the periodic signed thinning operator are provided. Based on this thinning operator, Chesneau and Kachour (2012) established a signed INAR(1) model. Motivated by the work of Chesneau and Kachour (2012), we introduce a periodic model, denoted by S-PINAR(1), with period s. In contrast to conventional INAR(1) models, these models are defined in \(\mathbb {Z}\) allowing for negative values both for the series and its autocorrelation function. For a proper \(\mathbb {Z}\)-valued time series, a distribution for the innovation term defined on \(\mathbb {Z}\) is required. The S-PINAR(1) model assumes a specific innovation distribution, the Skellam distribution. Regarding parameter estimation, two methods are considered: conditional least squares and conditional maximum likelihood. The performance of the S-PINAR(1) model is assessed through a simulation study.


Integer-valued autoregressive models Signed thinning operator Skellam distribution 


  1. Alzaid, A.A., Omair, M.A.: On the poisson difference distribution inference and applications. Bull. Malays. Math. Sci. Soc. 8(33), 17–45 (2010)MathSciNetzbMATHGoogle Scholar
  2. Alzaid, A.A., Omair, M.A.: An extended binomial distribution with applications. Commun. Stat.-Theory Methods 41(19), 3511–3527 (2012)MathSciNetCrossRefGoogle Scholar
  3. Alzaid, A.A., Omair, M.A.: Poisson difference integer-valued autoregressive model of order one. Bull. Malays. Math. Sci. Soc. 2(37), 465–485 (2014)MathSciNetzbMATHGoogle Scholar
  4. Andersson, J., Karlis, D.: A parametric time series model with covariates for integers in \(Z\). Stat. Model. 14(2), 135–156 (2014)MathSciNetCrossRefGoogle Scholar
  5. Bakouch, H.S., Kachour, M., Nadarajah, S.: An extended poisson distribution. Commun. Stat.-Theory Methods 45(22), 6746–6764 (2016)MathSciNetCrossRefGoogle Scholar
  6. Chesneau, C., Kachour, M.: A parametric study for the first-order signed integer-valued autoregressive process. J. Stat. Theory Pract. 6(4), 760–782 (2012)MathSciNetCrossRefGoogle Scholar
  7. Kachour, M., Truquet, L.: A \(p\)-order signed integer-valued autoregressive (SINAR(\(p\))) model. J. Time Ser. Anal. 2(3), 223–236 (2011)MathSciNetCrossRefGoogle Scholar
  8. Karlis, D., Ntzoufras, I.: Bayesian modelling of football outcomes: using the Skellam’s distribution for the goal difference. J. Manag. Math. 20(2), 133–145 (2009)zbMATHGoogle Scholar
  9. Kim, H., Park, Y.: A non-stationary integer-valued autoregressive model. Stat. Papers 49(3), 485–502 (2008)MathSciNetCrossRefGoogle Scholar
  10. Klimko, L.A., Nelson, P.I.: On conditional least squares estimation for stochastic processes. Ann. Stat. 6(3), 629–642 (1978)MathSciNetCrossRefGoogle Scholar
  11. Skellam, J.G.: The frequency distribution of the difference between two poisson variates belonging to different populations. J. Royal Stat. Soc. 109(3), 296 (1946)MathSciNetCrossRefGoogle Scholar
  12. Steutel, F.W., van Harn, K.: Discrete analogues of self-decomposability and stability. Ann. Probab. 7(5), 893–899 (1979)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Coimbra College of AgriculturePolytechnic Institute of CoimbraCoimbraPortugal
  2. 2.CIDMAUniversity of AveiroAveiroPortugal
  3. 3.CEMAT and ISTUniversity of LisbonLisbonPortugal

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