Statistical Papers

, Volume 60, Issue 6, pp 2119–2139 | Cite as

Mixed Poisson INAR(1) processes

  • Wagner Barreto-SouzaEmail author
Regular Article


Overdispersion is a phenomenon commonly observed in count time series. Since Poisson distribution is equidispersed, the INteger-valued AutoRegressive (INAR) process with Poisson marginals is not adequate for modelling overdispersed counts. To overcome this problem, in this paper we propose a general class of first-order INAR processes for modelling overdispersed count time series. The proposed INAR(1) processes have marginals belonging to a class of mixed Poisson distributions, which are overdispersed. With this, our class of overdispersed count models have the known negative binomial INAR(1) process as particular case and open the possibility of introducing new INAR(1) processes, such as the Poisson-inverse Gaussian INAR(1) model, which is discussed here with some details. We establish a condition to our class of overdispersed INAR processes is well-defined and study some statistical properties. We propose estimators for the parameters and establish their consistency and asymptotic normality. A small Monte Carlo simulation to evaluate the finite-sample performance of the proposed estimators is presented and one application to a real data set illustrates the usefulness of our proposed overdispersed count processes.


Autocorrelation Count process Markov chain Poisson-inverse Gaussian distribution Overdispersion 



I thank the two anonymous referees and the Associated Editor for their useful suggestions and comments that led to an improved version of this article. I also thank the financial support from CNPq (Brazil) and FAPEMIG (Brazil).


  1. Abraham B, Balakrishna N (2002) Inverse gaussian autoregressive models. J Time Ser Anal 20:605–618MathSciNetzbMATHGoogle Scholar
  2. Alamatsaz MH (1983) Completeness and self-decomposability of mixtures. Ann Inst Stat Math 35:355–363MathSciNetzbMATHGoogle Scholar
  3. Aly EEAA, Bouzar N (1994) Explicit stationary distributions for some galton-watson processes with immigration. Stoch Models 10:499–517MathSciNetzbMATHGoogle Scholar
  4. Al-Osh MA, Alzaid AA (1987) First-order integer valued autoregressive (INAR(1)) process. J Time Ser Anal 8:261–275MathSciNetzbMATHGoogle Scholar
  5. Anderson TW (1971) The statistical analysis of time series. Wiley, New YorkzbMATHGoogle Scholar
  6. Andersson J, Karlis D (2014) A parametric time series model with covariates for integers in \(\mathbb{Z}\). Stat Model 14:135–156MathSciNetGoogle Scholar
  7. Barreto-Souza W (2015) Zero-modified geometric INAR(1) process for modelling count time series with deflation or inflation of zeros. J Time Ser Anal 36:839–852MathSciNetzbMATHGoogle Scholar
  8. Barreto-Souza W, Bourguignon M (2015) A skew INAR(1) process on \(\mathbb{Z}\). Adv Stat Anal 99:189–208MathSciNetzbMATHGoogle Scholar
  9. Bisaglia L, Canale A (2016) Bayesian nonparametric forecasting for INAR models. Comput Stat Data Anal 100:70–78MathSciNetzbMATHGoogle Scholar
  10. Forst G (1979) A characterization of self-decomposable probabilities in the half-line. Zeit Wahrscheinlichkeitsth 49:349–352MathSciNetzbMATHGoogle Scholar
  11. Freeland RK, McCabe BPM (2004a) Analysis of low count time series data by Poisson autoregression. J Time Ser Anal 25:701–722MathSciNetzbMATHGoogle Scholar
  12. Freeland RK, McCabe BPM (2004b) Forecasting discrete valued low count time series. Int J Forecast 20:427–434Google Scholar
  13. Freeland RK, McCabe BPM (2005) Asymptotic properties of CLS estimators in the Poisson AR(1) model. Stat Prob Lett 73:147–153MathSciNetzbMATHGoogle Scholar
  14. Hamilton JD (1994) Time series analysis. Princeton University Press, PrincetonzbMATHGoogle Scholar
  15. Harvey AC, Fernandes C (1989) Time series models for count or qualitative observations. J Bus Econ Stat 7:407–417Google Scholar
  16. Jazi MA, Jones G, Lai CD (2012) First-order integer valued processes with zero inflated poisson innovations. J Time Ser Anal 33:954–963MathSciNetzbMATHGoogle Scholar
  17. Jung RC, Tremayne AR (2011) Useful models for time series of counts or simply wrong ones? Adv Stat Anal 95:59–91MathSciNetzbMATHGoogle Scholar
  18. Karlis D, Xekalaki E (2005) Mixed Poisson distributions. Int Stat Rev 73:35–58zbMATHGoogle Scholar
  19. Karlsen H, Tjostheim D (1988) Consistent estimates for the NEAR(2) and NLAR(2) time series models. J R Stat Soc Ser B 50:313–320MathSciNetGoogle Scholar
  20. McKenzie E (1985) Some simple models for discrete variate time series. Water Resour Bull 21:645–650Google Scholar
  21. McKenzie E (1986) Autoregressive moving-average processes with negative binomial and geometric marginal distributions. Adv Appl Probab 18:679–705MathSciNetzbMATHGoogle Scholar
  22. McKenzie E (1988) Some ARMA models for dependent sequences of Poisson counts. Adv Appl Probab 20:822–835MathSciNetzbMATHGoogle Scholar
  23. McKenzie E (2003) Discrete variate time series. In: Rao CR, Shanbhag DN (eds) Handbook of statistics. Elsevier, Amsterdam, pp 573–606Google Scholar
  24. Meintanis SG, Karlis D (2014) Validation tests for the innovation distribution in INAR time series models. Comput Stat 29:1221–1241MathSciNetzbMATHGoogle Scholar
  25. Nastić AS, Ristić MM (2012) Some geometric mixed integer-valued autoregressive (INAR) models. Stat Probab Lett 82:805–811MathSciNetzbMATHGoogle Scholar
  26. Nastić AS, Ristić MM, Djordjević MS (2016a) An INAR model with discrete Laplace marginal distributions. Braz J Probab Stat 30:107–126MathSciNetzbMATHGoogle Scholar
  27. Nastić AS, Laketa PN, Ristić MM (2016b) Random environment integer-valued autoregressive process. J Time Ser Anal 37:267–287MathSciNetzbMATHGoogle Scholar
  28. Nastić AS, Ristić MM, Janjić AD (2016c) A mixed thinning based geometric INAR(1) model. FilomatGoogle Scholar
  29. Pillai RN, Satheesh S (1992) \(\alpha \)-inverse Gaussian distributions. Sankhya A 54:288–290MathSciNetzbMATHGoogle Scholar
  30. Ridout MS (2009) Generating random numbers from a distribution specified by its Laplace transform. Stat Comput 19:439–450MathSciNetGoogle Scholar
  31. Ristić MM, Nastić AS, Ilić AVM (2013) A geometric time series model with dependent Bernoulli counting series. J Time Ser Anal 34:466–476MathSciNetzbMATHGoogle Scholar
  32. Ristić MM, Bakouch HS, Nastić AS (2009) A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. J Stat Plan Inference 139:2218–2226MathSciNetzbMATHGoogle Scholar
  33. Ristić MM, Nastić AS, Bakouch HS (2012) Estimation in an integer-valued autoregressive process with negative binomial marginals (NBINAR(1)). Commun Stat 41:606–618MathSciNetzbMATHGoogle Scholar
  34. Schweer S, Weiß CH (2014) Compound Poisson INAR(1) processes: stochastic properties and testing for overdispersion. Comput Stat Data Anal 77:267–284MathSciNetzbMATHGoogle Scholar
  35. Scotto MG, Weiß CH, Gouveia S (2015) Thinning-based models in the analysis of integer-valued time series: a review. Stat Model 15:590–618MathSciNetGoogle Scholar
  36. Steutel FW, van Harn K (1979) Discrete analogues of self-decomposability and stability. Ann Probab 7:893–899MathSciNetzbMATHGoogle Scholar
  37. Weiß CH (2008a) Thinning operations for modeling time series of counts-a survey. Adv Stat Anal 92:319–341MathSciNetGoogle Scholar
  38. Weiß CH (2008b) Serial dependence and regression of Poisson INARMA models. J Stat Plan Inference 138:2975–2990MathSciNetzbMATHGoogle Scholar
  39. Weiß CH (2009) Controlling jumps in correlated processes of Poisson counts. Appl Stoch Models Bus Ind 25:551–564MathSciNetzbMATHGoogle Scholar
  40. Weiß CH (2013) Integer-valued autoregressive models for counts showing underdispersion. J Appl Stat 40:1931–1948MathSciNetGoogle Scholar
  41. Weiß CH (2015) A Poisson INAR(1) model with serially dependent innovations. Metrika 78:829–851MathSciNetzbMATHGoogle Scholar
  42. Weiß CH, Homburg A, Puig P (2016) Testing for zero inflation and overdispersion in INAR(1) models. Stat PapGoogle Scholar
  43. Weiß CH, Kim HY (2013) Binomial AR(1) processes: moments, cumulants, and estimation. Statistics 47:494–510MathSciNetzbMATHGoogle Scholar
  44. Yang K, Wang D, Jia B, Li H (2016) An integer-valued threshold autoregressive process based on negative binomial thinning. Stat PapGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Departamento de EstatísticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

Personalised recommendations