Statistical Methods & Applications

, Volume 28, Issue 4, pp 625–653 | Cite as

BINAR(1) negative binomial model for bivariate non-stationary time series with different over-dispersion indices

  • Yuvraj SunecherEmail author
  • Naushad Mamode Khan
  • Miroslav M. Ristić
  • Vandna Jowaheer
Original Paper


The existing stationary bivariate integer-valued autoregressive model of order 1 (BINAR(1)) with correlated Negative Binomial (NB) innovations is capable of modelling stationary count series where the innovation terms of both series have same over-dispersion index. Such BINAR(1) may not be useful to model real-life series that are affected by common time-dependent covariates whereby the two series may display non-stationarity as well as different over-dispersion indices. In this paper, we propose a novel BINAR(1) model with the pair of innovations following a joint NB distribution that accommodates different over-dispersion indices. The estimation of parameters is conducted using generalized quasi-likelihood (GQL) approach that operates in two phases. Monte Carlo simulations are implemented to assess the performance of the proposed GQL under the wide range of combinations of the model parameters. This BINAR(1) model is also applied to analyze the daily series of day and night accident data in some regions of Mauritius.


BINAR(1) Non-stationary NB GQL Over-dispersion 

Mathematics Subject Classification

65C60 62J12 62H12 62J20 62J10 



  1. Cuadras CM (2002) Correspondence analysis and diagonal expansions in terms of distribution function. J Stat Plan Inference 103(1–2):137–150MathSciNetCrossRefGoogle Scholar
  2. Eagleson GK (2002) Polynomial expansions of bivariate distributions. Ann Math Stat 35(3):1208–1215MathSciNetCrossRefGoogle Scholar
  3. Jowaheer V, Sutradhar BC (2002) Analysing longitudinal count data with overdispersion. Biometrika 89(2):389–399MathSciNetCrossRefGoogle Scholar
  4. Jowaheer V, Mamode Khan NA, Sunecher Y (2017) A BINAR(1) time series model with cross-correlated COM–Poisson innovations. Commun Stat Theory Methods 47(5):1133–1154MathSciNetCrossRefGoogle Scholar
  5. Karlis D, Pedeli X (2013) Flexible bivariate INAR(1) processes using copulas. Commun Stat Theory Methods 42(4):723–740MathSciNetCrossRefGoogle Scholar
  6. Mallick TS, Sutradhar BC (2008) GQL versus conditional GQL inferences for non-stationary time series of counts with overdispersion. J Time Ser Anal 29(2):402–420MathSciNetCrossRefGoogle Scholar
  7. Mamode Khan NA, Sunecher Y, Jowaheer V (2016) Modelling a non-stationary BINAR(1) Poisson process. J Stat Comput Simul 86(15):3106–3126MathSciNetCrossRefGoogle Scholar
  8. Marshall AW, Olkin I (1985) A family of bivariate distributions generated by the bivariate Bernoulli distribution. J Am Stat Assoc 80(390):332–338MathSciNetCrossRefGoogle Scholar
  9. Marshall AW, Olkin I (1990) Multivariate distribution generated from mixtures of convolution and product families. In: Block HW, Sampson AR, Sanits TH (eds) Topics in statistical dependence, vol 16. Institute of Mathematical Statistics, Bethesda, pp 371–393CrossRefGoogle Scholar
  10. McKenzie E (1986) Autoregressive moving-average processes with negative binomial and geometric marginal distrbutions. Adv Appl Probab 18(3):679–705MathSciNetCrossRefGoogle Scholar
  11. Nastic AS, Laketa PN, Ristic MM (2016) Random environment integer-valued autoregressive process. J Time Ser Anal 37(2):267–287MathSciNetCrossRefGoogle Scholar
  12. Ng CM, Ong SH, Srivastava HM (2010) A class of bivariate negative binomial distributions with different index parameters in the marginals. Appl Math Comput 217(7):3069–3087MathSciNetzbMATHGoogle Scholar
  13. Pan W (2001) Akaike’s information criterion in generalized estimating equations. Biometrics 57(1):120–125MathSciNetCrossRefGoogle Scholar
  14. Pedeli X, Karlis D (2011) A bivariate INAR(1) process with application. Stat Model Int J 11(4):325–349MathSciNetCrossRefGoogle Scholar
  15. Pedeli X, Karlis D (2013) On composite likelihood estimation of a multivariate INAR(1) model. J Time Ser Anal 34(2):206–220MathSciNetCrossRefGoogle Scholar
  16. Popovic PM (2015) Random coefficient bivariate INAR(1) process. Facta Univer Ser Math Inform 30(3):263–280MathSciNetzbMATHGoogle Scholar
  17. Ristic MM, Nastic AS, Jayakumar K, Bakouch HS (2012) A bivariate INAR(1) time series model with geometric marginals. Appl Math Lett 25(3):481–485MathSciNetCrossRefGoogle Scholar
  18. Schweer S, Weib CH (2014) Compound Poisson INAR(1) processes: stochastic properties and testing for overdispersion. Comput Stat Data Anal 77(c):267–284MathSciNetCrossRefGoogle Scholar
  19. Sunecher Y, Mamodekhan NA, Jowaheer V (2016) Estimating the parameters of a BINMA Poisson model for a non-stationary bivariate time series. Commun Stat Simul Comput 46(9):6803–6827MathSciNetCrossRefGoogle Scholar
  20. Sunecher Y, Mamodekhan NA, Jowaheer V (2017) A GQL estimation approach for analysing non-stationary over-dispersed BINAR(1) time series. J Stat Comput Simul 87(10):1911–1924MathSciNetCrossRefGoogle Scholar
  21. Sutradhar BC (2008) Best practice recommendation for forecasting counts. Technical report, Department of Mathematics and Statistics, Memorial University of Newfoundland, CanadaGoogle Scholar
  22. Sutradhar BC, Jowaheer V, Rao P (2014) Remarks on asymptotic efficient estimation for regression effects in stationary and non-stationary models for panel count data. Braz J Probab Stat 28(2):241–254MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yuvraj Sunecher
    • 1
    Email author
  • Naushad Mamode Khan
    • 2
  • Miroslav M. Ristić
    • 3
  • Vandna Jowaheer
    • 2
  1. 1.University of Technology MauritiusPort LouisMauritius
  2. 2.University of MauritiusReduitMauritius
  3. 3.University of NišNišSerbia

Personalised recommendations