Bivariate Poisson Generalized Lindley Distribution and the Associated BINAR (1) Process

Modeling probabilistic behavior of two random variables simultaneously demanded the construction of bivariate random variables. Bivariate distributions may be thus considered as extensions of univariate distributions. Many approaches have been discussed in the literature for the construction of bivariate random variables. Balakrishna and Lai [2009 ] mentions many of those. Constructing discrete as well as continuous bivariate random variables using mixture methodology is discussed frequently in the statistical literature (see Karlis and Xekalaki [2005 ], Lai [2006 ], Sarabia Alegría et al. [2008 ], etc). The main advantage of this method is that it will have simple expressions for its marginal probability density function (pdf) and hence its moments, correlation, etc. Another method is to use a new class of distribution for the construction. The Sarmanov family ( Sarmanov [1966 ]) of distributions can be used to construct discrete as well as continuous bivariate distributions with a flexible covariance structure. Ting Lee [1996 ] studied various properties regarding the family and constructed bivariate distributions by using various marginal distributions and construction methods. Also the Farlie-Gumbel-Morgenstern (FGM) copula is a special case of Sarmanov family The INAR (1) model is suitable for time series count datasets, which exhibit over-dispersion, in many statistical and applied fields such as medical insurance, sports, finance, etc. Since the initial works of McKenzie [1985 ] and Al Osh and Alzaid [1987 ] on INAR (1) process with Poisson innovations, significant number of works has been introduced in literature having univariate innovation distributions (see, Livio et al. [2018 ], Altun [2020 ], Eliwa et al. [2020 ], Emrah and Mamode Khan [2021 ], Irshad et al. [2021a ], etc).