New bivariate Poisson extended exponential distributions and associated BINAR(1) processes with applications

Abstract

This study proposes two types of bivariate Poisson extended exponential distributions: the basic bivariate Poisson extended exponential distribution and the Sarmanov-based bivariate Poisson extended exponential distribution. The two bivariate Poisson extended exponential distributions are then introduced as joint innovation distributions in a bivariate first-order integer-valued autoregressive process based on binomial thinning. The model parameters are estimated using maximum likelihood in basic bivariate Poisson extended exponential and Sarmanov-based bivariate Poisson extended exponential distributions. Conditional maximum likelihood is applied to the bivariate first-order integer-valued autoregressive process. Simulation experiments are used to evaluate the performance of small and large samples. In addition, the newly developed bivariate first-order integer-valued autoregressive models are then applied to the Pittsburgh crime series data and candies data. We show that they fit better than existing bivariate first-order integer-valued autoregressive models described in the literature.