Poisson quasi-Lindley and Poisson-new XLindley univariate and bivariate models: derivation techniques and automobile insurance applications

The Poisson quasi-Lindley and the Poisson-new XLindley distributions are revisited, emphasizing on alternative derivation techniques. These distributions can be derived as Poisson mixtures when (i) the probability density function of the mixing distribution is known, (ii) the moment generating function of the mixing distribution is known, or (iii) the regression function of the mixing continuous random variable on the mixed discrete random variable is of a known form. Furthermore, they can be derived by the addition of independent random variables. An indication that the Poisson-new XLindley distribution is a member of the class of Poisson quasi- Lindley models is also given. An Extended Poisson quasi-Lindley (EPQL) distribution is constructed following the above derivation procedures and, as a generalized binomial distribution, it is extensively studied, highlighting its role as a marginal distribution in bivariate settings. Two general and structurally different bivariate Poisson quasi-Lindley and Poisson-new XLindley distributions are then introduced utilizing various techniques, including mixing, generalization, addition of independent bivariate random variables, regression functions, and conditional distributions. These bivariate models exhibit positive correlation and over-dispersed marginals. Several of their characteristics are derived, including probability generating functions, probabilities and their recurrences, moments, conditional distributions, and regression functions. The special feature of these general models is that several of their members, including bivariate Poisson-new XLindley distributions, are fitted satisfactorily to different sets of automobile insurance data previously used in the literature. In particular, members of the first bivariate framework are applied to three sets of data involving the number of claims and claim amounts, while members of the second framework are fitted to data concerning material damage and bodily injury from portfolios of liability insurance policies. Finally, suggestions for future research are also provided.