On The Poisson Difference Distribution Inference and Applications

The distribution of the disfference between two independent Poisson random variables involves the modified Bessel function of the frst kind. Using properties of this function, maximum likelihood estimates of the parameters of the Poisson difference were derived. Asymptotic distribution property of the maximum likelihood estimates is discussed. Maximum likelihood estimates were compared with the moment estimates in a Monte Carlo study. Hypothesis test- ing using likelihood ratio tests was considered. Some new formulas concerning the modiffed Bessel function of the frst kind were provided. Alternative formu- las for the probability mass function of the Poisson difference (PD) distribution are introduced. Finally, two new applications for the PD distribution are pre- sented. The frst is from the Saudi stock exchange (TASI) and the second is from Dallah hospital.