Poisson distribution is well used as a standard model for analyzing the count data in many fields such as environmental sciences, epidemiology, manufacturing applications, econometrics and finance. The importance for testing the fit of the observations arises from a Poisson distribution. In general, the tests related to the Index of Dispersion (ID) concerning the mean and variance is quite powerful against a wide variety of alternatives except when the ID is closer to 1 (Gurtler and ɺɺ Henze, 2000: 219; Karlis and Xekalaki, 2000: 381; Meintanis ...