We describe a methodology of constructing probability distributions on the nonnegative integers (aka counting distributions). Counting distributions have historic roots as they have been studied since the beginning of Probability Theory. The reason is their statistical importance and applicability in almost all societal and scientific areas. The methodology is based on considering natural exponential families of probability distributions. These families are uniquely determined by their variance functions. Then, counting distributions can be constructed from variance functions that show some regularity conditions. The counting distributions in this talk are constructed from polynomial variance functions with nonnegative coefficients. The usability of these new counting distributions is exhibited by various applications of data fitting, and insurance risk modeling.
This is joint work with Shaul K. Bar-Lev (Holon Institute of Technology, Holon, Israel).