Developing Underdispersed Discrete Distributions: A New Approach to Poisson Generalization

This thesis introduces a new family of discrete, mean-parametrized count distributions that can achieve arbitrarily low variance for a given mean. The properties of this family are then used to generalize the Poisson distribution, enabling it to model scenarios where the data is not equidispersed. The desirable underdispersion limit behavior of a shifted Bernoulli with any mean is achieved, a feature currently found only in the COM-Poisson distribution, which, however, lacks an explicit formula for the mean.