Discrete Power Distributions and Inference Using Likelihood

Abstract

Discrete power distributions are proposed and studied, by considering the positive jumps on the discontinuities of an original discrete distribution function. Inequalities in moments and distribution functions are studied, allowing the definition of discrete intermediate distributions that lie between an original distribution and a power distribution. Original uniform, binomial, Poisson, negative binomial, and hypergeometric distributions are considered, to propose new power and intermediate distributions. Stochastic orders and unimodality are discussed. Estimation problems using likelihood are investigated. Simulation experiments are performed, to evaluate the bias and the mean square error of the maximum likelihood estimates, that are numerically calculated, with classic tools for numerical optimization.

Keywords

• Asymptotics
• Inequalities
• Information
• Intermediate distributions
• Maximum likelihood estimation
• Power distributions
• Stochastic orders
• Unimodality