Mixtures of Gamma Distributions and Harris Infinite Divisibility

Harris infinite divisibility

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

We look at the relation between mixtures of gamma distributions and Harris infinite divisibility. We show that mixtures of gamma distributions with shape parameter 1/k, k > 1 integer, are Harris infinitely divisible. We also prove that Harris infinitely divisible distributions are geometrically infinitely divisible and that all Harris infinitely divisible laws are not mixtures of gamma distributions.