Discrete averaged mixing applied to the logarithmic distributions (Q2822867)

The paper contributes to the problem of classification of probability distributions into broader classes, what can be substantiated both mathematically and from the point of view of applications, and introduces a new family of mixtures of discrete distributions. In this paper the authors present a new type of mixing, namely addition (or subtraction) of the non-normalized components of ``standard'' distributions divided by the sum (or difference) of the normalizing constants. The family of distributions which can be considered to be results of this new type of mixing is called discrete averaged mixed distributions. The authors further analyze a subclass of the family, namely averaged mixed logarithmic (AML) distributions: their general form, parametric space, basic probabilistic characteristics as pmf, pgf, moments, and connections with other types of mixing and special cases which appeared in the literature.NEWLINENEWLINESeveral AML distributions have appeared in the literature in the past, however, without being considered as special cases of the AML family of distributions, as, e.g. the Plunkett-Jain logarithmic distribution. The other distributions are somewhat hidden, but, as the authors present in this paper, they also can be rewritten in the AML form, as, e.g., the Cane logarithmic distribution, the Galton-Watson logarithmic distribution, the Gross-Harris logarithmic distribution, the Holgate-logarithmic distribution, the Jackson-Nickols logarithmic distribution, the Kemp logarithmic distribution, the Kobayashi-Osawa logarithmic distribution, the Lotka logarithmic distribution, and the Rubinovitch logarithmic distribution.