Characterizing the Zeta Distribution via Continuous Mixtures

TL;DR

This paper introduces two new ways to characterize the Zeta distribution using continuous mixtures of Negative Binomial and Poisson distributions, providing insights into their structure and identifiability.

Contribution

It presents novel characterizations of the Zeta distribution as continuous mixtures, including cases with fixed and quasi-distributions, enhancing understanding of its properties.

Findings

Zeta distribution as mixture of Negative Binomial distributions for r >= 1

Zeta distribution as mixture of Poisson distributions

Identifiability of the distributions through unique mixing distributions

Abstract

We offer two novel characterizations of the Zeta distribution: first, as tractable continuous mixtures of Negative Binomial distributions (with fixed shape parameter, r > 0), and second, as a tractable continuous mixture of Poisson distributions. In both the Negative Binomial case for r >= 1 and the Poisson case, the resulting Zeta distributions are identifiable because each mixture can be associated with a unique mixing distribution. In the Negative Binomial case for 0 < r < 1, the mixing distributions are quasi-distributions (for which the quasi-probability density function assumes some negative values).