A goodness-of-fit test for the Zeta distribution with unknown parameter

TL;DR

This paper proposes a new goodness-of-fit test for the Zeta distribution with unknown parameters, utilizing a Stein-type characterization and birth-death process generator, demonstrating its consistency and validating it through simulations.

Contribution

It introduces a novel Stein-based test for Zeta distribution fitting with unknown parameters, including theoretical validation and simulation comparisons.

Findings

Test is omnibus consistent

Validates null distribution and bootstrap procedure

Outperforms existing Zeta-specific tests in simulations

Abstract

We introduce a new goodness-of-fit test for count data on $\mathbb{N}$ for the Zeta distribution with unknown parameter. The test is built on a Stein-type characterization that uses, as Stein operator, the infinitesimal generator of a birth-death process whose stationary distribution is Zeta. The resulting $L^2$-type statistic is shown to be omnibus consistent, and we establish the limit null behavior as well as the validity of the associated parametric bootstrap procedure. In a Monte Carlo simulation study, we compare the proposed test with the only existing Zeta-specific procedure of Meintanis (2009), as well as with more general competitors based on empirical distribution functions, kernel Stein discrepancies and other Stein-type characterizations.