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

Bruno Ebner; Daniel Hlubinka

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arXiv · 2512.24128

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

Bruno Ebner·Daniel Hlubinka

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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.

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Bruno Ebner, Daniel Hlubinka. 2025-12-30. A goodness-of-fit test for the Zeta distribution with unknown parameter. https://arxiv.org/abs/2512.24128

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