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

We introduce a new goodness-of-fit test for count data on N 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 L² 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 consistent procedure of Meintanis (2009), as well as with more general competitors based on empirical distribution functions, kernel Stein discrepancies and other Stein-type characterizations.