A Stein Characterization-type Omnibus Tests for the Discrete Pareto Distribution

The discrete Pareto (or Zeta, Zipf) distribution, arises naturally in modeling rank-frequency data across diverse fields such as linguistics, demography, biology, and computer science. Despite its widespread applicability, goodness-of-fit testing for the discrete Pareto distribution remains underdeveloped, particularly in the presence of heavy tails and infinite support. This article introduces a novel goodness-of-fit test based on a new Stein-type characterization of the discrete Pareto distribution, formulated using its probability generating function. The proposed method is applicable even when the shape parameter is unknown and avoids binning or smoothing techniques. We study the asymptotic properties of the test and assess its empirical size and power through extensive simulation experiments. The results show that the proposed test either outperforms or matches the performance of existing method across various alternatives. Applications to real datasets are provided to demonstrate its practical relevance and robustness.