Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.1808965 Statistical Theory Applied Statistics Statistics (Other) İstatistiksel Teori Uygulamalı İstatistik İstatistik (Diğer) Circular data analysis with wrapped Poisson–Bilal distribution: Properties and applications https://orcid.org/0009-0009-5733-6641 Abd El-Bar Ahmed Tanta University https://orcid.org/0000-0002-7839-9686 Bengalath Jabir Government Arts and Science College 03 15 2026 55 2 826 850 10 22 2025 03 06 2026 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

This article introduces the wrapped Poisson-Bilal distribution, a novel discrete model for analyzing circular data, which is produced by wrapping the discrete Poisson-Bilal distribution on the unit circle. The probability mass function of the proposed model can be expressed as a mixture of two wrapped geometric distributions, offering a simpler and more interpretable model for circular data. We provide closed-form formulas for essential distributional properties, such as characteristic function, trigonometric moments, and circular measures of location and dispersion, allowing this model to be readily utilized in a variety of applications. The parameter of the proposed model is estimated using two methods: the maximum likelihood estimation and the moment method. The effectiveness of the model was evaluated through simulation studies in various parameter settings. Furthermore, we applied the model to three real-world datasets, demonstrating its superior performance compared to existing models based on different information criteria. These results demonstrate the capacity of the model as a powerful tool for analyzing circular data.

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