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.
Bengalath et al. (Sun,) studied this question.