Wrapped Maxwell-Boltzmann distribution: properties and application in circular statistics
DOI :
https://doi.org/10.5269/bspm.79250Résumé
This paper introduces the Wrapped Maxwell-Boltzmann (WM-B) distribution, a novel circular probability model derived by wrapping the Maxwell distribution onto the unit circle. Through Poisson summation and subsequent normalization, the density is analytically simplified to its first-order form: a cosine perturbation of the circular uniform distribution. We derive its key distributional properties, including trigonometric moments, mean direction, circular variance, and entropy, and establish the non-negativity condition for the scale parameter as σ ≥ 0.2Ï€. Six methods for parameter estimation are investigated: Maximum Likelihood (MLE), Maximum Product of Spacings (MPSE), Least Squares (LS), Weighted Least Squares (WLS), Cramer von Mises (CvM), and Bayesian Estimation. Simulation studies demonstrate that the MPSE method is the most efficient for σ = 0.75, exhibiting the lowest bias and Root Mean Squared Error (RMSE). The model’s empirical relevance is confirmed through application to two real-life datasets: wind direction and pigeon homing experiment data. For the wind direction data, the fitted parameter σ = 2.23 yielded a Watson goodness-of-fit p-value of 0.475, indicating model adequacy. The WM-B distribution offers a mathematically simple and practically effective tool, demonstrating superior or competitive performance against established circular models in practical applications.
Références
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25. Noori, N. A., Abdullah, K. N., et al., Development and applications of a new hybrid Weibull-inverse Weibull distribution, Modern Journal of Statistics, 1, 1, 80–103, (2025). DOI: https://doi.org/10.64389/mjs.2025.01112
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27. Orji, G. O., Etaga, H. O., Almetwally, E. M., Igbokwe, C. P., Aguwa, O. C., and Obulezi, O. J., A new odd reparameterized exponential transformed-x family of distributions with applications to public health data, Innovation in Statistics and Probability, 1, 1, 88–118, (2025). DOI: https://doi.org/10.64389/isp.2025.01107
28. Nwankwo, B. C., Obiora-Ilouno, H. O., Almulhim, F. A., SidAhmed Mustafa, M., and Obulezi, O. J., Group acceptance sampling plans for type-I heavy-tailed exponential distribution based on truncated life tests, AIP Advances, 14, 3, (2024). DOI: https://doi.org/10.1063/5.0194258
29. Nwankwo, M. P., Alsadat, N., Kumar, A., Bahloul, M. M., and Obulezi, O. J., Group acceptance sampling plan based on truncated life tests for Type-I heavy-tailed Rayleigh distribution, Heliyon, 10, 19, (2024). DOI: https://doi.org/10.1016/j.heliyon.2024.e38150
30. Obulezi, O. J., Obiora-Ilouno, H. O., Osuji, G. A., Kayid, M., and Balogun, O. S., Weibull Sine Generalized Distribution Family: Fundamental Properties, Sub-model, Simulations, with Biomedical Applications, Electronic Journal of Applied Statistical Analysis, 18, 01, 183–212, (2025). DOI: https://doi.org/10.1285/i20705948v18n1p183
2. Chesneau, C., Tomy, L., and Jose, M., Wrapped modified Lindley distribution, Journal of Statistics and Management Systems 24(5), 1025–1040, (2021). DOI: https://doi.org/10.1080/09720510.2020.1796313
3. Fisher, N. I., Lewis, T., and Embleton, B. J. J., Statistical analysis of spherical data, Cambridge University Press, (1993).
4. Gatto, R., and Jammalamadaka, S. R., A saddlepoint approximation for testing exponentiality against some increasing failure rate alternatives, Statistics & probability letters 58(1), 71–81, (2002). DOI: https://doi.org/10.1016/S0167-7152(02)00117-7
5. Glimm, E., Fisher, NI: Statistical Analysis Of Circular Data. Cambridge University Press, Cambridge, UK 1995. 277 pp.,£16.95, Biometrical Journal 38(3), 314–314, (1996).
6. Heyes, S. B., Zokaei, N., and Husain, M., Longitudinal development of visual working memory precision in childhood and early adolescence, Cognitive Development 39, 36–44, (2016). DOI: https://doi.org/10.1016/j.cogdev.2016.03.004
7. Jammalamadaka, S. R., and Kozubowski, T. J., A general approach for obtaining wrapped circular distributions via mixtures, Sankhya A 79(1), 133–157, (2017). DOI: https://doi.org/10.1007/s13171-017-0096-4
8. Jammalamadaka, S. R., and Sengupta, A., Topics in circular statistics, vol. 5, world scientific, (2001).
9. Joshi, S., and Jose, K. K., Wrapped lindley distribution, Communications in Statistics-Theory and Methods 47(5), 1013–1021, (2018). DOI: https://doi.org/10.1080/03610926.2017.1280168
10. Keller, A. Z., and Kamath, A. R. R., Alternative reliability models for mechanical systems, in Proceedings of the 3rd International Conference on Reliability and Maintainability, 411–415, (1982).
11. Kirschner, S., and Tomasello, M., Joint drumming: Social context facilitates synchronization in preschool children, Journal of experimental child psychology 102(3), 299–314, (2009). DOI: https://doi.org/10.1016/j.jecp.2008.07.005
12. Mardia, K. V., Directional statistics and shape analysis, Journal of applied Statistics 26(8), 949–957, (1999). DOI: https://doi.org/10.1080/02664769921954
13. Mardia, K. V., Bookstein, F. L., and Moreton, I. J., Statistical assessment of bilateral symmetry of shapes, Biometrika, 285–300, (2000). URL: https://www.jstor.org/stable/2673464
14. Mardia, K. V., Statistics of directional data, Journal of the Royal Statistical Society Series B: Statistical Methodology 37(3), 349–371, (1975). DOI: https://doi.org/10.1111/j.2517-6161.1975.tb01550.x
15. Maxwell, J. C., V. Illustrations of the dynamical theory of gases.—Part I. On the motions and collisions of perfectly elastic spheres, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 19(124), 19–32, (1860). DOI: https://doi.org/10.1080/14786446008642818
16. Pewsey, A., Neuhauser, M., and Ruxton, G. D., Circular statistics in R, OUP Oxford, (2013). 17. Reif, F., Fundamentals of statistical and thermal physics, Waveland Press, (2009).
18. Rentsch, S., and Rand, M. K., Eye-hand coordination during visuomotor adaptation with different rotation angles, PLoS One 9(10), e109819, (2014). DOI: https://doi.org/10.1371/journal.pone.0109819
19. Semary, H. E., Alshad, K. B., Bengalath, J., and Alghamdi, S. M., Semicircular Maxwell–Boltzmann distribution: Application to posterior corneal curvature data, Journal of Radiation Research and Applied Sciences 18(2), 101471, (2025). DOI: https://doi.org/10.1016/j.jrras.2025.101471
20. Schmidt-Koenig, K. (1963). On the role of the loft, the distance and site of release in pigeon homing (the ”cross-loft experiment”). The Biological Bulletin, 125(1), 154–164. doi: https://doi.org/10.2307/1539298
21. Obulezi, O. J., Obulezi distribution: a novel one-parameter distribution for lifetime data modeling, Modern Journal of Statistics, 2, 1, 32–74, (2026). DOI: https://doi.org/10.64389/mjs.2026.02140
22. Chesneau, C., Theory on a new bivariate trigonometric Gaussian distribution, Innovation in Statistics and Probability, 1, 2, 1–17, (2025). DOI: https://doi.org/10.64389/isp.2025.01223
23. Gemeay, A. M., Moakofi, T., Balogun, O. S., Ozkan, E., and Hossain, M. M., Analyzing real data by a new heavy-tailed statistical model, Modern Journal of Statistics, 1, 1, 1–24, (2025). DOI: https://doi.org/10.64389/mjs.2025.01108
24. Mousa, M. N., Moshref, M. E., Youns, N., and Mansour, M. M. M., Inference under Hybrid Censoring for the Quadratic Hazard Rate Model: Simulation and Applications to COVID-19 Mortality, Modern Journal of Statistics, 2, 1, 1–31, (2026). DOI: https://doi.org/10.64389/mjs.2026.02113
25. Noori, N. A., Abdullah, K. N., et al., Development and applications of a new hybrid Weibull-inverse Weibull distribution, Modern Journal of Statistics, 1, 1, 80–103, (2025). DOI: https://doi.org/10.64389/mjs.2025.01112
26. Onyekwere, C. K., Aguwa, O. C., and Obulezi, O. J., An updated lindley distribution: Properties, estimation, acceptance sampling, actuarial risk assessment and applications, Innovation in Statistics and Probability, 1, 1, 1–27, (2025). DOI: https://doi.org/10.64389/isp.2025.01103
27. Orji, G. O., Etaga, H. O., Almetwally, E. M., Igbokwe, C. P., Aguwa, O. C., and Obulezi, O. J., A new odd reparameterized exponential transformed-x family of distributions with applications to public health data, Innovation in Statistics and Probability, 1, 1, 88–118, (2025). DOI: https://doi.org/10.64389/isp.2025.01107
28. Nwankwo, B. C., Obiora-Ilouno, H. O., Almulhim, F. A., SidAhmed Mustafa, M., and Obulezi, O. J., Group acceptance sampling plans for type-I heavy-tailed exponential distribution based on truncated life tests, AIP Advances, 14, 3, (2024). DOI: https://doi.org/10.1063/5.0194258
29. Nwankwo, M. P., Alsadat, N., Kumar, A., Bahloul, M. M., and Obulezi, O. J., Group acceptance sampling plan based on truncated life tests for Type-I heavy-tailed Rayleigh distribution, Heliyon, 10, 19, (2024). DOI: https://doi.org/10.1016/j.heliyon.2024.e38150
30. Obulezi, O. J., Obiora-Ilouno, H. O., Osuji, G. A., Kayid, M., and Balogun, O. S., Weibull Sine Generalized Distribution Family: Fundamental Properties, Sub-model, Simulations, with Biomedical Applications, Electronic Journal of Applied Statistical Analysis, 18, 01, 183–212, (2025). DOI: https://doi.org/10.1285/i20705948v18n1p183
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2026-03-11
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