Wrapped Maxwell-Boltzmann distribution: properties and application in circular statistics

Chinonso  Eze; Doaa Ahmed; Charity  Onwuamaeze; Eviano   Joel; Okechukwu J. Obulezi; Ehab  Almetwally; Mohammed Elgarhy

doi:10.5269/bspm.79250

Chinonso Michael Eze Department of Statistics, University of Nigeria, Nsukka, Nigeria https://orcid.org/0000-0003-2755-2182
Doaa M. H. Ahmed Department of Insurance and Risk Management, Faculty of Business, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, 11432, Saudi Arabia https://orcid.org/0009-0005-7569-4830
Charity Uchenna Onwuamaeze Department of Statistics, University of Nigeria, Nsukka, Nigeria https://orcid.org/0009-0005-0745-785X
Eviano Israel Joel Department of Statistics, Dennis Osadebay University, Asaba https://orcid.org/0009-0004-3487-1549
Okechukwu J. Obulezi Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Nigeria
Ehab M. Almetwally Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia https://orcid.org/0009-0000-6949-9805
Mohammed Elgarhy Department of Basic Sciences, Higher Institute of Administrative Sciences, Belbeis, AlSharkia, Egypt https://orcid.org/0000-0002-1333-3862

Abstract

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 ≥ 0.2π. 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.

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