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

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.