Geodesic Normal distribution on the circle

TL;DR

This paper introduces and analyzes the geodesic Normal distribution on the circle, comparing it with the von Mises distribution, and explores parameter estimation methods with simulations.

Contribution

It provides a detailed study of the geodesic Normal distribution on the circle, including properties, comparisons with von Mises, and maximum likelihood estimation techniques.

Findings

The geodesic Normal distribution has properties similar to Gaussian on the real line.

Comparisons show differences in intrinsic and extrinsic means and variances.

Maximum likelihood estimation for parameters is feasible and demonstrated with simulations.

Abstract

This paper is concerned with the study of a circular random distribution called geodesic Normal distribution recently proposed for general manifolds. This distribution, parameterized by two real numbers associated to some specific location and dispersion concepts, looks like a standard Gaussian on the real line except that the support of this variable is $[0,2\pi)$ and that the Euclidean distance is replaced by the geodesic distance on the circle. Some properties are studied and comparisons with the von Mises distribution in terms of intrinsic and extrinsic means and variances are provided. Finally, the problem of estimating the parameters through the maximum likelihood method is investigated and illustrated with some simulations.