Geodesic Normal distribution on the circle

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, 2π) 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. Circular statistics deal with random variables taking values on hyperspheres and can be included in the broader field of directional statistics. Applications of circular statistics are numerous and can be found, for example, in fields such as climatology (wind direction data [MJ00]), biology (pigeons homing performances [Wat83]) or earth science (earthquake locations occurence and other data types, see [MJ00] for examples) among others. A circular distribution is a probability distribution function (pdf) which mass is concentrated on the circumference of a unit circle. The support of a random variable θ representing an angle measured in radians may be taken to [0, 2π) or [−π, π). We will focus here on continuous circular distributions, that is on absolutely continuous (w.r.t. the Lebesgue measure on the circumference) distributions. A pdf of a circular random variable has to fulfill the following axioms (i) f(θ) ≥ 0. (ii) ∫ 2π 0 f(θ)dθ = 1 (iii) f(θ) = f(θ + 2kπ) for any integer k (i.e. f is periodic).