Distributions for spherical data based on nonnegative trigonometric sums

A family of distributions for a random pair of angles that determine a point on the surface of a three-dimensional unit sphere (three-dimensional directions) is proposed. It is based on the use of nonnegative double trigonometric (Fourier) sums (series). Using this family of distributions, data that possess rotational symmetry, asymmetry or one or more modes can be modeled. In addition, the joint trigonometric moments are expressed in terms of the model parameters. An efficient Newton-like optimization algorithm on manifolds is developed to obtain the maximum likelihood estimates of the parameters. The proposed family is applied to two real data sets studied previously in the literature. The first data set is related to the measurements of magnetic remanence in samples of Precambrian volcanics in Australia and the second to the arrival directions of low mu showers of cosmic rays.