Pseudo-likelihood Estimator of the Bivariate Von-Mises Cosine Model

Directional statistics are very useful tools to model the phenomenon that are characterized by the angles. Recently, various disciplines including biology, astronomy, meteorology and bioinformatics have paid attention to use these distributions. Particularly, it was shown in biological researches that there are two pair angles describing, relatively, the complete geometrical and spatial structures of a protein in the three dimensional space. There is a distribution, called bivariate Von-Mises, to represent the position of the atoms based upon the values of these angles in a probabilistic manner. In this paper, considering an especial case of this density (cosine model), the properties of distribution including the numbers of modes and its approximation by the bivariate normal distribution are first studied. Then, to estimate the parameters using the pseudo-likelihood method is described. The theoretical materials are evaluated in simulation studies and then the application of the cosine model in a real example is presented.