Approximating Directional Densities by Sequences of Exponential Families

This paper develops approximation techniques for directional densities by nite dimensional exponential densities. The methodology is to use expansions with respect to classical spherical harmonics followed by estimating the unknown parameters by maximum likelihood. The advantage of this approach is that one estimates directional probability densities by probability densities while at the same time making use of higher orders of smoothness. In studying rates of convergence, the diiculty of estimating the density depends on the smoothness of the underlying density. Min-imax rates of convergence in terms of the Kullback-Leibler information divergence are obtained for densities in Sobolev spaces. In addition, automatic algorithms for model selection in terms of basis inclusion/exclusion are adapted for the directional framework. Numerical results using both simulated and real data are included for assessing nite sample performance.