A New Estimation Methodology for Standard Directional Distributions

One of the major stumbling blocks in the use of standard directional distributions is difficulty in implementing the well-established maximum likelihood estimation method. The root of the problem is that their normalizing constants are complicated. To circumvent this problem, sometimes a member from a non-exponential family is used and then a relevant member from the exponential family is matched. These alternative distributions have simple moment estimators though the matching can be somewhat arbitrary. Firstly, we resolve the estimation problem by introducing the score matching estimators for directional distributions. Secondly, we resolve the matching problem optimally by introducing the score matching approximation. We give some insight into these proposals through the basic directional distributions, though the results are optimal and universally applicable for distributions on compact manifolds.