Score matching estimators for directional distributions

Abstract

One of the major problems for maximum likelihood estimation in the well-established directional models is that the normalising constants can be difficult to evaluate. A new general method of "score matching estimation" is presented here on a compact oriented Riemannian manifold. Important applications include von Mises-Fisher, Bingham and joint models on the sphere and related spaces. The estimator is consistent and asymptotically normally distributed under mild regularity conditions. Further, it is easy to compute as a solution of a linear set of equations and requires no knowledge of the normalizing constant. Several examples are given, both analytic and numerical, to demonstrate its good performance.