  The von Mises-Fisher distribution is one of the most widely used probability
distributions to describe directional data. Finite mixtures of von Mises-Fisher
distributions have found numerous applications. However, the likelihood
function for the finite mixture of von Mises-Fisher distributions is unbounded
and consequently the maximum likelihood estimation is not well defined. To
address the problem of likelihood degeneracy, we consider a penalized maximum
likelihood approach whereby a penalty function is incorporated. We prove strong
consistency of the resulting estimator. An Expectation-Maximization algorithm
for the penalized likelihood function is developed and simulation studies are
performed to examine its performance.
