Penalized Maximum Likelihood Estimator for Mixture of von Mises-Fisher Distributions

TL;DR

This paper introduces a penalized maximum likelihood estimator for finite mixtures of von Mises-Fisher distributions, addressing the issue of unbounded likelihood and demonstrating its consistency and practical performance.

Contribution

It proposes a novel penalized likelihood approach for mixture models of von Mises-Fisher distributions, ensuring well-defined estimation and proving its strong consistency.

Findings

The penalized estimator is strongly consistent.

An EM algorithm for the penalized likelihood is developed.

Simulation studies show effective performance of the method.

Abstract

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.