Alternatives to the EM Algorithm for ML-Estimation of Location, Scatter Matrix and Degree of Freedom of the Student-$t$ Distribution

TL;DR

This paper introduces three new algorithms as alternatives to the EM algorithm for maximum likelihood estimation of the Student-$t$ distribution parameters, especially focusing on the often-overlooked degrees of freedom parameter, with theoretical guarantees and practical applications.

Contribution

The paper develops and analyzes three novel algorithms for ML estimation of Student-$t$ parameters, including the degrees of freedom, which do not rely on the EM framework and ensure decreasing objective function.

Findings

All three algorithms guarantee a decrease in the likelihood function per iteration.

Numerical simulations demonstrate the effectiveness of the algorithms.

Application to image data with Student-$t$ noise shows practical utility.

Abstract

In this paper, we consider maximum likelihood estimations of the degree of freedom parameter $\nu$, the location parameter $\mu$ and the scatter matrix $\Sigma$ of the multivariate Student-$t$ distribution. In particular, we are interested in estimating the degree of freedom parameter $\nu$ that determines the tails of the corresponding probability density function and was rarely considered in detail in the literature so far. We prove that under certain assumptions a minimizer of the negative log-likelihood function exists, where we have to take special care of the case $\nu \rightarrow \infty$, for which the Student-$t$ distribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixed $\nu$, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix $\nu$, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for $\nu$. We show how the objective function behaves for the different updates of $\nu$ and prove for all three algorithms that it decreases in each iteration step. We compare the algorithms as well as some accelerated versions by numerical simulation and apply one of them for estimating the degree of freedom parameter in images corrupted by Student-$t$ noise.