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In this paper, we consider maximum likelihood estimations of the degree offreedom parameter $\nu$, the location parameter $\mu$ and the scatter matrix$\Sigma$ of the multivariate Student-$t$ distribution. In particular, we areinterested in estimating the degree of freedom parameter $\nu$ that determinesthe tails of the corresponding probability density function and was rarelyconsidered in detail in the literature so far. We prove that under certainassumptions a minimizer of the negative log-likelihood function exists, wherewe have to take special care of the case $\nu \rightarrow \infty$, for whichthe Student-$t$ distribution approaches the Gaussian distribution. Asalternatives to the classical EM algorithm we propose three other algorithmswhich cannot be interpreted as EM algorithm. For fixed $\nu$, the firstalgorithm is an accelerated EM algorithm known from the literature. However,since we do not fix $\nu$, we cannot apply standard convergence results for theEM algorithm. The other two algorithms differ from this algorithm in theiteration step for $\nu$. We show how the objective function behaves for thedifferent updates of $\nu$ and prove for all three algorithms that it decreasesin each iteration step. We compare the algorithms as well as some acceleratedversions by numerical simulation and apply one of them for estimating thedegree of freedom parameter in images corrupted by Student-$t$ noise.

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