  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.
