In this paper, we consider maximum likelihood estimations of the degree of freedom parameter ν, the location parameter μ and the scatter matrix Σ of the multivariate Student t distribution. In particular, we are interested in estimating the degree of freedom parameter ν 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 ν, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix ν, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for ν. We show how the objective function behaves for the different updates of ν 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.

在本文中，我们考虑了多元学生t分布的自由度参数ν，位置参数μ和散射矩阵Σ的最大似然估计。特别地，我们对估计确定相应概率密度函数的尾部的自由度参数ν感兴趣，并且迄今为止在文献中很少对其进行详细考虑。我们证明在某些假设下，存在负对数似然函数的极小值，在这种情况下，我们必须特别注意\（\ nu \ rightarrow \ infty \）的情况，为此，学生t分布接近高斯分布。作为经典EM算法的替代方法，我们提出了三种其他不能解释为EM算法的算法。对于固定ν，第一种算法是文献中已知的加速EM算法。但是，由于我们不固定ν，因此无法将标准收敛结果应用于EM算法。其他两个算法在ν的迭代步骤中与此算法不同。我们展示了目标函数对于ν的不同更新的行为并针对所有三种算法证明其在每个迭代步骤中都会减少。我们通过数值模拟比较了算法和一些加速版本，并应用其中一种估计由Student t噪声破坏的图像的自由度参数。