Alternatives of the EM Algorithm for Estimating the Parameters of the Student-t Distribution
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In this paper, we consider maximum likelihood estimation of the degree of freedom parameter ν, the location paramter μ and the scatter matrix Σ of the multivariate Student-t distribution. In particular, we are interested in approximating the degree of freedom parameter ν that determines the tails of the corresponding probability density function. We prove that under certain assumptions a minimizer of the likelihood function exists, where we have to take special care of the case ν→∞, 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 by numerical examples and apply one of these algorithms for estimating the degree of freedom parameter in images corrupted by Student-t noise.
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