Alternatives to the EM algorithm for ML estimation of location, scatter matrix, and degree of freedom of the Student t distribution
Marzieh Hasannasab, Johannes Hertrich, Friederike Laus, Gabriele Steidl · Numerical Algorithms · 2020
Abstract In this paper, we consider maximum likelihood estimations of the degree of freedom parameterν, the location parameterμand the scatter matrixΣof the multivariate Studenttdistribution. In particular, we are interested in estimating the degree of freedom parameterνthat 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 $ u \rightarrow \infty $ ν→∞ , for which the Studenttdistribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixedν, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fixν, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step forν. We show how the objective function behaves for the different updates ofνand 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 Studenttnoise.