Abstract
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.
Highlights
The motivation for this work arises from certain tasks in image processing, where the robustness of methods plays an important role
In the Gaussian setting, their approach is equivalent to minimum mean square error estimation, and more general, the resulting estimator can be seen as a particular instance of a best linear unbiased estimator (BLUE)
For unknown degrees of freedom, there exist an accelerated version of the EM algorithm, the so-called Expectation/Conditional Maximization Either (ECME) algorithm [20] which differs from our algorithm
Summary
The motivation for this work arises from certain tasks in image processing, where the robustness of methods plays an important role. Based on an ML approach the authors of [16] introduced a so-called generalized myriad filter that estimates both the location and the scale parameter of the Cauchy distribution They used the filter in a nonlocal denoising approach, where for each pixel of the image they chose as samples of the distribution those pixels having a similar neighborhood and replaced the initial pixel by its filtered version. For denoising images corrupted by additive Cauchy noise, a similar approach was addressed in [17] based on ML estimation for the family of Student-t distributions, of which the Cauchy distribution forms a special case. The authors call this approach generalized multivariate myriad filter. We come back to the original motivation of this paper and estimate the degree of freedom parameter ν from images corrupted by one-dimensional Student-t noise
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