On the MAP estimation in the context of elliptical distributions

The purpose of this paper is to study the estimation problem of a multivariate elliptically symmetric random variable corrupted by a multivariate elliptically symmetric noise. In this study, the maximum a posteriori (MAP) approach is presented, extending recent works by Alecu et al. [1] and Selesnick [2, 3]: (i) the estimation is performed in a multivariate context, (ii) the corrupting noise is not limited to be Gaussian. This paper also extends our previous work that dealt with the minimum mean square error (MMSE) approach [4]. The MMSE is briefly recalled and the MAP is derived. Then the practical use of the MAP in a general setting is discussed and compared to that of the MMSE and of the Wiener estimator. Several examples illustrate the behaviors of these estimators and exhibit their performances.