  Elliptically symmetric distributions are widely used in portfolio modeling,
as well as in signal processing applications for modeling impulsive background
noises. Of particular interest are algorithms for covariance estimation and
subspace detection in such backgrounds.
  This article tackles the issue of correctly estimating the covariance matrix
associated to such models and detecting additional signal superimposed on such
distributions. A particular attention is given to the proper accounting of the
circular symmetry for the subclass of complex elliptical distributions in the
case of complex signals.
  In particular Tyler's estimator is shown to be a maximum likelihood estimate
over all elliptical models, and its extension to the complex case is shown to
be a maximum likelihood estimate for the subclass of complex elliptical models
(CES); other M-estimators are also shown to be maximum likelihood estimates
over some restricted classes of elliptical models. The extension of Tyler's and
other M-estimators to constrained covariance estimation is also discussed, in
particular for toeplitz constrains.
  Finally likelihood ratio signal detection tests associated to the various
estimators introduced in this article are also discussed.
