  The traditional class of elliptical distributions is extended to allow for
asymmetries. A completely robust dispersion matrix estimator (the `spectral
estimator') for the new class of `generalized elliptical distributions' is
presented. It is shown that the spectral estimator corresponds to an
M-estimator proposed by Tyler (1983) in the context of elliptical
distributions. Both the generalization of elliptical distributions and the
development of a robust dispersion matrix estimator are motivated by the
stylized facts of empirical finance. Random matrix theory is used for analyzing
the linear dependence structure of high-dimensional data. It is shown that the
Marcenko-Pastur law fails if the sample covariance matrix is considered as a
random matrix in the context of elliptically distributed and heavy tailed data.
But substituting the sample covariance matrix by the spectral estimator
resolves the problem and the Marcenko-Pastur law remains valid.
