The traditional class of elliptical distributions is extended to allow forasymmetries. A completely robust dispersion matrix estimator (the `spectralestimator') for the new class of `generalized elliptical distributions' ispresented. It is shown that the spectral estimator corresponds to anM-estimator proposed by Tyler (1983) in the context of ellipticaldistributions. Both the generalization of elliptical distributions and thedevelopment of a robust dispersion matrix estimator are motivated by thestylized facts of empirical finance. Random matrix theory is used for analyzingthe linear dependence structure of high-dimensional data. It is shown that theMarcenko-Pastur law fails if the sample covariance matrix is considered as arandom matrix in the context of elliptically distributed and heavy tailed data.But substituting the sample covariance matrix by the spectral estimatorresolves the problem and the Marcenko-Pastur law remains valid.
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