Frahm, Gabriel (2004) Generalized Elliptical Distributions: Theory and Applications. PhD thesis, Universität zu Köln.
The thesis recalls the traditional theory of elliptically symmetric distributions. Their basic properties are derived in detail and some important additional properties are mentioned. Further, the thesis concentrates on the dependence structures of elliptical or even meta-elliptical distributions using extreme value theory and copulas. Some recent results concerning regular variation and bivariate asymptotic dependence of elliptical distributions are presented. Further, the traditional class of elliptically symmetric distributions is extended to a new class of `generalized elliptical distributions' to allow for asymmetry. This is motivated by observations of financial data. All the ordinary components of elliptical distributions, i.e. the generating variate, the location vector and the dispersion matrix remain. Particularly, it is proved that skew-elliptical distributions belong to the class of generalized elliptical distributions. The basic properties of generalized elliptical distributions are derived and compared with those of elliptically symmetric distributions. It is shown that the essential properties of elliptical distributions hold also within the broader class of generalized elliptical distributions and some models are presented. Motivated by heavy tails and asymmetries observed in financial data the thesis aims at the construction of a robust dispersion matrix estimator in the context of generalized elliptical distributions. A `spectral density approach' is used for eliminating the generating variate. It is shown that the `spectral estimator' is an ML-estimator provided the location vector is known. Nevertheless, it is robust within the class of generalized elliptical distributions. The spectral estimator corresponds to an M-estimator developed 1983 by Tyler. But in contrast to the more general M-approach used by Tyler the spectral estimator is derived on the basis of classical maximum-likelihood theory. Hence, desired properties like, e.g., consistency, asymptotic efficiency and normality follow in a straightforward manner. Not only caused by the empirical evidence of extremes but also due to the inferential problems occuring for high-dimensional data the performance of the spectral estimator is investigated in the context of modern portfolio theory and principal components analysis. Further, methods of random matrix theory are discussed. They are suitable for analyzing high-dimensional covariance matrix estimates, i.e. given a small sample size relative to the number of dimensions. It is shown that the Marchenko-Pastur law fails if the sample covariance matrix is used in the context of elliptically of even generalized elliptically distributed and heavy tailed data. But substituting the sample covariance matrix by the spectral estimator resolves the problem and the classical arguments of random matrix theory remain valid.
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