Tyler's M-estimator, random matrix theory, and generalized elliptical distributions with applications to finance

In recent publications standard methods of random matrix theory were applied to principal components analysis of high-dimensional financial data. We discuss the fundamental results and potential shortcomings of random matrix theory in the light of the stylized facts of empirical finance. Especially, our arguments are based on the impact of nonlinear dependencies such as tail dependence. After a brief discussion of the stylized facts we present the class of multivariate generalized elliptical distributions. This class allows for the modeling of various anomalies frequently observed in financial data. Thus it will serve as a general model for the investigation of standard methods of random matrix theory. It is shown that the Marčenko-Pastur law generally fails when analyzing the empirical distribution function of the eigenvalues given by the sample covariance matrix of generalized elliptically distributed data. As an alternative we derive a random matrix referred to as the spectral estimator which is distribution-free within the class of generalized elliptical distributions. Moreover, we show that the spectral estimator corresponds to Tyler's M-estimator and many important properties of the spectral estimator can be obtained from the corresponding literature. Substituting the sample covariance matrix by the spectral estimator resolves the problems which are due to the stylized facts and the Marčenko-Pastur law remains valid. This holds even if the data are not generalized elliptically distributed but mutually independent.