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Konstantin Glombek, Kohln University Print
Friday, 15 February 2013, 14:30 - 15:30

Konstantin Glombek, Kohln University

High-Dimensionality in Statistics and Portfolio Optimization

Abstract: Many challenges in multivariate analysis face the problem of dealing with samples whose dimension is of the same order as their size. This high-dimensional setting often leads to inconsistencies or degenerated distributions of certain estimators. In particular, estimators which are based on the sample covariance matrix are affected as the eigenvalues of this matrix behave differently under high-dimensionality than the ones of the population covariance matrix. But the eigenvalues of certain estimators for scatter also exhibit a remarkable behavior in the classical setting when the sample size is much larger than the dimension. It is shown that the empirical distribution of the eigenvalues of Tyler’s M-estimator, suitably standardized, converges in probability to the semicircle law under spherical sampling and assuming that the sample dimension and size tend to infinity while their ratio tends to zero. Another concern is statistical inference for high-dimensional global minimum variance portfolios. The standard estimators for the variance and mean of the portfolio return of this portfolio are investigated concerning consistency and asymptotic distribution under high-dimensionality. The corresponding Sharpe ratio and weights are considered as well. Particular emphasis is put on the statement of hypotheses in the high-dimensional setting.

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