Elliptical Distributions

This book discusses elliptical distributions as well as their properties and provides a few applications. Firstly, basic concepts for multivariate random variables are introduced. Besides the expectation and the covariance matrix the first part of the book deals with important numerical measures of association. After the Pearson product-moment correlation coefficient two rank correlation coefficients follow: Spearman's rho and Kendall's tau. Several properties of these three measures are being presented. They are applied for the analysis of a financial data set. In the second part of the book, spherical distributions are introduced in order to be able to provide elliptical distributions and several of their properties afterwards. The relationship between this class of distributions and the well-known ones (for example the normal distribution) is shown. This book pays special attention to the relationship between rho and tau for elliptical random variables since it can be used to obtain a robust estimator for rho. Finally, two methods to transform non-positive semidefinite correlation matrices to positive semidefinite ones are being presented. These concepts are illustrated by applying them to an example.