Preliminaries

Matrix variate distributions have been studied by statisticians for a long time. The first results on this topic were published by Hsu and Wishart. These distributions provedto be useful in statistical inference. For example, the Wishart distribution is essential when studying the sample covariance matrix in the multivariate normal theory. Random matricescan also be used to describe repeated measurements on multivariate variables. In this case,the assumption of the independence of the observations, a commonly used condition in statistical analysis, is often not feasible. When analyzing data sets like these, the matrix variate elliptically contoured distributions can be used to describe the dependence structure of the data. This is a rich class of distributions containing the matrix variate normal, contaminated normal, Cauchy and Student’s t-distributions. The fact that the distributions in this class possess certain properties, similar to those of the normal distribution, makes them especially useful. For example, many testing procedures developed for the normal theory to test various hypotheses can be used for this class of distributions, too. In this chapter, we present a general introduction into the theory of matrix variate elliptically contoured distributions and provide an extensive literature review. Furthermore, someuseful results from matrix algebra and functional equation are presented which are used in other chapters of the book.