From Normality to Skewed Multivariate Distributions: A Personal View

An overview of the development of multivariate distributions' theory is given starting with T. W. Anderson's monograph [1]. In the book, multivariate analysis methods were presented in the compact matrix language for the first time. The methods were developed for normally distributed populations. Ten years later, the class of elliptically contoured distributions (from now on, elliptical distributions) was described by Kelker [22]. Elliptical distributions formed a new class of symmetric multivariate distributions which included the normal distribution. To model skewed data, the theory of multivariate Edgeworth type expansions was developed in the 1980s. Ten years later, A. Azzalini and his colleagues introduced multivariate skew elliptical distributions by transforming symmetric elliptical distributions with an additional parameter vector. In the 1990s, copula theory became important in modeling multivariate data. With copulas, it is possible to build multivariate distributions with a given set of marginals and a stipulated correlation structure. These models were intensively developed from the second half of the 1980s through the 1990s. Special attention has been paid by several authors to the Gaussian and t copulas, as well as to skew normal and skew t copulas.