Skew-symmetric and generalized skew-elliptical distributions

A popular approach to model departures from normality consists of modifying a symmetric probability density function (pdf) of a random variable, or of a random vector in the multivariate setting, in a multiplicative fashion, thereby introducing skewness. This idea has been in the literature for a long time, but it has been thoroughly implemented for the univariate normal distribution by Azzalini (1985, 1986), yielding the so-called skew-normal distribution. An extension to the multivariate case was then introduced by Azzalini and Dalla Valle (1996). Statistical applications of the multivariate skew-normal distribution were presented by Azzalini and Capitanio (1999), who also briefly discussed an extension to elliptical densities. Since then, several authors have tried to generalize these results to skewing arbitrary symmetric pdf’s with very general forms of multiplicative functions.