The skew-normal distribution

The book reviews the state-of-the-art advances in skew-elliptical distributions and provides many new developments in a single volume, collecting theoretical results and applications previously scattered throughout the literature. The main goal of this research area is to develop flexible parametric classes of distributions beyond the classical normal distribution. The first chapter of this book introduces the skew-normal distribution (SN distribution) in the univariate and multivariate case studying in depth their most interesting and relevant features. In the first part of the chapter definition, properties, the moment generating function and the cumulative function give the characterization of the univariate skew-normal distribution and its appealing theoretical features. A wide discussion about inferential problems connected to the methods of estimation of the parameters finishes with the solutions given in literature to face the troubles. The statistical tests developed to assess the hypothesis of skew-normality in literature are discussed at the end of the first part of the chapter. In the second part of the chapter, the multivariate skew-normal distribution that naturally extends the univariate case is presented starting from its genesis to its immediate and relevant properties as for example cumulative distribution function and moment generating function. It is also emphasized that a number of properties can be extended from the scalar case to the multivariate case. The discussion about the good tractability of linear and quadratic forms highlights as a relevant result the extension of Fisher-Cochran theorem to the SN case. In addition, the conditional distribution associate to a bivariate skew-normal variable is given and it is underlined that conditional mean and variance can be written as a function of the hazard function of the standard normal density. Finally, some results about the SN class in some contexts are analyzed: in particular, the problem of reliability and that of finding regions of assigned probability and minimum volume.