The distribution theory literature connected to the multivariate\ud skew-normal distribution has grown rapidly in recent years, and a\ud number of extensions and alternative formulations have been put\ud forward. Presently there are various coexisting\ud proposals, similar but not identical, and with rather unclear\ud connections. The purpose of this paper is to unify these\ud proposals under a new general formulation, clarifying at the same\ud time their relationships. The final part sketches an extension of the argument to the skew-elliptical family

A new class of multivariate skew-normal distributions, fundamental skew-normal distributions and their canonical version, is developed. It contains the product of independent univariate skew-normal distributions as a special case. Stochastic representations and other main properties of the associated distribution theory of linear and quadratic forms are considered. A unified procedure for extending this class to other families of skew distributions such as the fundamental skew-symmetric, fundamental skew-elliptical, and fundamental skew-spherical class of distributions is also discussed.

Linear mixed models (LMM) are frequently used to analyze repeated measures data, because they are more flexible to modelling the correlation within-subject, often present in this type of data. The most popular LMM for continuous responses assumes that both the random effects and the within-subjects errors are normally distributed, which can be an unrealistic assumption, obscuring important features of the variations present within and among the units (or groups). This work presents skew-normal liner mixed models (SNLMM) that relax the normality assumption by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in mixed models. The MCMC scheme is derived and the results of a simulation study are provided demonstrating that standard information criteria may be used to detect departures from normality. The procedures are illustrated using a real data set from a cholesterol study.Bayesian inference, MCMC, Gibbs sampler, multivariate skew-normal distribution, skewness,

Parametric families of multivariate nonnormal distributions have received considerable attention in the past few decades. The authors propose a new definition of a selection distribution that encompasses many existing families of multivariate skewed distributions. Their work is motivated by examples that involve various forms of selection mechanisms and lead to skewed distributions. They give the main properties of selection distributions and show how various families of multivariate skewed distributions, such as the skew‐normal and skew‐elliptical distributions, arise as special cases. The authors further introduce several methods of constructing selection distributions based on linear and nonlinear selection mechanisms.

For statistical inference connected to the scalar skew-normal distribution, it is known that the so-called centred parametrization provides a more convenient parametrization than the one commonly employed for writing the density function. We extend the definition of the centred parametrization to the multivariate case, and study the corresponding information matrix