Multivariate Skew-normal Distributions Research Articles

A note on an equivalence between chi-square and generalized skew-normal distributions

In this note, we establish an equivalence between chi-square and generalized skew-normal distributions. This result is based on a distributional invariance property of even functions in generalized skew-normal random vectors. It extends the chi-square properties related to univariate and multivariate skew-normal distributions.

A multivariate skew normal distribution

In this paper, we define a new class of multivariate skew-normal distributions. Its properties are studied. In particular we derive its density, moment generating function, the first two moments and marginal and conditional distributions. We illustrate the contours of a bivariate density as well as conditional expectations. We also give an extension to construct a general multivariate skew normal distribution.

A Bayesian interpretation of the multivariate skew-normal distribution

This paper provides a unified treatment and a Bayesian interpretation of two different classes of multivariate skew-normal distributions proposed by Azzalini and Dalla Valle (Biometrika 83 (1996) 715) and Gupta et al. (Tech. Rep., Cimat, Mexico (2001)). We show that the above classes of distributions can be viewed as particular cases of a more general family, which naturally arise in constrained modelling. Our approach can be viewed as a direct extension to the multivariate case of the O'Hagan and Leonard (Biometrika 63 (1976) 201) paper, where the authors construct, in the scalar case, a skew prior distribution for the location parameter of a Gaussian random variable, using a simple hierarchical argument.

Definition and probabilistic properties of skew-distributions

The univariate and multivariate skew-normal distributions have a number of intriguing properties. It is shown here that these hold for a general class of distributions, defined in terms of independence conditions on signs and absolute values. For this class, two stochastic representations become equivalent, one using conditioning on the positivity of a random vector and the other employing a vector of absolute values. General methods for computing moments and for obtaining the density function of a general skew-distribution are given. The case of spherical and elliptical distributions is briefly discussed.