A new class of skewed distributions, with a matrix skewness parameter, called extended mean mixtures of multivariate normal (EMMN) distributions, is constructed. The family of EMMN distributions includes the SN and MMN distributions as special cases. Some basic properties of this family, such as characteristic function, moment generating function, affine transformation and canonical forms of the distributions are derived. An EM-type algorithm is developed to carry out the maximum likelihood estimation of the parameters. Two special cases of this family are studied in detail. A simulation is carried out to examine the performance of the estimation method, and the flexibility is illustrated by fitting a special case of this family to a real data. Finally, the theoretical formula of the multivariate tail conditional expectation of the EMMN distribution is derived.

The kurtosis and skewness of distributions are important measures that can describe the shape of a distribution, and there have been many results for symmetric distributions, but there are still many difficulties and challenges in the characterization of skew distributions. Based on the results of Mardia's and Song's kurtosis measures of elliptical distributions obtained by Zografos ^[1], we generalize the results and study some measures for elliptical and skew-elliptical distributions. We also derive the expressions of moments of skew-elliptical distributions in terms of the ones of skew-normals and take skew- $t$, skew-Pearson type Ⅶ and skew-Pearson type Ⅱ distributions as examples.

In this paper we derive explicit formulas of tail conditional expectation ( $\text{TCE}$) and tail variance ( $\text{TV}$) for the class of location-scale mixtures of elliptical distributions, which includes the generalized hyper-elliptical ( $\text{GHE}$) distribution. We also develop portfolio risk decomposition with $\text{TCE}$ for multivariate location-scale mixtures of elliptical distributions. To illustrate our findings, we focus on the generalized hyperbolic ( $\text{GH}$) family which is a popular subclass of the $\text{GHE}$ for stocks modelling.

In this paper, we investigate some multivariate integral stochastic orderings of skew-normal random vectors. We derive the results of the sufficient and/or necessary conditions by applying an identity for $Ef(\mathbf{Y})-Ef(\mathbf{X})$, where $\mathbf{X}$ and $\mathbf{Y}$ are multivariate skew-normal random vectors, $f$ satisfies some weak regularity condition. The integral orders considered here are the componentwise convex, copositive, completely-positive orderings and their corresponding increasing ones as well as linear forms of stochastic orderings, which play a vital role in transforming the unmanageable multivariate components into an easy-to-handle univariate variable.