Some properties of the unified skew-normal distribution

TL;DR

This paper explores various properties of the unified skew-normal distribution, including moments, skewness, kurtosis, log-concavity, and behavior under conditioning, filling important gaps in its theoretical understanding.

Contribution

It provides new analytical expressions for moments up to the fourth order and investigates key properties like log-concavity and interval conditioning closure.

Findings

Derived moments up to the fourth order.

Expressions for Mardia's skewness and kurtosis.

Established log-concavity and conditioning properties.

Abstract

For the family of multivariate probability distributions variously denoted as unified skew-normal, closed skew-normal and other names, a number of properties are already known, but many others are not, even some basic ones. The present contribution aims at filling some of the missing gaps. Specifically, the moments up to the fourth order are obtained, and from here the expressions of the Mardia's measures of multivariate skewness and kurtosis. Other results concern the property of log-concavity of the distribution, and closure with respect to conditioning on intervals.