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, closure with respect to conditioning on intervals, and a possible alternative parameterization.

中文翻译：

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统一偏态正态分布的一些性质

对于被不同地表示为统一偏斜正态、封闭偏斜正态和其他名称的多元概率分布族，许多属性已经为人所知，但还有许多属性是未知的，甚至是一些基本的属性。目前的贡献旨在填补一些缺失的空白。具体来说，获得四阶矩，并从这里获得 Mardia 的多元偏度和峰度度量的表达式。其他结果涉及分布的对数凹度属性、关于区间条件的闭包，以及可能的替代参数化。