Exact distribution of the Gini Concentration index from a Skew normal sample

he skew-normal distribution and the fundamental skew-normal distribution are two generalizations of the normal one. The former can model moderate skewness and kurtosis, while the latter is very useful when dealing with ordering constraints. In this paper we demonstrate that the exact sampling distribution of the Gini Concentration Index from a skew-normal is a fundamental skew-normal distribution. Implications of this result are twofold. In the first place, it easily applies to the case of sampling from a normal distribution, which can be regarded as a skew-normal distribution with shape parameter equal to zero. In the second place, the result can be easily generalized to other measures of inequality which can be represented as ratios of linear combinations of order statistics. The paper also recalls some relevant properties of the skew-normal distributions and motivates its use in statistical modelling.