The variance of Gini's mean difference and its estimators

The use of Gini's mean difference as an index of variability has, until now, been restricted because of some difficulties arising in computing and estimating the variance of its estimator Delta^. The aim of this paper is to cope with these issues. Considering the mean deviation S(x) of a r.v. X about a given value x, the Gini's mean difference Delta results to be the expected value of S(x). Moreover, denoting by F the expected value of S^2(x) and by Delta^ the sample mean difference without repetition, Var(Delta^) can be expressed as a function of the variance of X, Delta and F. Two estimators for Var(Delta^) are obtained: starting from the natural estimator, whose asympthotic unbiasedness is shown, an unbiased estimator is then derived.