The variance of Gini’s mean difference and its estimators

Abstract

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 d. The aim of this paper is to cope with these issues. Considering the mean deviation D(x) of a r.v. X about a given value x, the Gini’s mean difference d results to be the expected value of D(x). Moreover, denoting by f the expected value of D2(x) and by d the sample mean difference without repetition, Var(d) can be expressed as a function of the variance of X, say s2, d and f. Two estimators for Var(d) are obtained: starting from the natural estimator, whose asympthotic unbiasedness is shown, an unbiased estimator is then derived.