Stochastic ordering of variability measure estimators

Variability measures, such as the variance or the Gini mean difference, are widely used to summarize the dispersion of random variables. In the statistical setting, it is quite natural to assume that if a random quantity has more variability, then the estimators of its variability measures should be greater in some stochastic sense. Stochastic orders can be used to give inequalities in this regard, confirming, or not, this suitable property. This paper is devoted to the stochastic comparison of some variability measure estimators; conditions such that some of these estimators are comparable in the usual stochastic order and in the increasing convex order whenever the involved random variables have different variability are provided. Special attention is devoted to the cases of sample variance and Gini mean differences, and to the case of simple random samples.