Representation Theorems for measures of location and measures of dispersion

In this paper we characterize classes of statistical functionals through some results which have been inspired by a classical theorem on mean values due to de Finetti, Nagumo, and Kolmogorov. All the functionals are order preserving w.r.t. particular stochastic orderings. The first result is a quasi-linear representation for functionals assuming, together with monotonicity w.r.t. first degree stochastic dominance and associa- tivity, a particular continuity condition which can be interpreted as a mild type of robustness. This result is used in a "dual" way to characterize other measures of location, like median, quantiles, trimmed means, and Winsorized means. In the second part of the paper our aim is to characterize some measures of dispersion of a distribution around its expected value which are order preserving w.r.t. the so-called dilation ordering. Most statistical indices of variability can be obtained in this way. This and a "dual" theorem also account for several mea- sures of inequality, which are order preserving with respect to the concentration ordering based on the Lorenz curve, like Gini's celebrated index.