Some results on the representation of measures of location and spread as L-functionals

Abstract

Random variables are compared w.r.t. their location or their dispersion by means of various types of stochastic orderings. First degree stochastic dominance is relevant to location and the spread ordering, defined via comparisons of all the corresponding interquantile distances, to dispersion. Our first result is a characterization of some measures of location as L-functionals, and was inspired by de Finetti's representation theorem for quasi-linear means. The main result of the paper is a representation theorem for some spread measures as integrals with respect to the measure defined by the quantile function. A ‘spread distance’, namely a metric linked to the ordering, is introduced and continuity of the functional with respect to this metric is required to be one of the characteristic properties that enable the representation to be made.