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Given a random variable $X$ and considered a family of its possibledistortions, we define two new measures of distance between $X$ and each itsdistortion. For these distance measures, which are extensions of the Gini'smean difference, conditions are determined for the existence of a minimum, or amaximum, within specific families of distortions, generalizing some resultspresented in the recent literature.

Generalized Gini's mean difference through distortions and copulas, and related minimizing problems