Robust inequality comparisons based on ordinal attributes with Kolm-independent measures

The literature on health inequality with ordinal attributes is benefiting from the development of inequality measures, which are useful in any wellbeing assessment involving ordinal variables (e.g. subjective wellbeing). Lv, Wang, and Xu ("On a new class of measures for health inequality based on ordinal data", Journal of Economic Inequality, 2015) recently characterized a new class of this type of inequality measures axiomatically. In addition to their appealing functional forms, these measures are the only ones in the literature satisfying a property of independence, inspired by Kolm ("Unequal inequalities I", Journal of Economic Theory, 1976). As acknowledged by the authors, it is reasonable to be concerned about the robustness of inequality comparisons with ordinal attributes to the several alternative suitable measures within the class. This note derives the stochastic dominance condition whose fulfilment guarantees that all inequality measures within the class rank a pair of distributions consistently; thereby providing an empirically implementable robustness test for this class of measures.