Extending the approaches to polarization ordering of ordinal variables

The partial dominance orderings introduced in Allison and Foster (J.Health Econom. 23(3), 505–524, 2004) and Chakravarty and Maharaj (Int. J. Econom. Theory 11(2), 231–246, 2015) are applicable to ordinal variables with identical median categories. In the present paper, we extend these approaches to distributions having different medians. We introduce the notions of “Slide” and “Addition of Unpopulated Categories at Extremities”(AUCE) and construct new distributions with matching median categories from distributions having non-identical medians. In general, these two notions are sufficient to generate distributions with matching median categories and in some cases, Slide alone suffices. If both Slide and AUCE are required to generate the new distributions, then these distributions may not be ranked by the dominance ordering of Allison and Foster but may be ranked by that of Chakravarty and Maharaj. On the other hand, if Slide alone is used, then the generated distributions may be ranked by both criteria. We characterize the class of polarization measures that ranks the original distributions in accordance with the dominance orderings of the constructed distributions. We observe that a number of well-known polarization measures belongs to this class. Empirical applicability of the theory is illustrated using data on male and female educational attainments in India.