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.

中文翻译：

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扩展序数变量极化排序的方法

在Allison和福斯特引入的部分显性序（J.Health ECONOM。23 Chakravarty和玛哈拉吉（中间体J. ECONOM理论（3），505-524，2004）和11（2），231-246，2015）是适用于中位数类别相同的序数变量。在本文中，我们将这些方法扩展到具有不同中位数的分布。我们介绍了“幻灯片”和“四肢增加不受欢迎的类别”的概念”（AUCE），并从中位数不相同的分布中构造出匹配中位数类别的新分布。通常，这两个概念足以生成具有匹配中值类别的分布，在某些情况下，仅使用Slide就足够了。如果同时需要Slide和AUCE来生成新的分布，则这些分布可能不会按Allison和Foster的优势排序来排名，而可能会按Chakravarty和Maharaj的排名来排序。另一方面，如果仅使用“幻灯片”，则可以通过两个标准对生成的分布进行排名。我们描述了极化度量的类别，该类别根据构造的分布的优势顺序对原始分布进行排序。我们观察到许多众所周知的极化措施都属于此类。