Modally Focused, Likelihood Based, Inequality Measurement in Multivariate Ordered Categorical Paradigms: A Note

The increasing use of multivariate ordered categorical data in the social sciences presents a challenge for those concerned with measuring inequality. The absence of cardinal measure and the ambiguities inherently associated with artificial attribution of scale to ordinal categories, precludes the use of standard distance-based inequality measures. However, these issues have been surmounted in the univariate world by employing notions of probabilistic distance (the increasing likelihood that some outcome between two categories will occur the bigger is the categorical gap between them) and measuring aggregate probabilistic distance from a Median category focus point. Unfortunately, in multivariate environments the median outcome is not uniquely defined, however the modal category is, thus providing a readymade reference point from which to measure probabilistic distance. In addition, as the most frequently observed outcome, its density value provides a natural measure of the extent of commonality or equality in the population, further rationalizing its use as a point of reference for inequality measurement. This note develops modally focused inequality measures for multivariate ordered categorical environments together with their asymptotic distributions for inference purposes and discusses their axiomatic foundations.