Extending the approaches to polarization ordering of ordinal variables

Multidimensional polarization for ordinal data

“…While Kobus (2015) relaxes the assumption of a unique median and even more conclusiveness can be reached with Mendelson's generalization, still the reliance on a common crossing is a limiting feature of the AF approach, namely, distributions that do not cross cannot be compared. Abul Naga and Yalcin (2010) and Sarkar and Santra (2016) shed some light on median-independent orderings. This is definitely an important research direction for multidimensional polarization for ordinal data.…”

Measuring Welfare, Inequality and Poverty with Ordinal Variables

“…Then, Sarkar and Santra (2020) show that different combinations of slides and additions can homogenise medians (and potentially other quantiles). In some cases, only slides may be needed, in others only additions, and in a third kind a specific combination thereof will be required.…”

“…Increased clustering (or increased bipolarity, e.g. Apouey, 2007;Chakravarty and Maharaj, 2015;Sarkar and Santra, 2020): for every 𝒑, 𝒒 ∈ ℙ 𝐶,𝛼 , 𝐼(𝒑) ≥ 𝐼(𝒒) if 𝒑 is obtained from 𝒒 through a sequence of simple progressive transfers on either side of the median (but not across). Apouey (2007) then showed that her proposal class (21) satisfies both transfers axioms (aversion to median-preserving spreads and increased clustering) and other standard desirable properties.…”

“…That is, these transformations play the same role as rescaling, translations, and the like in the world of quantitative data, and without the need to rely on arbitrary scales for categories, which is most remarkable. However, not all inequality measures satisfying aversion to quantile-preserving spreads are insensitive to the transformations proposed by Sarkar and Santra (2020) to homogenise the quantile of interest. Only those inequality indices which are (1) averse to quantile-preserving spreads and (2) insensitive to quantile-homogenising transformations are suitable for inequality comparisons with distributions without common quantiles based on the notion of quantilepreserving spreads.…”

“…Then the prominent measurement approach based on quantile-preserving spreads, chiefly median-preserving spreads, is described in detail, crucially distinguishing between explicit and implicit quantiles, as well as between inequality and bipolarisation concepts (works reviewed include Mendelson, 1987;Apouey, 2007;Abul Naga and Yalcin, 2008;Lazar and Silber, 2013;Kobus, 2015;Chakravarty and Maharaj, 2015). Drawing from Sarkar and Santra (2020), a special subsection explains proposals to measure inequality and bipolarisation when distributions do not share quantiles of interest (e.g. the median).…”