Measuring kurtosis by right and left inequality orders

The question of how to measure kurtosis in both symmetric and asymmetric distributions is addressed using the kurtosis diagram of Zenga (2006). Kurtosis is related to inequality at either side of the median, and we establish a hierarchy of kurtosis orderings in which the kurtosis diagram stands at the weakest level. A sufficient condition for constructing kurtosis measures compatible with such ordering is provided. The merits of the proposed approach in both clarifying and formalizing the idea of kurtosis are evaluated and examples are discussed throughout.