Right and left kurtosis measures: large sample estimation and an application to financial returns

Abstract

Although the standard fourth moment coefficient is routinely computed as “the kurtosis” of a distribution, the measure is not easily interpreted and has been a subject of considerable debate in statistical literature. The financial community has recently joined in the debate, calling for more robust estimators of kurtosis in distributions of stock market returns. For these reasons, we here consider alternative measures of right and left kurtosis, which arise from a recent characterization of kurtosis as inequality at either side of the median. Based on Gini's coefficient of concentration, the new measures apply to both symmetric and asymmetric distributions, their interpretation is clear and they are consistent with common risk perceptions of investors and risk managers. In this contribution, we show that the theory of L‐statistics provides a natural framework for the construction of empirical estimators of the proposed measures and the derivation of their asymptotic properties under mild moment requirements. A real data example illustrates the potential of these estimators in financial contexts, in which the existence of higher moments is still an open question. Copyright © 2014 John Wiley & Sons, Ltd