Measuring statistical heterogeneity: The Pietra index
Abstract
There are various ways of quantifying the statistical heterogeneity of a given probability law: Statistics uses variance  which measures the law’s dispersion around its mean; Physics and Information Theory use entropy  which measures the law’s randomness; Economics uses the Gini index  which measures the law’s egalitarianism. In this research we explore an alternative to the Gini indexthe Pietra indexwhich is a counterpart of the KolmogorovSmirnov statistic. The Pietra index is shown to be a natural and elemental measure of statistical heterogeneity, which is especially useful in the case of asymmetric and skewed probability laws, and in the case of asymptotically Paretian laws with finite mean and infinite variance. Moreover, the Pietra index is shown to have immediate and fundamental interpretations within the following applications: renewal processes and continuous time random walks; infiniteserver queueing systems and shot noise processes; financial derivatives. The interpretation of the Pietra index within the context of financial derivatives implies that derivative markets, in effect, use the Pietra index as their benchmark measure of statistical heterogeneity.
 Publication:

Physica A Statistical Mechanics and its Applications
 Pub Date:
 January 2010
 DOI:
 10.1016/j.physa.2009.08.006
 Bibcode:
 2010PhyA..389..117E