The arithmetic-harmonic inequality index: Theory, inference, and finite-sample analysis

Abstract

We investigate the arithmetic-harmonic inequality (AHI) index, a bounded and scale-invariant measure of dispersion for positive random variables, defined through the interplay between the mean and its reciprocal. We derive analytical expressions for the AHI index within the generalized inverse Gaussian (GIG) family, encompassing the inverse Gaussian and gamma distributions as important special cases. We study the associated estimator, obtain a tractable expression for its expectation, establish its asymptotic properties, and derive explicit first-order bias approximations. A Monte Carlo study is conducted to evaluate the finite-sample performance of the estimator under various scenarios. An application to GDP per capita data for countries in the Americas illustrates the role of the AHI index within the broader Atkinson family across several values of the inequality-aversion parameter. The results show the good performance of the AHI index as a tractable and interpretable measure of economic dispersion.