Edgeworth expansions and normalizing transforms for inequality measures

Abstract

Finite sample distributions of studentized inequality measures differ substantially from their asymptotic normal distribution in terms of location and skewness. We study these aspects formally by deriving the second order expansion of the first and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coefficients. In the second part we improve over first-order theory by deriving Edgeworth expansions and normalizing transforms. These normalizing transforms are designed to eliminate the second order term in the distributional expansion of the studentized transform and converge to the Gaussian limit at rate O(n−1). This leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n−3/2). We illustrate our procedure with an application to regional inequality measurement in Côte d'Ivoire.