Edgeworth expansions and normalizing transforms for inequality measures

Abstract

Finite sample distributions of studentized inequality measures di¤er 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 …rst and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coe¢ cients. In the second part we improve over …rst-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 con…dence 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.

Parametric Inference for Index Functionals

“…Similar conclusions are also drawn in Davidson and Flachaire (2007); Cowell and Flachaire (2007); Davidson (2009);Davidson (2010) and Davidson (2012). This has for example motivated Schluter and van Garderen (2009) and Schluter (2012), using the results of Hall (1992), to propose normalizing transformations of inequality measures using Edgeworth expansions, to adjust asymptotic Gaussian approximations.…”

On the problem of inference for inequality measures for heavy‐tailed distributions

“…For instance, Schluter and van Garderen (2009) have shown that the actual (finite sample) densities of the estimators are substantially skewed and far from normal. Such distributions not only include the class of heavy-tailed distributions, whose tail decays like a power function, but also, for instance, the lognormal distribution, whose tail decays exponentially fast, provided the shape parameter is sufficiently large.…”

Statistical Methods for Distributional Analysis

“…Several approaches have been proposed in the literature to obtain more reliable inference. Schluter andvan Garderen (2009) andSchluter (2012) propose normalizing transformation of the index, before to use the bootstrap, in order to use a statistic with a distribution closer to the Normal. Let g denote a transformation of the index W ; a standard bootstrap confidence interval can be obtained on the transformed index g(W ) and, therefore, on the untransformed index by inverting the relation between the welfare index and the parameters.…”