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

Abstract

We consider the class of heavy-tailed income distributions and show that the shape of the income distribution has a strong effect on inference for inequality measures. In particular, we demonstrate how the severity of the inference problem responds to the exact nature of the right tail of the income distribution. It is shown that the density of the studentized inequality measure is heavily skewed to the left, and that the excessive coverage failures of the usual confidence intervals are associated with excessively low estimates of both the point measure and the variance. For further diagnostics, the coefficients of bias, skewness and kurtosis are derived and examined for both studentized and standardized inequality measures. These coefficients are also used to correct the size of confidence intervals. Exploiting the uncovered systematic relationship between the inequality estimate and its estimated variance, variance stabilizing transforms are proposed and shown to improve inference significantly.

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.…”

Wealth Inequality: A Survey

“…Second, simple 'plug in' estimators of inequality functionals exhibit finite sample bias and their sampling error remains large and difficult to estimate reliably even in large samples (Cowell and Flachaire, 2007;Schluter and van Garderen, 2009;Schluter, 2012).…”

“…A different strategy for inference is developed in Schluter and van Garderen () and Schluter () who propose normalizing transformation of the index before application of the bootstrap.…”

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.…”

“…The other columns show the results for the alternative bootstrap methods presented above. Results obtained by the approach proposed by Schluter (2012) are presented in the third column (varstab), bootstrapping a variance stabilizing transform of the Theil index. In the fourth column, the semi-parametric bootstrap proposed by Davidson and Flachaire (2007) and Cowell and Flachaire (2007) is used to generate bootstrap samples (semip), with k = n 1/2 and h = 0.6.…”