Estimation of quantile versions of the Lorenz curve and the Gini index for the generalized Pareto distribution

This paper concerns the estimation of quantile versions of the Lorenz curve and the Gini index in the case of the generalized Pareto distribution. These curves and indices, unlike the Lorenz curve and the Gini index, are also defined for distributions whose expected value is not finite. The quantile versions of the Lorenz curve and the Gini index of the generalized Pareto distribution depend only on the shape parameter. Accuracy of the shape parameter estimators, recommended in the literature and those whose accuracy has not been studied so far, is compared in simulations. The accuracy of the plug-in estimators of the quantile versions of the Lorenz curve and the Gini index is also studied. Based on the simulations performed, if the sample size is not too large, we recommend using Zhang's estimator of the shape parameter in the estimation of quantile versions of the Lorenz curve and the Gini index. In case the shape parameter is small we also recommend the IPO estimator. The applications of the described methods in the real data analysis are also presented.