Inequality (concentration) curves such as Lorenz, Bonferroni, Zenga curves, as well as a new inequality curve -- the $D$ curve, are broadly used to analyse inequalities in wealth and income distribution in certain populations. Quantile versions of these inequality curves are more robust to outliers. We discuss several parametric estimators of quantile versions of the Zenga and $D$ curves. A minimum distance (MD) estimator is proposed for these two curves and the indices related to them. The consistency and asymptotic normality of the MD estimator is proved. The MD estimator can also be used to estimate the inequality measures corresponding to the quantile versions of the inequality curves. The estimation methods considered are illustrated in the case of the Weibull model, which is often applied to the precipitation data or times to the occurrence of a certain event.

翻译：不平等（集中）曲线，如洛伦兹曲线、博纳费罗尼曲线、曾加曲线，以及一种新的不平等曲线——D曲线，被广泛用于分析特定群体中财富和收入分配的不平等。这些不平等曲线的分位数版本对异常值更具稳健性。我们讨论了曾加曲线和D曲线分位数版本的几种参数估计方法。针对这两条曲线及其相关指数，提出了一种最小距离（MD）估计量。证明了该MD估计量的一致性和渐近正态性。该MD估计量还可用于估计对应不平等曲线分位数版本的不平等度量指标。文中以威布尔模型为例说明了所考虑的估计方法，该模型常用于降水数据或某事件发生时间的数据分析。