We develop a unified nonparametric framework for sharp partial identification and inference on inequality indices when the data contain coarsened observations of the variable of interest. We characterize the extremal allocations for all Schur-convex inequality measures, and show that sharp bounds are attained by distributions with finite support. This reduces the computational problem to finite-dimensional optimization, and for indices admitting linear-fractional representations after suitable ordering of the data (including the Gini coefficient and quantile ratios), we express the bound problems as linear or quadratic programs. We then establish $\sqrt{n}$ inference for the upper and lower bounds using a directional delta method and bootstrap confidence intervals. In applications, we compute sharp Gini bounds from household wealth data with mixed point and interval observations and use historical U.S. grouped income tables to bound time series for the Gini and quantile ratios.

翻译：我们针对当数据包含目标变量的粗化观测时，提出了一个统一的非参数框架，用于对不平等指数进行尖锐的部分识别与推断。我们刻画了所有Schur-凸不平等度量的极值分配，并证明尖锐的界可通过具有有限支撑的分布达到。这将计算问题简化为有限维优化，并且对于在数据适当排序后允许线性分式表示的指数（包括基尼系数和分位数比率），我们将界问题表述为线性或二次规划。随后，我们利用方向Delta方法和自助置信区间，为上下界建立了$\sqrt{n}$推断。在应用中，我们根据包含点观测和区间观测混合的家庭财富数据计算了尖锐的基尼系数界，并利用美国历史分组收入表对基尼系数和分位数比率的时间序列进行了界定。