Bounds on inequality with incomplete data

We develop a unified, nonparametric framework for sharp partial identification and inference on inequality indices when income or wealth are only coarsely observed -- for example via grouped tables or individual interval reports -- possibly together with linear restrictions such as known means or subgroup totals. First, for a broad class of Schur-convex inequality measures, we characterize extremal allocations and show that sharp bounds are attained by distributions with simple, finite support, reducing the underlying infinite-dimensional problem to finite-dimensional optimization. Second, for indices that admit linear-fractional representations after suitable ordering of the data (including the Gini coefficient, quantile ratios, and the Hoover index), we recast the bound problems as linear or quadratic programs, yielding fast computation of numerically sharp bounds. Third, we establish $\sqrt{n}$ inference for bound endpoints using a uniform directional delta method and a bootstrap procedure for standard errors. In ELSA wealth data with mixed point and interval observations, we obtain sharp Gini bounds of 0.714--0.792 for liquid savings and 0.686--0.767 for a broad savings measure; historical U.S. income tables deliver time-series bounds for the Gini, quantile ratios, and Hoover index under grouped information.