The Interpolation of the Lorenz Curve and Gini Index from Grouped Data

Abstract

ECONOMISTS OFTEN SUMMARIZE the income distribution by the Lorenz curve and Gini index. A variety of parametric methods (e.g., [1 and 8]) have been developed to estimate these measures from the grouped income data governments provide (e.g., [3 and 12]). Previously, one of the authors developed a distribution-free approach [5] which yielded accurate bounds on the Gini index. While analogous bounds on the Lorenz curve can be obtained [5 and 10], the resulting curve is not smooth so a method of interpolation is needed. The purpose of this paper is to adapt an old technique of numerical analysis, Hermite interpolation [7 and 13], to our problem and to show that it usually works in theory and in practice. Our paper was motivated by the work of Brittain [2] who also used numerical methods. Unfortunately, his procedure often resulted in estimates of the Gini index which were inconsistent with the above-mentioned bounds. Although the piecewise Hermite interpolation yielded accurate estimates of the Gini index, it is not always convex as the Lorenz curve must be. Section 5 states conditions for the interpolated curve to be convex or at least increasing over an interval. While these conditions are usually satisfied by real data, a theoretical example illustrates how an error may arise.