THE MEASUREMENT of economic inequality is a timely and important topic. Often the Gini index or the entire Lorenz curve is used; however, the relative mean deviation (or Pietra ratio) has been used by Schutz [9] and Budd [1] to study United States data. Eltet6 and Frigyes [2] developed new measures to aid in their analysis of Hungary's income distribution, and Kondor [6] has shown that these new indices are related to the relative mean deviation. In order to draw valid conclusions from actual samples, one needs to know the sampling distribution of the statistic used to estimate the measure of inequality. The purpose of the present paper is to adapt methods used by the author [3 and 4] in another context to obtain the large sample theory of the mean deviation, Pietra ratio, and the measures of Eltet6 and Frigyes. Since several of these measures estimate some parameters of the underlying income distribution function, the asymptotic theory of the estimators is more complicated than might appear at first glance.

[This paper presents a method for obtaining the large sample distribution of the mean deviation and related measures of income inequality.]

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.

[This paper considers the interpolation of percentiles when the value of the c.d.f. F(x) is specified at n points. Upper and lower bounds on the percentile are obtained assuming that data is generated by a unimodal density. Sharper bounds are derived when the average of the observations in each interval is given. In addition to improving the standard linear interpolation, our results indicate that much information can be gained by reporting group means as well as frequency counts.]