Large Sample Properties of Two Inequality Indices

Abstract

THIS PAPER PROVIDES the large sample properties of two inequality indices, Atkinson's (1970) index and the generalized entropy index (Cowell and Kuga (1981), Shorrocks (1984)). The large sample properties of the equally-distributed-equivalent (e.d.e.) income measures associated with these indices are also derived. The asymptotic distributions of these inequality and welfare measures allow the construction of confidence intervals and tests for differences in the measures; these confidence intervals and tests are asymptotically distribution-free. Given the increasing availability of large micro-data sets on incomes, these results are directly applicable to empirical work on income distributions. The sampling distributions of some inequality indices, such as the Gini coefficient and Pietra index, are known.' But the ethical bases of these indices have been subject to criticism, and some researchers may prefer the Atkinson or generalized entropy indices on ethical grounds. These researchers are confronted with a choice between using an ethically unattractive index with a known sampling distribution, or using an ethically preferable index whose sampling properties are unknown. The increasing use of the Atkinson and generalized entropy indices in empirical analyses of income distributions suggests many are choosing the second option. It then becomes important to understand the sampling properties of these indices, so that statistical inference procedures may be applied. The analysis in this paper complements the recent research on sampling properties and inference for Lorenz curves. For example, Beach and Davidson (1983), and Gastwirth and Gail (1985) propose tests for equality of Lorenz curves, and Beach and Richmond (1985) provide joint confidence intervals for the Lorenz ordinate vector. Bishop, Chakraborti, and Thistle (1987, 1988a, 1988b) and Bishop, Formby, and Thistle (1989) provide a number of extensions. These papers utilize results from the theory of linear functions of order statistics.2 However, this approach is not directly applicable to the Atkinson and generalized entropy indices; these indices are not linear functions of order statistics. The Atkinson and generalized entropy indices are functions of fractional and negative moments of the distribution, and their large sample properties follow from the properties of the sample moments.