Joint Confidence Intervals for Income Shares and Lorenz Curves

Abstract

In a recent paper, Beach and Davidson [1983] extended the principles of statistical inference to Lorenz curves and income shares by establishing the (asymptotic) distribution and covariance structure of a vector of Lorenz curve ordinates corresponding to a set of quantile abscissae. One can thus use the Lorenz curve and income shares, no longer just as descriptive devices for summarizing distribution information, but now also as analytical tools for comparing alternative distributions and carrying out conventional statistical inference on them. It would clearly be desirable, however, to extend and complement this work by providing a method for computing a set of joint confidence intervals about these income shares and for illustrating graphically a joint confidence band about a set of estimated Lorenz curve ordinates. The present paper provides a new and simple technique for doing this. When analyzing differences between sets of income shares or Lorenz curves, one typically wants to elicit more detailed information than is provided by a standard joint chi-square test on the overall set of shares. That is, when the hypothesis of two sets of shares being the same is rejected, we generally want to know further which particular differences in shares are different from zero, and of these nonzero differences which are positive and which negative (Savin [1980]). A natural way of providing such information is through multiple comparison procedures developed in this paper. In a recent work, Richmond [1982] has developed a general method of finding multiple comparison intervals that is particularly suited to the present problem and in this situation appears to dominate other conventional alternatives. In contrast, for example, to Scheff&'s S-projection method, Richmond's approach focuses on a set of primary points of interest (income shares) among a more general set of points (including the Lorenz curve ordinates) and optimizes the joint confidence interval lengths on this primary set of points. Since estimated Lorenz curve ordinates are built up from underlying income shares, these shares (perhaps further supplemented) are natural choices for the primary set. The method, however, also provides a joint confidence band that geometrically bounds the set of Lorenz curve ordinates (and interpolated segments between them). The present