Asymptotic confidence bands for the Lorenz and Bonferroni curves based on the empirical Lorenz curve

Abstract

We construct asymptotic confidence bands for the Lorenz and Bonferroni curves that are fundamental tools for analyzing data arising in economics and reliability. The width of the obtained confidence bands is regulated by weight functions depending on the available information about the underlying distribution function. We show that, in some instances, on deleting the smallest and largest observations, the empirical Lorenz and Bonferroni curves become better estimators of the corresponding theoretical ones, and also provide a complete description of such instances. In the process of constructing confidence bands, we prove weighted weak approximation results for the Lorenz and Bonferroni processes, as well as for the Vervaat process that plays a fundamental role in obtaining the main results. We also present examples that indicate the optimality of results.