Uniform Convergence of Smoothed Distribution Functions with an Application to Delta Method for the Lorenz Curve
In this note, we present some theoretical results useful for inference on a population Lorenz curve for income and expenditure distributions, when the population density of the distribution is not (uniformly) bounded away from zero, and potentially has thick tails. Our approach is to define Hadamard differentiability in a slightly nonstandard way, and using it to establish a functional delta method for the Lorenz map. Our differentiability concept is nonstandard in that the perturbation functions, which are used to compute the functional derivative, are assumed to satisfy certain limit conditions. These perturbation functions correspond to a (nonparametric) distribution function estimator. Therefore, as long as the employed estimator satis.es the same limit conditions, which we verify in this paper, the delta method and corresponding asymptotic distribution results can be established.