Goodness-of-fit tests for Pareto and Log-normal distributions

The Zenga (1984) inequality curves λ(p) and Z(p) have some interesting properties. The former is constant in p for Type I Pareto distributions, while the latter is constant in p for Log- Normal distributions. After discussing in detail these aspects, these characterizing behaviors will be exploited to obtain graphical and analytical tools for tail analysis, estimation and goodness of fit tests. Order statistics-based estimators of the curves will be presented and discussed; furthermore, a testing procedure for Pareto-type behavior and one for Log- normality based on a regression of λ(p) and Z(p) against p will be introduced. The properties of the proposed estimation and testing strategies are analyzed theoretically and by means of simulations; comparisons with competing testing strategies are presented. An application to data sets on city sizes, facing the debated issue of distinguishing Pareto-type tails from Log-normal tails, illustrates how the proposed method works in practice.