Characterizing systems of distributions by quantile measures

In: Statistica Neerlandica: journal of the Netherlands Society for Statistics and Operations Research, Band 50, Heft 3, S. 417-430

Abstract

Modelling an empirical distribution by means of a simple theoretical distribution is an interesting issue in applied statistics. A reasonable first step in this modelling process is to demand that measures for location, dispersion, skewness and kurtosis for the two distributions coincide. Up to now, the four measures used hereby were based on moments.In this paper measures are considered which are based on quantiles. Of course, the four values of these quantile measures do not uniquely determine the modelling distribution. They do, however, within specific systems of distributions, like Pearson's or Johnson's; they share this property with the four moment‐based measures.This opens the possibility of modelling an empirical distribution—within a specific system—by means of quantile measures. Since moment‐based measures are sensitive to outliers, this approach may lead to a better fit. Further, tests of fit—e.g. a test for normality—may be constructed based on quantile measures. In view of the robustness property, these tests may achieve higher power than the classical moment‐based tests.For both applications the limiting joint distribution of quantile measures will be needed; they are derived here as well.