Statistical inference for the relative density

Abstract

Social scientists are increasingly interested in techniques for comparing changes in distributional shape in addition to mean levels. One such technique is based on the relative distribution, a nonparametric summary of the information required for scale-invariant comparisons between two distributions. The relative distribution is being used by social scientists to represent and analyze distributional differences, enabling researchers to move well beyond comparisons of means and variances in a simple intuitive way. The authors develop a nonparametric estimator for the relative density function. They study its asymptotic properties, derive computable expressions for the asymptotic variance, and consider local bandwidth selection. They also illustrate how the relative density can be decomposed into a component due to location differences and a component due to shape differences. This decomposition identifies that component of interdistributional dissimilarity due to interdistributional inequality. The methods are illustrated by comparing the earnings distributions of working women to that of working men based on the 1990 census and to women from 1967 to 1996.