Lorenz, Gini, Bonferroni and Quantile Regression

                            Kjell Doksum
                                                          
                              Abstract

Lorenz and Bonferroni introduced measures of the concentration of income
that indicate how much the incomes below the uth quantile fall short of
the egalitarian situation where everyone has the same income.  As u
changes,these measures become curves on [0,1].Gini introduced an index
that is the average over u of the difference between the Lorenz curve and
its egalitarian version.Bonferroni similarly introduced an index based on
the Bonferroni curve.In this paper we consider the situation where the
income distribution depends on covariates.  In this case the Lorenz and
Bonferroni curves as well as the Gini and Bonferroni indices are functions
of the covariates.We consider the estimation of these functions for
parametric,semiparametric and nonparametric models.  In particular,we
consider a semiparametric model involving regression coefficients and an
unknown baseline income distribution.In this model we find partial
likelihood estimates of the regression coefficients and the baseline
distribution that can be used to construct estimates of the Gini and
Bonferroni indices. We also consider nonparametric models where
nonparametric quantile regression estimators can be used to estimate
Lorenz,Gini,and Bonferroni regression.  This is joint work with Rolf Aaberge 
and Steinar Bjerve.