  We consider concepts and models for measuring inequality in the distribution
of resources with a focus on how inequality varies as a function of covariates.
Lorenz introduced a device for measuring inequality in the distribution of
income that indicates how much the incomes below the u$^{th}$ quantile fall
short of the egalitarian situation where everyone has the same income. Gini
introduced a summary measure of inequality that is the average over u of the
difference between the Lorenz curve and its values in the egalitarian case.
More generally, measures of inequality are useful for other response variables
in addition to income, e.g. wealth, sales, dividends, taxes, market share and
test scores. In this paper we show that a generalized van Zwet type dispersion
ordering for distributions of positive random variables induces an ordering on
the Lorenz curve, the Gini coefficient and other measures of inequality. We use
this result and distributional orderings based on transformations of
distributions to motivate parametric and semiparametric models whose regression
coefficients measure effects of covariates on inequality. In particular, we
extend a parametric Pareto regression model to a flexible semiparametric
regression model and give partial likelihood estimates of the regression
coefficients and a baseline distribution that can be used to construct
estimates of the various conditional measures of inequality.
