DOI: 10.4225/03/5897b8d62d391
title: Comparison of nonparametric Lorenz curves and regression functions under inequality constraints

description: Comparison of two populations with respect to their income inequalities is an important topic in economic and social studies. The Lorenz curve is a useful measure of income inequality. The Lorenz ordinate L(x) at x (0≤ x ≤1) is deﬁned as the proportion of the total income that is owned by the lowest-earning 100x percent of people. It turns out that L(x) is an increasing convex function deﬁned in the interval [0,1], with L(0) = 0 and L(1) = 1. If the Lorenz curve of population 1 never lies below that of population 2, then population 1 displays less income inequality than population 2, and we say that population 1 Lorenz dominates population 2. 
In general, income inequality studies are performed under the assumption that the income distribution has a known parametric form. By contrast, in this thesis we allow the income distribution to take any arbitrary shape. Since, in practice, the income distribution is not known, the methods that we propose are based on more realistic assumptions. 
The majority of the Lorenz dominance tests proposed in the literature compare two Lorenz curves at a ﬁnite number of points on the x-axis. Recently, Bhattacharya (2007) proposed a Lorenz dominance test that compares two Lorenz curves on the entire x-axis, without using any scale function. In Chapter 3 of this thesis, a consistent test is developed for Lorenz dominance, based on a class of weight functions. The hypotheses for the test are formulated by comparing two Lorenz curves over the entire x-axis. A simulation study is performed to compare the performance of our proposed test with that of Bhattacharya’s (2007) test. The results of this study show that our scale-based test performs better than the test without a scale function. 
If the Lorenz curves for two populations do not intersect, then Lorenz dominance is easy to interpret and provides a simple way of ranking income distributions with respect to income inequality. However, experience shows that Lorenz curves often do cross each other, and hence, the ranking of populations based on Lorenz dominance is impossible. A common practice for ranking populations when the Lorenz curves intersect is to use summary meas¬ures of inequality such as the Gini coefﬁcient. However, any ranking based on a single inequality measure will have limitations. 
In Chapter 4, we introduce a new testing procedure for ranking two income distributions when the corresponding Lorenz curves may intersect. The idea is to allow the Lorenz curves to cross, but to impose restrictions on the possible difference between them. To this end, we will introduce the ideas of non-inferiority and superiority in order to reformulate the inference problem. 
The testing problem is formed by comparing two Lorenz curves at a ﬁnite number of points on the x-axis. We test against the hypothesis that the Lorenz curve of population 2 does not lie below that of population 1 by more than a speciﬁed margin (non-inferiority margin) at any point on the x-axis, and that Lorenz curve of population 2 lies above that of population 1 by more than a speciﬁed margin (superiority margin) for at least one point on the x-axis. A bootstrap algorithm is proposed to implement the test. In a simulation study, we observe that the type-I error rates for the test are close to the nominal level. We then illustrate the usefulness of our proposed method using an empirical example. 
Finally, in Chapter 5, tests for detecting differences between two univariate nonparametric regression curves are developed. The aim with this new method is to establish that one treatment is not inferior to another for the whole population, and also that it is superior for at least a part of the population, when the treatment effect is represented by a nonparametric regression curve. The inference problem is formulated as a test against the alternative hypo¬thesis (a) that the regression curve for population 1 does not fall below that for population 2 by more than a speciﬁed small amount, at any value of the covariate, and (b) that the former exceeds the latter, at some values of the covariate, by more than a speciﬁed amount. The test statistic is easy to compute, and tables of asymptotic critical values are also provided. 
Because the asymptotic test is conservative, a less conservative bootstrap test is proposed and is shown to be asymptotically valid. In a simulation study, we observe that the type-I error rates for these tests are close to the nominal level, and that the bootstrap test exhibits a higher estimated power, as expected.