Page 194 - Contributed Paper Session (CPS) - Volume 6

CPS1881 Mike S.C. et al.

Likelihood ratio tests for Lorenz Dominance
Mike S. Chang; Michelle Liou; Philip E. Cheng
Institute of Statistical Science Academia Sinica, Taipei, Taiwan, Republic of China

Abstract
The notion of Lorenz dominance (LD) between two Lorenz curves (LC; Lorenz,
1905) is useful for evaluating welfare redistribution toward decreasing the
inequality or ranking income distributions based on expected utility. Early
studies of estimation and testing for the LD property were mostly discussed
based on a grid of finite points on the unit interval. Extending the inference
from a finite grid to the entire unit interval, consistent tests for the LD
hypothesis were recently developed upon functions of the empirical Lorenz
processes. The asymptotic distributions of the test statistics depend on the
unknown distributions and theoretical test p-values were empirically assessed
using the bootstrap method. In view of the intersection (once) property of a
pair of crossing Lorenz curves discussed in the 1980s, elementary properties
of crossing and dominant Lorenz curves will be examined in this study. It is
found that distinct patterns of the difference curve between a pair of curves
can be used to characterize the LD and the crossing Lorenz curves (CLC)
conditions through inequalities of their quantile functions. We will use these
patterns to construct likelihood ratio (LR) tests tailored for the LD and the CLC
hypotheses, respectively. The proposed LR tests are consistent with respect to
the standard test levels and critical regions based on approximate chi-square
distributions. A simulation study is conducted using Log-normal, Pareto and
Weibull distributions, and reliable performance of the LR tests are obtained.
An exceptional difficult case is found under the LD hypothesis when two
lognormal distributions with close standard deviation parameters are rather
close to each other, for which larger sample sizes are useful for making a test
decision. An empirical study is carried out for certain pairs of real GDP annual
data of 133 countries from “The Penn World Table (Mark 5): An Expanded Set
of International Comparisons, 1950-1988”.

Keywords
Crossing Lorenz curves; Likelihood ratio test; Lorenz dominance

1. Introduction
The Lorenz curve provides a graph of overall income proportions shared
by cumulative proportions of people acquiring incomes from low to high
levels. The notion of Lorenz dominance (LD) between two curves (one curve is

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