Page 195 - Contributed Paper Session (CPS) - Volume 6

CPS1881 Mike S.C. et al.
everywhere above the other) was useful for evaluating welfare redistribution
toward decreasing the inequality through a progressive transfer, or for ranking
income distributions according to expected utility (Atkinson, 1970). Inference
for the LD property has been extensively discussed in the literature since
1980s. Asymptotic normality of estimating Lorenz curve vector ordinates and
testing LD by significant differences among pairwise comparisons of Lorenz
ordinates were essentially based on a grid of finite points on the unit interval.
Extending the inference of LD from a finite grid to the entire unit interval, a
consistent test was developed for the LD hypothesis based on functions of the
empirical Lorenz processes (Barrett et al, 2014). The theoretical test p-values
were empirically assessed using the bootstrap method because the asymptotic
distributions of the test statistics depend on the unknown distributions.
The goal of this study is to investigate a distribution-free test scheme for
the LD hypothesis. This begins with an elementary fact that the difference
curve of a pair of Lorenz curves is a combination (or continuation) of concave
and convex curves depending on the condition of crossing Lorenz curves,
denoted by CLC, or dominant curves, the LD case. Distinct aspects of the
difference curve expressed by inequality patterns of the quantile functions
naturally characterize the CLC and LD conditions. Thus, the quantile inequality
patterns can be used to construct likelihood ratio (LR) tests under the CLC and
the LD hypotheses, respectively. The LR test for the LD hypothesis is by design
a reduced form of that for the CLC hypothesis, it is convenient to first test the
latter when it is applicable, otherwise, test only the former; and the separate
test results can be summarized to support a decision. The proposed LR tests
are consistent with respect to the critical regions and test levels based on
approximate chi-square distributions. The proposed LR tests are examined
using a simulation study and a real GDP per capita data analysis.
The proposed test design is laid out in two sections. In Section 2, it is
illustrated that distinct inequality patterns between paired quantiles of the two
distribution under comparison are exhibited under the CLC and LD hyptheses.
From these patterns, LR tests for the CLC and LD hypotheses can be separately
constructed by comparing the sample quantiles against the expected
quantiles under the hypotheses. In Section 3, a simulation study of pairs of
Lorenz curves from a few common distribution families is conducted to
evaluate the effectiveness of the proposed LR tests under the CLC and LD
hypotheses. The proposed LR tests are also used to investigate potential CLC
and LD conditions among a few real GDP per capita data of 133 countries
recorded across a few years in the Penn World Table (Summers and Heston,
1991). A pair of yearly data was tested to exhibit the CLC hypotheses and
another pair exhibited the LD property, the proposed LR tests were effectively
used and conclusive. The bootstrap method was also applied to testing the
same two paired data, and found not quite satisfactory with the CLC case, and

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