Page 196 - Contributed Paper Session (CPS) - Volume 6

CPS1881 Mike S.C. et al.
not satisfatory with the LD case when two curves were quite close to each
other. It is worth noting that the proposed LR tests derived from count
statistics of bivariate data are applicable to random samples of any two
distributions whether they are independent or not. We conclude this study
with a brief explanation that the LR tests are developed along with the
rationale of a nonparametric sign test for the difference curve between two
Lorenz curves.

2. Methodology
The Lorenz curve:
Assume that a non-negative random variable X has an absolutely
continuous distribution with continuous probability density f, finite mean =

2
E and variance = Var. Let { , = 1, ⋯ , } and {, = 1, ⋯ , } denote

random samples of variables X and Y with cumulative distribution functions
(cdf) F and G, and probability density functions (pdf) f and g, respectively. The
1
̂
empirical cdf of the { } is defined by () = ∑ ( ≤ ), 0 ≤ < ∞; the

=1

quantile function and the empirical quantile function are defined by −1 () =
̂
̂
inf{: () ≥ } and −1 () = inf{: () ≥ }, 0 ≤ ≤ 1, respectively. Similar
notations for the variable Y are defined by analogy.
Definition 1. The Lorenz curve of the random variable is, for 0 ≤ ≤ 1,
1
() = 0 −1 (),
∫

and the empirical Lorenz curve is

̂
̂
() = 1 ∫ −1 (), (2.1)

̂ 0
1
̅
where ̂ = ∑ (= ) is the sample mean.
=1

Likelihood ratio (LR) test statistics:
Consider the null hypothesis of CLC between distribution functions F
0
and G, and the alternative hypothesis of LD. Specifically, let = 01 ∪
02
1
0
present two distinct cases of CLC: denotes the case of a single intersection,
01
say, crosses once from above; and denotes the situation of having

02
two or more intersection points, which rarely occurs in practice.
∗
Case 1. Under , there exists a unique ∈ (0, 1) such that () ≡ () −

01
∗
() > 0, ∈ (0, ); ( ) = 0; and () < 0, ∈ ( , 1). By convexity of the
∗
∗

Lorenz curve, () is concave on [0, ), and convex on ( , 1]. That is,
∗
∗
′′
() = 1 − 1 < 0, (2.2)
( −1 ()) ( −1 ())
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