author's proof - DARP

AUTHOR'S PROOF
Reference distributions and inequality measurement
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
Q2
Fig. 2 Quantile approach with
the most equal reference
distribution
x
Q3
°
°
°
°
Δ x 3
°
x 3
F −1 *
(·)
1
2.2 Reference distributions 96
The use of the J index also requires the specification of a reference distribution: there 97
are several possibilities. 98
The most equal reference distribution Let us assume that the most equal income 99
distribution is when the same amount is given to each individuals:
100 Q4
( ) i
F∗
−1 =ˆμ for i = 1,...,n (4)
n + 1
If we use this (egalitarian) distribution as the reference distribution in Eq. 3,thenwe 101
find the standard Generalised Entropy inequality measure: 8 102
1
n∑
[( ) α xi
J α =
− 1]
,α̸= 0, 1 (5)
nα(α − 1) ˆμ
UNCORRECTED PROOF
i=1
So the GE inequality measures are divergence measures between the EDF and the 103
most equal distribution, where everybody gets the same income. They tell us how far 104
a distribution is from the most equal distribution. A sample with a smaller index has 105
amoreequal distribution. 106
Figure 2 presents the quantile approach for this case. We can see that the EDF is 107
always above (below) the reference distribution for large (small) values of incomes. 108
It makes clear that large (small) values of α would be more sensitive to changes in 109
high (small) incomes. 110
8 The limiting forms are J 0 =−n 1 ∑ ( )
ni=1
log xi
, J ˆμ 1 = n 1 ∑ ( )
ni=1 x i
ˆμ log xi
. ˆμ