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AUTHOR'S PROOF
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1 Article Title Reference distributions and inequality measurement
2 Article Sub-Title
3 Article Copyright -
Year
Springer Science+Business Media New York 2013
(This will be the copyright line in the final PDF)
4 Journal Name The Journal of Economic Inequality
5
Family Name
6 Particle
Flachaire
7 Given Name Emmanuel
8 Corresponding Suffix
9 Author
Organization Aix-Marseille Université
10 Division Greqam
11 Address 2 rue de la Charité, 13002, Marseille, France
12 e-mail emmanuel.flachaire@univ-cezanne.fr
13
Family Name
14 Particle
Cowell
15 Given Name Frank A.
16 Suffix
Author
17 Organization London School of Economics
18 Division Sticerd
19 Address Houghton Street, London, WC2A 2AE, UK
20 e-mail None
21
Family Name
22 Particle
Bandyopadhyay
23 Given Name Sanghamitra
24 Suffix
Author
25 Organization University of London
26 Division Queen Mary
27 Address Mile End Road, London, E1 4NS, UK
28 e-mail None
29
30 Schedule Revised
Received 3 May 2011
31 Accepted 7 December 2012

AUTHOR'S PROOF
32 Abstract We investigate a general problem of comparing pairs of distribution
which includes approaches to inequality measurement, the
evaluation of “unfair” income inequality, evaluation of inequality
relative to norm incomes, and goodness of fit. We show how to
represent the generic problem simply using (1) a class of
divergence measures derived from a parsimonious set of axioms
and (2) alternative types of “reference distributions.” The problems
of appropriate statistical implementation are discussed and
empirical illustrations of the technique are provided using a variety
of reference distributions.
33 Keywords
separated by ' - '
34 Foot note
information
Divergence measures - Generalised entropy measures - Income
distribution - Inequality measurement - D63 - C10

AUTHOR'S PROOF
J Econ Inequal
DOI 10.1007/s10888-012-9238-z
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
Reference distributions and inequality measurement
Frank A. Cowell · Emmanuel Flachaire ·
Sanghamitra Bandyopadhyay
Received: 3 May 2011 / Accepted: 7 December 2012
© Springer Science+Business Media New York 2013
Abstract We investigate a general problem of comparing pairs of distribution which 1
includes approaches to inequality measurement, the evaluation of “unfair” income 2
inequality, evaluation of inequality relative to norm incomes, and goodness of fit. 3
We show how to represent the generic problem simply using (1) a class of divergence 4
measures derived from a parsimonious set of axioms and (2) alternative types of 5
“reference distributions.” The problems of appropriate statistical implementation 6
are discussed and empirical illustrations of the technique are provided using a variety 7
of reference distributions. 8
Keywords Divergence measures · Generalised entropy measures · 9
Income distribution · Inequality measurement 10
JEL Classification D63 · C10 11
1 Introduction 12
UNCORRECTED PROOF
There is a broad class of problems in distributional analysis that involve comparing 13
two distributions. This may involve judging whether a functional form is a good 14
fit to an empirical distribution; it may involve computation of the divergence of 15
F. A. Cowell Q1
Sticerd, London School of Economics, Houghton Street, London WC2A 2AE, UK
E. Flachaire (B)
Greqam, Aix-Marseille Université, 2 rue de la Charité 13002 Marseille, France
e-mail: emmanuel.flachaire@univ-cezanne.fr
S. Bandyopadhyay
Queen Mary, University of London, Mile End Road, London E1 4NS, UK

AUTHOR'S PROOF
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
F. A. Cowell et al.
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an empirical distribution from a theoretical economic model; it may involve the
ethical evaluation of an empirical distribution with reference to some norm or ideal
distribution. This paper shows how the class of problems can be characterised in a
way that has a natural interpretation in terms of familiar analytical tools.
This is not some recondite or abstruse topic. Several authors have explicitly
characterised inequality using this two-distribution paradigm: an inequality measure
is defined in terms of the divergence of an empirical income distribution from
an equitable reference distribution [5, 20, 28]. Furthermore several recent papers
have revived interest in the idea of inequality evaluations with reference to “norm
incomes” or a reference distribution; 1 in particular some authors have focused on
replacing a perfectly egalitarian reference distribution with one that takes explicit
account of fairness [3, 19]. In recent contributions Magdalou and Nock [27] have also
examined the concept of divergence between any two income distributions and its
economic interpretation and Cowell et al. [8] show how similar concepts can be used
to formulate an approach to the measurement of goodness of fit. 2
In this paper we use an aprioriapproach to the problem that allows one to
construct a distance concept that is appropriate for characterising the divergence
between the empirical distribution function and a proposed reference distribution.
The approach is related to results in information theory and is adaptable to other
fields in economics that make use of models of distributions [10]. The paper is
structured as follows. Section 2 introduces the concept of divergence and a set of
principles for distributional comparisons in terms of divergence; we show how these
principles characterise a class of measures and discuss different concepts of reference
distribution that are relevant for different versions of the generic problem under
consideration. Section 3 discusses issues of implementation and Section 4 performs
a set of experiments and applications using the proposed measures and UK income
data. Section 5 concludes.
2 The approach
The essence of the problem is the characterisation of the divergence of an income
distribution from a reference distribution: this divergence can be seen as an aggregation
of discrepancies in different parts of the distribution. To fix ideas we
may interpret this in terms of a classic modelling problem: comparing an Empirical
Distribution Function (EDF) ˆF (x) = 1 ∑ n
n+1 i=1 ι (x i ≤ x) 3 and a theoretical reference
distribution F ∗ . For instance, Fig. 1 presents an EDF (dotted line) and a theoretical
UNCORRECTED PROOF
1 Almås et al. [4] compare “actual and equalizing earnings”; their work is related to Paglin’s Gini [29]
and Wertz’s Gini [32]. Also see Jenkins and O’Higgins [25] and Garvy [24].
2 The Cowell et al. [8] approach differs from that developed here in that it deals with the problem of
continuous reference distributions on unbounded support.
3 ι is an indicator function such that ι (S) = 1 if statement S is true and ι (S) = 0 otherwise. We use 1
n+1
rather than 1 n to avoid a problem where i = n. Had we used i n in Eq. 1 then y n would automatically
be set to sup (X) where X is the support of F ∗ .

AUTHOR'S PROOF
Reference distributions and inequality measurement
Fig. 1 Quantile approach
x
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
Q2
Q3
°
°
°
°
Δ x 3
°
x 3 F −1 *
(·)
1
reference distribution (solid line) on a reverse graph, with q denoting proportions 50
of the population on the horizontal axis and the income quantiles on the vertical 51
axis. There are two obvious ways to describe discrepancies between the EDF and the 52
reference distribution: 53
1. The standard approach in the goodness-of-fit literature is based on the “horizon- 54
tal” differences between ˆF (x) and F ∗ (x) for any given x. 55
2. An alternative approach is to compare, for given q values, the corresponding q- 56
quantiles given by the EDF and F ∗ . Denote by {x (1) , x (2) , ..., x (n) } the members 57
of the sample in increasing order; the corresponding values given by the EDF 58
are the adjusted sample proportion q ={ 1
n+1 , 2
n+1 ,..., n
} and, for each q, the 59
n+1
corresponding value for the reference distribution is equal to 60
UNCORRECTED PROOF
y i = F −1
∗
( i
n + 1
)
. (1)
Here we look at the “vertical” differences between the x (i) and y i . As we will 61
see below it is this alternative view that is more fruitful for application to the 62
inequality-measurement problem. 63
2.1 Aggregation of information 64
A divergence measure for the EDF and the reference distribution aggregrates 65
discrepancies between the quantiles of the two distributions, for each value of q. 66
Each discrepancy concerns an income pair (x i , y i )—the ith quantile in the two 67
distributions—and the profile of these pairs captures all the essential information 68
for the problem to which the divergence measure will be applied. We suggest that 69
an appropriate divergence measure should satisfy the following principles: (1) If two 70

AUTHOR'S PROOF
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
F. A. Cowell et al.
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profiles are equivalent in terms of divergence and the discrepancy for pair i is the
same in each profile, then a local variation at i simultaneously in the two profiles has
no overall effect. (2) If there is zero discrepancy at each of two pairs in the profile
then moving x-income and y-income simultaneously from one pair to the other
has no effect on divergence. (3) Divergence remains unchanged by a uniform scale
change to x-values and y-values simultaneously. (4) Given two discrepancy profiles
with the same aggregate divergence, rescaling all the income discrepancies in each
profile by the same factor results in two new profiles that are equivalent in terms of
divergence. 4
Using the result in Theorem 1 (in the Appendix) and normalising on the case
where both the observed and the reference distribution exhibit complete equality,
these principles gives the following class of measures:
( ) 1
J α x, y :=
nα(α − 1)
[
n∑ [ ] α [ ] ]
1−α xi yi
− 1 . (2)
μ 1 μ 2
i=1
( )
where α takes any real value 5 and J α x, y ≥ 0 for arbitrary x and y. 6 There is an
analogy with the Kullback and Leibler [26] index of relative entropy [10].
So a divergence measure for the EDF and a theoretical reference distribution
(
F ∗
would be given by replacing x i and y i in Eq. 2, respectively, by EDF∗
−1 i
)
n+1 = x(i)
(
and F∗
−1 i
)
:
7
n+1
1
J α =
nα(α − 1)
⎡
n∑
i=1
⎣[
x(i)
] [ α
(
F
−1 i
) ] 1−α
∗ n+1
− 1⎤
⎦ ,α̸= 0, 1. (3)
ˆμ μ (F ∗ )
The J index can be used to measure the divergence between an empirical income
distribution, given by a sample of individual incomes, and any theoretical reference
distribution. It requires the choice of specific values for the parameter α according
to the judgment that one wants to make about the relative importance of different
types of discrepancy: choosing a large positive value for α puts weight on parts of the
distribution where the observed incomes x i greatly exceed the corresponding values
y i in the reference distribution; choosing a substantial negative value puts weight on
cases where the opposite type of discrepancy arises.
UNCORRECTED PROOF
4 SeeAxioms4,5,6,7intheAppendix.
5 ( )
J 0 x, y =−
1 ∑ ( )
ni=1 y i
n μ 2
log xi
μ 1
/ y ( )
i
μ 2
, J 1 x, y =
1 ∑ ( ni=1 x i
n μ 1
log xi
μ 1
/ y i
μ 2
).
6 ( ) ∑
To see this write J α x, y as ni=1 y i
[
nμ 2
ψ (qi ) − ψ (1) ] , where q i := x iμ 2
y i μ 1
,ψ(q) :=
qα
α[α−1] Because
ψ is convex function, for any (q 1 , ..., q n ) and any set of non negative weights (w 1 , ..., w n ) that sum
to 1, ∑ n
i=1 w i ψ (q i ) ≥ ψ (∑ n )
i=1 w i q i . Letting wi = y i / [nμ 2 ] and using the definition of q i we can see
that w i q i = x i / [nμ 1 ] so we have ∑ n
i=1 w i ψ (q i ) ≥ ψ (1) and the result follows. See also [6].
7 The limiting forms are J 0 =−n
1
( ( ))
1 ∑ i
ni=1 x (i)
n ˆμ log x (i)
ˆμ / F−1 ∗ n+1
μ(F ∗) .
(
∑ ni=1
F∗
−1 i
n+1
μ(F ∗)
)
log
(
( ) i
x (i)
ˆμ / F−1 ∗ n+1
μ(F ∗)
)
, J 1 =

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

AUTHOR'S PROOF
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F. A. Cowell et al.
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The most unequal reference distribution Rather than selecting the most equal
distribution as a reference distribution, we can reverse the standard approach by
using the most unequal distribution as a reference distribution. The most unequal
income distribution is when one person gets all the income and the others zero:
( ) {
i
F∗
−1
0 for i = 1,...,n − 1
=
n + 1 n ˆμ for i = n
If we use this distribution as the reference distribution in Eq. 3, then we have: 9
[( )
1 max α xi
J α =
− 1]
,α

AUTHOR'S PROOF
Reference distributions and inequality measurement
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
Q2
where 10 138
{
(1 − p)/(1 − k) if i ≤⌈n(1 − k)⌉
c i =
(10)
p/k
if i > ⌈n(1 − k)⌉
There are two interesting special cases. If p = k, everybody gets the same income 139
value, ˆμ, and the reference distribution is the most equal distribution. If k = 1/n and 140
p = 1, only one individual gets all the income, n ˆμ, and the reference distribution is 141
the most unequal distribution. 142
In practice, k and p have to be fixed: (1) to avoid the first drawback, k and p 143
should be independent of the sample size, with k > 1/n and p > 1/n ; (2) to avoid 144
the second drawback, zero incomes are not allowed in the reference distribution, that 145
is, if p = 1 we have k = 1.Finally,tomakeourindexJ α,k,p useful in practice, we need 146
to use constant values, such that 147
1/n < k < 1 and 1/n < p < 1, or p = k = 1. (11)
In empirical studies, we could use several values of k and p. For instance, k = 1 − 148
p = 0.05, 0.01, 0.005, correspond to the reference distributions with the top 5 %, 1 % 149
and 0.5 % getting, respectively, 95 %, 99 % and 99.5 % of the total income. 150
Other reference distributions Clearly, other reference distributions could be used. 151
For instance, if we assume that productive talents are distributed in the population 152 Q4
according to a continuous distribution of talents F ∗ and that wages should be related 153
to talent, a situation in which everyone received the same income to everybody 154
might be considered as unfair. In this case one might use F ∗ as the reference 155
distribution and make use of the index (Eq. 3): any deviation from F ∗ would come 156
from something else than talent. If total income is finite, it makes sense to use 157
a distribution defined on a finite support. For instance, we could use a Uniform 158
distribution or a Beta distribution with two parameters, which can provide a variety 159
of appropriate shapes. 160
3 Implementation 161
UNCORRECTED PROOF
It is necessary to establish the existence of an asymptotic distribution for J α and J α,k,p 162
in order to justify its use in practice. If the most equal distribution is taken as the 163
reference distribution (k = p = 1), the index J α,1,1 is nothing but the standard GE 164
inequality measure, which is asymptotically Normal and has well-known statistical 165
properties. 11 If a continuous distribution is taken as the reference distribution, it can 166
be shown that the limiting distribution of nJ α is that of 167
[ ∫
1 1
B 2 (t)dt
2μ F∗ 0 F∗
−1 (t) f∗ 2(F−1
∗ (t)) − 1 (∫ 1
) 2
]
B(t)dt
μ F∗ 0 f ∗ (F∗
−1
(12)
(t))
10 The limiting forms are J 0,k,p =−n 1 ∑ ( )
ni=1 x(i)
c i log , J c i ˆμ 1,k,p = n
1 ∑ ni=1 x (i)
ˆμ
( )
log
x(i)
. c i ˆμ
11 Among others, see Cowell and Flachaire [9], Davidson and Flachaire [12], Schluter and van
Garderen [31], Schluter [30], Davidson [11].

AUTHOR'S PROOF
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
F. A. Cowell et al.
168 where f ∗ is the density of distribution F ∗ and B(t) is a Brownian bridge. This random
169 variable can have an infinite expectation. It is only if F ∗ has a bounded support
170 that the limiting distribution has reasonable properties—see Cowell et al. [8] and
171 Davidson [11] for more details. If we use a continuous parametric reference distri-
172 bution, since total income is finite, it makes sense to use a distribution F ∗ defined
173 on a bounded support only. For instance, one could use a Uniform distribution or
174 a Beta distribution with two parameters, which can provide many different shapes.
175 The same approach can be used for nJ α,k,p , noting that the last statistic is equivalent
176 to the statistic defined in (9) in Cowell et al. [8], where 2i/(n + 1) is replaced by c i
177 defined in Eq. 10.
178 In the two cases, the limiting distribution of J α and J α,k,p exists, but is not
179 tractable. This is enough to justify the use of bootstrap methods for making inference.
180 To compute a bootstrap confidence interval, we generate B bootstrap samples by
181 resampling from the original data, and then, for each resample, we compute the
182 index J. We obtain B bootstrap statistics, Jα b , b = 1,...,B. The percentile bootstrap
183 confidence interval is equal to
CI perc = [ c b 0.025 ; ]
cb 0.975
(13)
184 where c b 0.025 and cb 0.975
are the 2.5 and 97.5 percentiles of the EDF of the bootstrap
185 statistics—for a comprehensive discussion on bootstrap methods, see Davison and
186 Hinkley [17], Davidson and MacKinnon [15]. For well-known reasons [14, 17] the
187 number B should be chosen so that (B + 1)/100 is an integer: here we set B = 999
188 unless otherwise stated.
189 To be used in practice, we need to determine the finite sample properties of J α
190 and J α,k,p . The coverage error rate of a confidence interval is the probability that
191 the random interval does not cover the true value of the parameter. A method of
192 constructing confidence intervals with good finite sample properties should generate
193 a coverage error rate close to the nominal rate. For a confidence interval at 95 %,
194 the nominal coverage error rate is equal to 5 %. We use Monte-Carlo simulation
195 to approximate the coverage error rate bootstrap confidence intervals in several
196 experimental designs.
197 In our experiments, samples are drawn from a lognormal distribution. For fixed
198 values of α, k, p and n, we draw 10,000 samples. For each sample we compute J α or
199 J α,k,p and its confidence interval at 95 %. The coverage error rate is computed as
200 the proportion of times the true value of the inequality measure is not included in
201 the confidence intervals. 12 Confidence intervals perform well in finite samples if the
202 coverage error rate is close to the nominal value, 0.05.Table1 presents the coverage
203 error rate of bootstrap confidence intervals at 95 % of J α and J α,k,p for several
204 reference distributions. The standard GE measures use the most equal reference
205 distribution, it corresponds to J α,1,1 .When“thetop1%richestgets99%ofthe
206 income” is the reference distribution we use the index J α,0.01,0.99 ; when “the top
207 5 % richest gets 99 % of the income” is the reference distribution we use J α,0.05,0.99 .
208 In addition, we examine J α with two continuous (bounded) parametric reference
209 distributions, the Beta(1,1) distribution which is equal to the Uniform(0,1), and the
UNCORRECTED PROOF
12 The true values are computed replacing x (i) in Eq. 3 by F −1 ( n+1 i ),whereF is the distribution of x,
that is, the lognormal distribution in our experiments.

AUTHOR'S PROOF
Reference distributions and inequality measurement
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
Q2
Q5 t1.1 Table 1 Coverage error rate of bootstrap confidence intervals at 95 % of J α and J α,k,p , 10,000
replications, 499 bootstraps, and x ∼ Lognormal(0, 1)
t1.2 α −1 0 0.5 1 2
Equal reference distribution
Standard GE measures (k = p = 1)
t1.3 n = 100 0.0753 0.0734 0.0832 0.0912 0.1166
t1.4 n = 200 0.0747 0.0667 0.0713 0.0785 0.1024
t1.5 n = 500 0.0669 0.0673 0.0716 0.0781 0.0983
t1.6 n = 1000 0.0658 0.0606 0.0642 0.0709 0.0878
t1.7 n = 2000 0.0567 0.0565 0.0620 0.0658 0.0831
t1.8 n = 5000 0.0557 0.0562 0.0606 0.0672 0.0809
Unequal reference distributions
Top 5 % gets 99 % of the income (k = 0.05, p = 0.99)
t1.9 n = 100 0.0722 0.0887 0.0983 0.1005 0.0395
t1.10 n = 200 0.0597 0.0662 0.0733 0.0813 0.0482
t1.11 n = 500 0.0594 0.0577 0.0638 0.0681 0.0584
t1.12 n = 1000 0.0553 0.0543 0.0572 0.0619 0.0575
t1.13 n = 2000 0.0581 0.0557 0.0552 0.0590 0.0564
t1.14 n = 5000 0.0526 0.0531 0.0562 0.0588 0.0569
Top 1 % gets 99 % of the income (k = 0.01, p = 0.99)
t1.15 n = 100 0.2220 0.2221 0.2172 0.1347 0.0325
t1.16 n = 200 0.1601 0.1689 0.1705 0.1265 0.0275
t1.17 n = 500 0.0998 0.1117 0.1188 0.1064 0.0346
t1.18 n = 1000 0.0703 0.0788 0.0867 0.0878 0.0422
t1.19 n = 2000 0.0581 0.0642 0.0682 0.0717 0.0486
t1.20 n = 5000 0.0558 0.0598 0.0616 0.0627 0.0504
Continuous reference distributions
Beta(1,1)
t1.21 n = 100 0.0830 0.0877 0.0923 0.0981 0.1162
t1.22 n = 200 0.0703 0.0756 0.0805 0.0865 0.1029
t1.23 n = 500 0.0689 0.0740 0.0778 0.0847 0.1011
t1.24 n = 1000 0.0650 0.0674 0.0710 0.0766 0.0905
t1.25 n = 2000 0.0605 0.0632 0.0645 0.0700 0.0838
t1.26 n = 5000 0.0623 0.0638 0.0660 0.0715 0.0824
Beta(2,2)
t1.27 n = 100 0.0778 0.0841 0.0896 0.0945 0.1122
t1.28 n = 200 0.0680 0.0730 0.0764 0.0832 0.1002
t1.29 n = 500 0.0694 0.0722 0.0762 0.0829 0.0988
t1.30 n = 1000 0.0611 0.0656 0.0682 0.0742 0.0885
t1.31 n = 2000 0.0574 0.0626 0.0636 0.0679 0.0834
t1.32 n = 5000 0.0584 0.0632 0.0651 0.0694 0.0816
UNCORRECTED PROOF
Beta(2,2) which is a symmetric inverted-U-shape distribution. Table 1 shows that 210
the finite sample properties of the indices with alternative reference distributions are 211
not very diferent from those of the standard GE measures, except for J α,0.01,0.99 when 212
n ≤ 500. The coverage error rate is close to 0.05 for very large samples. For small 213
and moderate samples, further investigation is required to improve the finite sample 214
properties, with, for instance, a fast double or triple bootstrap [13, 16]. 215

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216
217
218
219
220
4 Application
Let us compare the performance of J α,k,p with that of conventional GE inequality
measures using UK income data as a case study. 13 Table 2 presents the results of
indices J α and J α,k,p estimated with three different types of reference distribution,
along with bootstrap confidence intervals at 95 %.
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
Equality The top panel of Table 2 presents estimates of J α,k,p using an “equality” Q4
reference distribution. Clearly, when we select the most equal distribution as the
reference distribution, i.e. k = p = 1, the index J α,k,p is reduced to the standard GE
inequality measure. Estimates for standard GE measures, J α,1,1 are tabulated in the
first row, for values of α ranging from −1 to 2. 14 When α = 1, J α,1,1 is the Theil
index. For values of α = 0.5, 1, 1.5, 2, J α,1,1 represents transformed Atkinson indices
[7]. All estimates of standard GE measures increase considerably between 1979 and
1988, suggesting a significant rise in inequality in the 80s.
Extreme inequality The key point highlighted earlier was that changing the refer- Q4
ence distribution from which we measure the distance of the empirical distribution
opens up the possibility for researchers to choose the exact distribution from which
they wish to measure distance of the empirical distribution. While standard GE
indices tell us about the distance of the empirical distribution from an equal reference
distribution, one can change the focus to that of its distance from an unequal
reference distribution. The second panel of Table 2 presents estimates of J α,k,p using
several “extreme inequality” reference distributions. The interpretation of the size
of the J α,k,p index now is the reverse of the interpretation of standard GE measures.
For a standard GE inequality measure, a small value of J α,1,1 corresponds to the
empirical distribution being close to the equal reference distribution compared to
that of a large value of J α,1,1 . However, for an unequal reference distribution a small
value of J α,k,p corresponds to the empirical distribution being close to the particular
“extreme inequality” reference distribution that has been specified.
To illustrate we focus on two different unequal reference distributions: one, where
the top 1 % of the income distribution receive 99 % of the income, and second,
where the top 5 % of the income distribution receive 99 % of the income. From
Table 2 we can see that, with one exception, the values of J α,k,p have dropped
between years 1979 and 1988: in other words, it is almost always true that the distance
from the “extreme inequality” reference distribution has decreased. The exception
is the case (k = 0.05, p = 0.99,α = 2) where the movement relative to the reference
distribution is not significant. The implication is that UK inequality grew during the
1980s whether one interprets this in terms of distance from equality, or as distance
from a reference unequal distribution, except for one case. This case concerns top-
UNCORRECTED PROOF
13 The application uses the “before housing costs” income variable of the Family Expenditure Survey
for years 1979 and 1988 [18], deflated and equivalised using the McClements adult-equivalence
scale. We exclude households with self-employed individuals as reported incomes are known to be
misrepresented. The years 1979 and 1988 have been chosen to represent the maximum recorded
difference in inequality across the available years, post-1975.
14 A large value of α implies greater weight on parts of the distribution where the observed incomes
are vastly different from the corresponding values in the reference distribution.

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Reference distributions and inequality measurement
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
Q2
Q5 t2.1 Table 2 Inequality indices Jα,k,p and Jα computed with different reference distributions
t2.2 α −1 0 0.5 1 2
Equal reference distribution
Standard GE measures (k = p = 1)
t2.3 1979 0.1218 0.1056 0.1046 0.1066 0.1201
t2.4 [0.1119;0.1355] [0.1016;0.1097] [0.1005;0.1086] [0.1023;0.1111] [0.1132;0.1271]
t2.5 1988 0.1836 0.1541 0.1543 0.1618 0.2096
t2.6 [0.1685;0.2018] [0.1468;0.1613] [0.1460;0.1634] [0.1508;0.1728] [0.1843;0.2381]
Unequal reference distributions
Top 1 % gets 99 % of the income (k = 0.01, p = 0.99)
t2.7 1979 15.29 3.370 2.906 4.403 55.39
t2.8 [14.46;16.21] [3.315;3.427] [2.887;2.926] [4.390;4.419] [55.05;55.75]
t2.9 1988 11.70 3.086 2.795 4.341 57.97
t2.10 [10.59;12.792] [2.982;3.182] [2.749;2.836] [4.300;4.378] [57.32;58.66]
Top 5 % gets 99 % of the income (k = 0.05, p = 0.99)
t2.11 1979 3.803 2.080 2.271 3.768 44.43
t2.12 [3.708;3.907] [2.057;2.106] [2.254;2.288] [3.747;3.789] [44.08;44.74]
t2.13 1988 3.194 1.915 2.151 3.631 44.14
t2.14 [3.088;3.293] [1.882;1.945] [2.123;2.175] [3.591;3.665] [43.47;44.73]
Continuous reference distributions
Beta(1,1) or Uniform(0,1)
t2.15 1979 0.0320 0.0406 0.0483 0.0613 0.1457
t2.16 [0.0308;0.0333] [0.0391;0.0421] [0.0465;0.0501] [0.0589;0.0638] [0.1311;0.1648]
t2.17 1988 0.0339 0.0418 0.0486 0.0591 0.1125
t2.18 [0.0313;0.0373] [0.0383;0.0461] [0.0444;0.0536] [0.0538;0.0655] [0.1014;0.1276]
UNCORRECTED PROOF

AUTHOR'S PROOF
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F. A. Cowell et al.
UNCORRECTED PROOF
Q5 t2.1 Table 2 (continued)
t2.2 α −1 0 0.5 1 2
Beta(2,2)
t2.3 1979 0.0115 0.0132 0.0143 0.0158 0.0199
t2.4 [0.0105;0.0127] [0.0121;0.0146] [0.0131;0.0158] [0.0144;0.0175] [0.0181;0.0222]
t2.5 1988 0.0210 0.0243 0.0267 0.0299 0.0405
t2.6 [0.0180;0.0242] [0.0204;0.0283] [0.0221;0.0316] [0.0242;0.0362] [0.0300;0.0524]
Beta(2,5)
t2.7 1979 0.0116 0.0138 0.0153 0.0173 0.0237
t2.8 [0.0109;0.0124] [0.0129;0.0147] [0.0143;0.0163] [0.0162;0.0185] [0.0219;0.0256]
t2.9 1988 0.0121 0.0142 0.0157 0.0175 0.0231
t2.10 [0.0100;0.0145] [0.0116;0.0172] [0.0127;0.0192] [0.0141;0.0217] [0.0177;0.0294]
t2.11 Data are from the Family Expenditures Surveys in UK. Bootstrap confidence intervals at 95 % are given in brackets

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JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
Q2
sensitive inequality where, in terms of “distance from maximum inequality,” the 253
change in the distribution is inconclusive. 254
Theoretical distribution Finally let us consider how inequality changed using a con- 255 Q4
tinuous reference distribution F ∗ . The last panel of Table 2 tabulates the results for 256
three F ∗ from the Beta distribution family. Did UK income inequality, interpreted 257
as a distance from a Beta-family reference distribution increase We can see that 258
the values of J α are not statistically different between 1979 and 1988 when the 259
Beta(1,1) (uniform) or Beta(2,5) (unimodal, right skewed) is used as the reference 260
distribution distribution, while they are statistically different when the Beta(2, 2) 261
(unimodal symmetric) is used as the reference distribution. 262
The estimates of the standard GE inequality measures J α,1,1 and of those of J α,k,p 263
and J α in Table 2 provide us with different information about divergence of the 264
empirical distribution from the chosen reference distribution. By varying the values 265
of k and p, one can specify the exact skewness of the reference distribution one would 266
like to measure distance of the empirical distribution from. Likewise, by varying the 267
values of α one can focus on different parts of the income distribution. A large value 268
of α implies a greater weight on parts of the distribution where the observed incomes 269
are vastly different from the corresponding values in the reference distribution. 270
Finally, one can choose specific parametric distributions which correspond to the 271
relevant reference distribution that the researcher is interested in. 272
5 Conclusion 273
The problem of comparing pairs of distributions is a widespread one in distributional 274
analysis. It is often treated on an ad-hoc basis by invoking the concept of norm 275
incomes and an arbitrary inequality index. 276
Our approach to the issue is a natural generalisation of the concept of inequal- 277
ity indices where the implicit reference distribution is the trivial perfect-equality 278
distribution. Its intuitive appeal is supported by the type of axiomatisation that 279
is common in modern approaches to inequality measurement and other welfare 280
criteria. The axiomatisation yields indices that can be interpreted as measures of 281
divergence and are related to the concept of divergence entropy in information 282
theory [10]. Furthermore, they offer a degree of control to the researcher in that 283
the J α indices form a class of measures that can be calibrated to suit the nature of the 284
economic problem under consideration. Members of the class have a distributional 285
interpretation that is close to members of the well-known generalised-entropy class 286
of inequality indices. 287
In effect the user of the J α -index is presented with two key questions: 288
UNCORRECTED PROOF
1. the income discrepancies underlying inequality are with reference to what 289
2. to what kind of discrepancies do you want the measure to be particularly 290
sensitive 291
As our empirical illustration has shown, different responses to these two key ques- 292
tions provide different interpretations from the same set of facts. 293

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F. A. Cowell et al.
294
Appendix: Axiomatic foundation
295 For convenience we work with z :=(z 1 , z 2 , ..., z n ), where each z i is the ordered pair
296 (x i , y i ), i = 1, ..., n and belongs to a set Z , which we will take to be a connected
297 subset of R + × R + . For any z ∈ Z n denote by z (ζ,i) the member of Z n formed
298 by replacing the ith component of z by ζ ∈ Z . The divergence issue focuses on the
299 discrepancies between the x-values and the y-values. To capture this introduce a
300 discrepancy function d : Z → R such that d (z i ) is strictly increasing in |x i − y i |.The
301 solution is in two steps: (1) characterise a weak ordering ≽ on Z n (z ≽ z ′ means “the
302 income pairs in z are at least as close according to ≽ as the income pairs in z ′ ”) (2)
303 use the function representing ≽ to generate the index J.
304
Axiom 1 (Continuity) ≽ is continuous on Z n .
305 Axiom 2 (Monotonicity) If z, z ′ ∈ Z n differ only in their ith component then
d (x i , y i ) < d ( 306
x
i ′, i) y′ ⇐⇒ z ≻ z ′ .
307 Axiom 3 (Symmetry) For any z, z ′ ∈ Z n such that z ′ is obtained by permuting the
308 components of z: z ∼ z ′ .
309 Axiom 3 implies that we may order the components of z such that x 1 ≤ x 2 ≤ ... ≤
310 x n and y 1 ≤ y 2 ≤ ... ≤ y n .
311 Axiom 4 (Independence) For z, z ′ ∈ Z n such that: z ∼ z ′ and z i = z
i ′ for some i then
z (ζ,i) ∼ z ′ (ζ,i) for all ζ ∈ [ ] [ ]
312 z i−1 , z i+1 ∩ z
′
i−1
, z
i+1
′ .
313 Axiom 5 (Zero local discrepancy) Let z, z ′ ∈ Z n be such that, for some i and j, x i =
314 y i , x j = y j , x
i ′ = x i + δ, y
i ′ = y i + δ, x ′ j = x j − δ, y ′ j = y j − δ and, for all k ̸= i, j, x ′ k =
315 x k , y ′ k = y k.Thenz ∼ z ′ .
316 Axiom 6 (Income scale irrelevance) For any z, z ′ ∈ Z n such that z ∼ z ′ , tz ∼ tz ′ for
317 all t > 0.
UNCORRECTED PROOF
318 Axiom 7 (Discrepancy scale irrelevance) Suppose there are z 0 , z ′ 0 ∈ Z n such that
z 0 ∼ z ′ 0 . Then for all t > 0 and z, z′ such that d (z) = td (z 0 ) and d ( z ′) = td ( 319
z 0) ′ : z ∼ z ′ .
320
321
322
323
324
325
326
327
Lemma 1 Given Axioms 1 to 5 (a) ≽ is representable by ∑ n
i=1 φ i (z i ) , ∀z ∈ Z n where,
for each i, φ i : Z → R is a continuous function that is strictly decreasing in |x i − y i |
and (b) φ i (x, x) = a i + b i x.
Proof Axioms 1 to 5 imply that ≽ can be represented by a continuous function :
Z n → R that is increasing in |x i − y i |, i = 1, ..., n. Part (a) of the result follows from
Axiom 4 and Theorem 5.3 of [23]. Now take z
( ′ and z as specified in Axiom 5: z ∼ z ′ if
and only if φ i (x i + δ, x i + δ) − φ i (x i , x i ) − φ j x j + δ, x j + δ ) (
+ φ j x j + δ, x j + δ ) = 0
which implies
φ i (x i + δ, x i + δ) − φ i (x i , x i ) = f (δ)
328 for any x i and δ. The solution to this Pexider equation implies (b).

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Reference distributions and inequality measurement
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Q2
Lemma 2 Given Axioms 1 to 6 ≽ is representable by φ
( ∑n
i=1 x ih i
(
xi
y i
))
where h i is a 329
real-valued function. 330
Proof Using the function introduced in the proof of Lemma 1 Axiom 6 implies 331
(z) = ( z ′) and (tz) = ( tz ′) ; since this has to be true for arbitrary z, z ′ we have 332
(tz)
(z) = ( tz ′)
(z ′ ) = ψ (t)
where ψ is a continuous function R → R. Hence,usingtheφ i given in Lemma 1 we 333
have for all i : φ i (tz i ) = ψ (t) φ i (z i ) or, equivalently φ i (tx i , ty i ) = ψ (t) φ i (x i , y i ) . In 334
view of Aczél and Dhombres [2], page 346 there must exist c ∈ R and a function 335
h i : R + → R such that 336
( )
φ i (x i , y i ) = xi c h xi
i . (14)
y i
From Lemma 1 and Eq. 14: φ i (x i , x i ) = xi ch i (1) = a i + b i x i , which, ifφ i (x, x) is non- 337
constant in x, implies c = 1. Noting that Lemma 1 implies that ≽ is also representable 338
by φ (∑ n
i=1 φ i (z i ) ) (whereφ : R → R is continuous and monotonic increasing, and 339
taking Eq. 14 with c = 1 gives the result. ⊓⊔ 340
Theorem 1 Given Axioms 1 to 7 ≽ is representable by φ (∑ n
)
i=1 xα i y1−α i where α ̸= 1 341
is a constant. 342
Proof (Cf [21]) Take the special case where, in distribution z ′ 0
the income discrep- 343
ancy takes the same value r for all n income pairs. If (x i , y i ) represents a typical 344
component in z 0 then z 0 ∼ z ′ 0
implies 345
( n∑ ( ) )
xi
r = ψ x i h i (15)
y
i=1 i
where ψ is the solution in r to 346
n∑
( )
xi
n∑
x i h i = x i h i (r) (16)
y i
UNCORRECTED PROOF
i=1
i=1
In Eq. 16 we take the x i as fixed weights. Using Axiom 7 in Eq. 15 requires 347
( n∑ (
tr = ψ x i h i t x ) )
i
,forallt > 0. (17)
y
i=1
i
Using Eq. 16 we have 348
(
n∑
n∑ ( )
x i h i
(tψ
)) xi
n∑
(
x i h i = x i h i t x )
i
(18)
y
i=1
i=1 i y
i=1
i
Introduce the following change of variables 349
( )
xi
u i := x i h i , i = 1, ..., n (19)
y i

AUTHOR'S PROOF
JrnlID 10888_ArtID 9238_Proof# 1 - 08/01/13
F. A. Cowell et al.
350
and write the inverse of this relationship as
351
352
353
354
355
356
357
358
x i
= ψ i (u i ) , i = 1, ..., n
y i
(20)
Substituting Eqs. 19 and 20 into Eq. 18 we get
(
n∑
n∑
))
n∑
x i h i
(tψ u i = x i h i (tψ i (u i )) .
i=1
i=1
i=1
(21)
Also define the following functions θ 0 (u, t) := ∑ n
i=1 x ih i (tψ (u)), θ i (u, t) :=
x i h i (tψ i (u)) , i = 1, ..., n. Substituting these into Eq. 21:
( n∑
)
n∑
θ 0 u i , t = θ i (u i , t)
i=1
i=1
which has as a solution
θ i (u, t) = b i (t) + B (t) u, i = 0, 1, ..., n
where b 0 (t) = ∑ n
i=1 b i (t) ([1], p. 142). Therefore we have
(
h i t x )
i
= b ( )
i (t)
xi
+ B (t) h i , i = 1, ..., n (22)
y i x i
y i
From Eichhorn [22], Theorem 2.7.3 the solution to Eq. 22 is of the form
{ }
βi v
h i (v) =
+ γ i ,α̸= 1
β i log v + γ i α = 1
(23)
where β i > 0 is arbitrary. Lemma 2 and Eq. 23 give the result.
⊓⊔
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UNCORRECTED PROOF

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