EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 3, 2019, 689-708
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Decomposing the Quantile Ratio Index with
Applications to Australian Income and Wealth Data

Luke A. Prendergast1, Robert G. Staudte1,∗

1 Department of Mathematics and Statistics, College of Science, Health and Engineering,
La Trobe University, Melbourne, Australia

Abstract. The quantile ratio index is a simple and effective measure of relative inequality for
income data that is resistant to outliers. A useful property of this index is investigated here: given
a partition of the income distribution into a union of sets of symmetric quantiles, one can find the
inequality for each set and readily combine them in a weighted average to obtain the index for the
entire population. When applied to data for various years, one can track how these contributions
to inequality vary over time, as illustrated here for Australian Bureau of Statistics income and
wealth data.

2010 Mathematics Subject Classifications: 62P20, 91B82

Key Words and Phrases: Confidence intervals, Gini index, inequality measures, quantile density,
robust statistics

1. Introduction

It is desirable to break down a measure of inequality for a population of incomes into
contributions to inequality from sub-populations. A natural partition of a population for
the quantile ratio index (QRI) of [14] is provided by unions of symmetric quantiles. It
allows one to determine how inequality in the middle half, for example, affects the QRI
compared to how inequality between the smallest and largest quartiles does.

1.1. Background

Given the vast literature on inequality measures and many possible decompositions of
them, it is reasonable to ask why should we start over in these endeavours with ratios
of quantiles or averages of ratios of symmetric quantiles? The answer is two-fold: first,
estimation of the traditional inequality measures is quite difficult. It has been shown by
[5] that many inequality measures have unbounded influence functions, which leads their

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3436

Email addresses: luke.prendergast@latrobe.edu.au (L. Prendergast),
r.staudte@latrobe.edu.au (R. Staudte)

http://www.ejpam.com 689 c© 2019 EJPAM All rights reserved.



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 690

sample versions (estimators) to be overly sensitive to outliers, which are not uncommon in
income data. Moreover, these estimators can be extremely slow to converge to normality
(even when the population variance is finite), which undermines confidence intervals based
on the fact of asymptotic normality, see [12, Fig.7]. Second, only recently have good
inferential methods become available for ratios of quantiles. That is because confidence
intervals for quantiles or ratios of quantiles require good estimates of the quantile density,
recently found in [11]. These estimates are the basis for confidence intervals for ratios of
quantiles, [13], quantile versions of the Lorenz curve and associated inequality measures
in [12] and the quantile ratio index itself [14].

When we talk about ‘decomposability’ of inequality indices we usually mean an ANOVA
type breakdown of the inequality index between and within subpopulations, [2, 4, 6, 8,
10, 15–17, and references therein]. The QRI has not yet been shown to satisfy such a
decomposition, but readily lends itself to partitions of unions of symmetric quantiles. So,
for example, while one might use the popular P90/P10 ratio of percentiles to compare
large with small incomes, here we can apply the QRI to the union of the largest 20%
of incomes with the smallest 20%, say, and also see how this contributes to the overall
QRI for the entire population, as summarized in the weighted average (3). By estimating
these components of the QRI for data in different years, we can see which portions of the
population are changing over time, as we illustrate with Australian income and wealth
data in Section 3.

To make these statements precise, we next formally introduce the QRI. Let F be the
cumulative distribution function (cdf) describing a population of non-negative incomes
with possible positive mass on zero F (0) < 1/2, and define the quantile function of F by
Q(p) = inf{x : F (x) ≥ p}, 0 ≤ p < 1. Further define Q(1) = limp→1Q(p), which equals
+∞ if F has infinite support. Often we write xp for Q(p). Following [14], define the ratio
of symmetric quantiles by R(p) = Q(p/2)/Q(1− p/2) = xp/2/x1−p/2 for 0 ≤ p ≤ 1. A plot
of R(p) against p shows how the typical (median) income of those with the lowest 100p%
incomes, divided by typical (median) income of those with the highest 100p% incomes
varies with p.

Relative inequality in the population of incomes is measured by the quantile ratio
index, the area above R(p) and less than the horizontal line at one: I =

∫ 1
0 {1−R(p)} dp.

Each of the (1−R(p))s is itself a measure of relative inequality; for example, 1−R(0.2) =
1− x0.1/x0.9 is the ratio of percentiles P90/P10, after transformation to the unit interval
so that larger values indicate more relative inequality. Thus I is a simple average of
these relative inequality measures, one for each p, while most such measures, including the
Gini index, are a ratio of two measures, concentration and scale. It is shown in [14] that
I = I(F ) has a bounded influence function which explains the good robustness properties
of its estimator Î , defined below in Section 2. For extensive comparison of the Gini index
with Î and other outlier resistant measures of relative inequalty, see [12–14].



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 691

1.2. Derivation of the decomposition and examples

Partition the unit interval into symmetric unions of intervals as follows: given K ≥ 1
and 0 = p0 < p1 < · · · < pK−1 < pK = 1/2 define Ak = [pk−1, pk) ∪ (1 − pk, 1 − pk−1] for
k = 1, . . . ,K − 1 and let the last AK = [pK−1, 1− pK−1]. For our purposes it is useful to
think of AK as essentially the union of two intervals [pK−1, 1/2) and (1/2, 1− pK−1], the
central point 1/2 playing a trivial role in what follows. We call {A1, . . . , AK} a symmetric
K-partition of [0,1]. When K = 2 for any 0 < p1 < 1/2 the symmetric 2-partition consists
of A1 = [0, p1) ∪ (1− p1, 1] and A2 = [p1, 1− p1]; and in particular for p1 = 1/4 we obtain
the quartile partition: the set A1 describes the outer quartiles while A2 the inner quartiles
or central half. Another example is the quintile partition obtained by taking K = 3
and p1 = 0.2, p2 = 0.4. This symmetric 3-partition consists of A1 = [0, 0.2) ∪ (0.8, 1.0],
A2 = [0.2, 0.4) ∪ (0.6, 0.8] and A3 = [0.4, 0.6].

Next we derive the decomposition of I into a weighted sum of inequality contributions
in the partitioning of F inherited from a symmetric K-partition. To this end, when
X has cdf F , write X ∼ F ; and denote U ∼ U [0, 1] for U uniformly distributed on
[0,1]. Fix k and let Q{Ak} denote the image of Ak under Q = F−1. Also introduce
wk = Pr(X ∈ Q{Ak}) = Pr(U ∈ Ak) = 2(pk − pk−1). For simplicity of notation in the
coming paragraphs, temporarily let a = pk−1 and b = pk, so Ak = [a, b)∪ (1− b, 1−a] and
c = 2(b− a) = wk. The conditional distribution of F given X ∈ Q{Ak} has cdf Fk given
by:

Fk(x) =


F (x)−a

c , for Q(a) ≤ x < Q(b) ;
1
2 , for Q(b) ≤ x < Q(1− b) ;
1
2 + F (x)−(1−b)

c , for Q(1− b) ≤ x < Q(1− a) .

The quantile function Qk of Fk can be obtained by solving for x = Qk(u) in the above
expression to obtain:

Qk(u) =

{
Q(a+ cu), for 0 < u ≤ 1/2 ;
Q(cu+ 1 + a− 2b), for 1/2 < u < 1 .

Given X ∈ Q{Ak}, the conditional quantile inequality curve is defined for 0 < p < 1 by

Rk(p) =
Qk(p/2)

Qk(1− p/2)
=

Q(a+ c p/2)

Q(1− a− c p/2)
. (1)

Given X ∈ Q{Ak}, the conditional QRI is denoted Ik and determined by

1− Ik =

∫ 1

0
Rk(p) dp =

∫ 1

0

Q(a+ c p/2)

Q(1− a− c p/2)
dp =

1

c

∫ 2b

2a
R(u) du , (2)

where we have made the change of variable u = 2a + cp. Rewriting 1 − Ik in terms of
our earlier notation a = pk−1, b = pk and c = 2(b − a) = wk, multiplying both sides of
equation (2) by wk and summing over k leads to the weighted average:

I =
∑
k

wkIk . (3)



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 692

It is evident that I1 ≥ · · · ≥ IK because Q is non-decreasing. Moreover, we can interpret
each Ik as the conditional inequality, given X ∈ Q{Ak}. The product wkIk gives the
amount that the kth partition member contributes to I.

0.0 0.2 0.4 0.6 0.8 1.0

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

R

p

R

I = 0.6638

0.0 0.2 0.4 0.6 0.8 1.0

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

R1

p
R

1

I1 = 0.9171

0.0 0.2 0.4 0.6 0.8 1.0

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

R2

p

R
2

I2 = 0.6352

0.0 0.2 0.4 0.6 0.8 1.0

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

R3

p

R
3

I3 = 0.2144

Figure 1: Plots of the quantile ratio R(p) = Q(p/2)/Q(1−p/2) for the standard lognormal distribution F and,
for the quintile partition, Rk(p) defined by (1) for k = 1, 2, 3. The inequality measure Ik for each Rk is the
shaded area above its graph and below the horizontal line at one.

1.3. Examples of symmetric K-partitions and inequality decompositions
for the standard lognormal distribution

For F the standard lognormal distribution and the quintile partition defined by K = 3
with p1 = 0.2, p2 = 0.4, the graphs of R and R1, R2, R3 are shown in Figure 1. The shaded
areas above the inequality curves and below the horizontal lines at one are respectively
I = 0.6638, I1 = 0.9171, I2 = 0.6352 and I3 = 0.2144. By (3) I = 0.6638 = 0.4× 0.9171 +
0.4 × 0.6352 + 0.2 × 0.2144. For this last partition I1 and I2 each contribute 40% to the
overall index I while the middle group contributes only 20%. (If one desires all partition
members to contribute equally to I, one needs p1 = 1/6 and p2 = 1/3. Then I is the
average of I1 = 0.9334, I2 = 0.7325 and I3 = 0.3254.)

Next consider the decile partition pk = k/10, k = 1, 2, 3, 4. It has I = (I1 + I2 + · · ·+



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 693

I5)/5 = (0.9619 + 0.8723 + 0.7376 + 0.5327 + 0.2144)/5. From this decomposition, one can
recover results for the coarser quintile partition obtained at the start of this example:

I = 0.4

{
0.9619 + 0.8723

2

}
+ 0.4

{
0.7376 + 0.5327

2

}
+ 0.2 {0.2144} .

1.4. More examples of decompositions of I

In general the calculation of
∫ r
0 R(p) dp requires numerical integration, but for the

lognormal distribution closed form expressions for the components Ik of I are obtainable.
The quantile function of the lognormal distribution with parameters µ, σ on the log-scale
is Qµ,σ(p) = exp{µ + zpσ}, where zp = Φ−1(p) is the p-quantile of the standard normal
distribution having cdf Φ. Therefore Rσ(p) = exp{2σzp/2}. Using elementary calculations
relegated to Appendix A, one can show∫ r

0
Rσ(p) dp = 2 exp{2σ2}Φ

{
Φ−1

(r
2

)
− 2σ

}
. (4)

It follows immediately that I = I(σ) = 1 − 2 exp{2σ2}Φ(−2σ). This QRI is monotone
increasing from 0 to 1 as the shape parameter σ increases from 0 to ∞.

We can also find closed form expressions for the ingredients in the decomposition (3)
using (4). For a K-partition Ak defined by 0 = p0 < p1 < · · · < pK−1 < pK = 1/2 we
have weights wk = 2(pk − pk−1) and Ik determined by

1− Ik(σ) =
1

wk

∫ 2pk

2pk−1

Rσ(p) dp =
2 exp{2σ2}

wk

{
Φ(zpk − 2σ)− Φ(zpk−1

− 2σ)
}
. (5)

In the top left of Figure 2 are shown the graphs of I(σ), I1(σ)/2 and I2(σ)/2 for the
quartile partition. The inequality I1(σ) rises rapidly to 1 for σ < 2. Therefore for σ > 2
nearly all the change in the index I(σ) for the lognormal is due to that in the central half.

Quartile partition inequality graphs for the Type II Pareto distribution with shape
parameter a are quite different, see the top right of Figure 2. In this case I = I(a)
is monotone decreasing in a from a high of 1 to its asymptote value of 0.7016. The
contribution I1(a)/2, inequality due to the outer quartiles is almost constant, decreasing
from a high of 0.5 to a low of 0.4615 as a increases without bound. The contribution of
inequality in the middle half of the population I2(a)/2 descends from 0.5 to a low of 0.2401
with increasing a. Thus nearly all the change in inequality comes from the central half of
the population. The asymptotic values are those belonging to the exponential distribution,
which is the limit of the Pareto(a) distributions as a→∞.

The lower plots in Figures 2 exhibit different behavior from the previous plots, with I
descending from 1 to 0 as the shape parameter increases. The contributions to inequality
of the inner and outer quartiles in each case start at 0.5 and descend slowly to 0. For the
symmetric Beta(α, α) family, the graphs (not shown) are similar in shape to those for the
Gamma(α) family, but they descend to 0 faster as α increases.



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 694

0 2 4 6 8 10

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

Lognormal

σ

I

0 2 4 6 8 10

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

Type II Pareto

α

I

0 2 4 6 8 10

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

Weibull

β

I

0 2 4 6 8 10

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

Gamma

α

I

Figure 2: Plots in solid lines of the graphs of I versus the shape parameters for four standard families. For the
quartile partition, I = (I1 + I2)/2, and the dashed lines show the respective graphs of I1/2 defined by (2) and
the dotted lines those of I2/2.

Equi-K-partitions for large K.

In practice we are usually concerned only with small K-partitions. But what happens to
the decomposition (3) if we fix F and take an equi-K-partition defined by pk = k/(2K),
for k = 0, 1, 2, . . . ,K, and let K → ∞? There are then K partition members with equal
weights wk = 2(pk−pk−1) = 1/K. The inequality in the kth subpopulation Fk contributes
wkIk = Ik/K to the sum in (3); and, for k/K → p

1− Ik =
1

wk

∫ 2pk

2pk−1

R(u) du = K

∫ k/K

(k−1)/K
R(u) du→ R(p) .

Thus for a large equi-K-partition with k/K → p, the conditional inequality Ik is approx-
imately 1−R(p).

1.5. Summary of results to follow

In the next Section 2 we explain how to find asymptotic standard errors and confi-
dence intervals for the Îks, and confirm by simulation studies the reliability of coverage



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 695

probabilities for them for a wide range of possible income distributions. Applications of
the symmetric decomposition theory and methodology to Australian income and wealth
data are in Section 3. Details for two web-based applications that we have developed for
readers to further study the IQR and to estimate the IQR for their own data are found in
Section 4. Further applications and extensions are suggested in Section 5.

2. Inference for QRI component estimates

In this section we obtain large sample confidence intervals for the Iks using methodology
of [14, Section 3.2] to find such intervals for I. The basic idea there was to choose a positive
integer J , define a grid pj = (j − 1/2)/J , j = 1, . . . , J on the unit interval, and then

estimate I =
∫ 1
0 (1−R(p)) dp by the average Î(J) = {

∑J
j=1[1− R̂(pj)}/J where R̂(pj) is an

estimate of the quantile ratio R(pj) = Q(pj/2)/Q(1−pj/2). Using standard results for the
asymptotic normality and covariance structure of sample quantiles, nominal 100(1− α)%

confidence intervals for I of the form Î(J)±z1−α/2{V̂ar(Î(J))}1/2 are obtained. The required
quantile function estimates are the continuous Type 8 estimates recommended by [9] and
the quantile density estimates are those developed by [11]. These large sample intervals are
shown to have good coverage probabilities for nearly all of the distributions listed below
in Table 1 and n =100, 200, 500, 1000 and 5000. Further the choice of grid size J = 100
is large enough to obtain this good coverage. We have developed applications with the
R software [R 7], see [14] to compute the standard errors and confidence intervals. For
details see Section 4.

2.1. Estimating components of I

Given a symmetric K-partition with kth element Ak = [pk−1, pk) ∪ (1− pk, 1− pk−1],
we found that for wk = 2(pk − pk−1) the kth conditional inequality is determined by (2),
which is simply 1− Ik =

{∫ 2pk
2pk−1

R(u) du
}
/wk.

We can estimate Ik by first estimating the integral of R over the interval [2pk−1, 2pk].
To this end define a grid on it by pkj = 2pk−1 + wk(j − 1/2)/J for j = 1, . . . , J. (There
does not seem to be any benefit in allowing J to depend on k.) Our estimate of Ik is then

Î
(J)
k =

1

wk J

J∑
j=1

[1− R̂(pkj )] , (6)

where R̂(pkj ) is an estimate of the quantile ratio R(pkj ) = Q(pkj/2)/Q(1 − pkj/2). The
nominal 100(1− α)% confidence interval for Ik is then

Î
(J)
k ± z1−α/2{V̂ar(Î

(J)
k )}1/2 . (7)

Details of the formula for the asymptotic variance Var(Î
(J)
k ) of Î

(J)
k and how to estimate

it are essentially the same as those for Î(J) found in [14, Appendix A] and so are omitted
here.



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 696

Table 1: Simulated coverage probabilities for the quartile partition with sample sizes n = 100, 500 and 1000 for
various choices of standard income distributions F , also studied in [14, Tables 1 and 3]. A total of 1000 trials
were conducted.

n = 100 n = 500 n = 1000
# F I1 I2 I1 I2 I1 I2
1 Lognormal 0.966 0.960 0.970 0.968 0.955 0.957
2 Beta(0.1,0.1) 1.000 0.918 1.000 0.976 1.000 0.965
3 Beta(0.5,0.5) 0.930 0.926 0.940 0.941 0.952 0.927
4 Beta(1,1) 0.943 0.935 0.953 0.946 0.957 0.952
5 Beta(10,10) 0.969 0.972 0.976 0.960 0.972 0.961
6 χ2

1 0.957 0.973 0.961 0.964 0.946 0.940
7 χ2

4 0.966 0.959 0.963 0.955 0.960 0.958
8 χ2

25 0.974 0.965 0.973 0.955 0.959 0.945
9 Pareto(1) 0.966 0.989 0.966 0.963 0.959 0.962
10 Pareto(2) 0.965 0.976 0.962 0.960 0.954 0.958
11 Pareto(100) 0.952 0.965 0.955 0.969 0.956 0.948
12 Exp(1) 0.956 0.963 0.945 0.960 0.949 0.962
13 Weibull(0.5) 0.968 0.993 0.967 0.969 0.962 0.969
14 Weibull(2) 0.959 0.961 0.966 0.953 0.960 0.957
15 Weibull(10) 0.980 0.983 0.985 0.955 0.987 0.960
16 LN-Frechet 0.983 0.970 0.973 0.968 0.968 0.959

Simulated coverage probabilities for the interval in (7) for varying sample sizes n from
16 different possible income distributions F are listed in Table 1 for the quartile partition.
A total of 1000 trials were conducted for each choice of n and distribution and the nominal
coverage was set to 0.95. With the exception of the extreme U-shaped Beta(0.1, 0.1)
distribution, excellent coverages are achieved even for the smaller sample size n = 100 for
estimation of both I1 and I2. Coverage tends to be slightly conservative and is typically
closer to nominal for larger sample sizes. The simulations were repeated with J = 50 and
J = 200 and similar results were obtained.

For the quintile partition, simulated coverage probabilities are presented in Table 2.
These simulations and others not shown here, convince us that the interval estimators of
the Iks are reliable for a wide number of income distributions F .

An estimate of I itself can be obtained by applying the decomposition (9) to the Îks,
but its standard error is not readily obtainable from the standard errors of the compo-
nents, because although vector Î = (Î1, . . . , ÎK) is asymptotically multivariate normal, its
limiting covariance matrix limn→∞ n

1/2Cov(Î) is not diagonal. Exact expressions for Î
and subcomponents Îk based on ordered data are presented in Appendix B.



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 697

Table 2: Simulated coverage probabilities for the quintile partition with sample sizes n = 100, 500 and 1000. A
total of 1000 trials were conducted.

n = 100 n = 500 n = 1000
# F I1 I2 I3 I1 I2 I3 I1 I2 I3
1 Lognormal 0.976 0.965 0.964 0.966 0.958 0.959 0.968 0.969 0.943
2 Beta(0.1,0.1) 0.999 0.991 0.799 1.000 1.000 0.882 1.000 0.999 0.899
3 Beta(0.5,0.5) 0.926 0.934 0.935 0.937 0.937 0.941 0.944 0.933 0.925
4 Beta(1,1) 0.946 0.947 0.951 0.957 0.954 0.953 0.948 0.938 0.947
5 Beta(10,10) 0.955 0.968 0.961 0.984 0.960 0.948 0.975 0.970 0.961
6 χ2

1 0.955 0.967 0.979 0.947 0.944 0.958 0.962 0.963 0.955
7 χ2

4 0.973 0.962 0.959 0.953 0.945 0.943 0.971 0.953 0.952
8 χ2

25 0.957 0.974 0.969 0.978 0.954 0.955 0.971 0.960 0.959
9 Pareto(1) 0.968 0.982 0.987 0.967 0.962 0.959 0.962 0.958 0.966
10 Pareto(2) 0.949 0.966 0.972 0.950 0.948 0.955 0.962 0.968 0.955
11 Pareto(100) 0.955 0.968 0.958 0.958 0.962 0.949 0.948 0.954 0.950
12 Exp(1) 0.944 0.965 0.971 0.960 0.952 0.947 0.942 0.948 0.952
13 Weibull(0.5) 0.969 0.989 0.996 0.964 0.966 0.973 0.956 0.969 0.958
14 Weibull(2) 0.966 0.960 0.962 0.971 0.940 0.940 0.956 0.958 0.954
15 Weibull(10) 0.973 0.990 0.961 0.985 0.971 0.954 0.979 0.957 0.955
16 LN-Frechet 0.985 0.979 0.980 0.977 0.974 0.967 0.971 0.964 0.953

3. Applications of QRI decompositions

3.1. Example 1: Australian disposable weekly income

Measuring household and personal weekly income is a complicated task carried out by
governmental departments, including the Australian Bureau of Statistics (ABS), whose
reports are available at [1]. The gross household income per week is published, but house-
holds differ so much in size that the equivalized disposal weekly income (DWI) is also
found. The ABS defines the DWI as ‘... the amount of disposable cash income that
a single person household would require to maintain the same standard of living as the
household in question, regardless of the size or composition of the latter.’

Table 3 provides ABS grouped data on DWI for selected years, based on representative
samples of households converted to 2014 dollar values. Figure 3 is our depiction of these
data with kernel density plots. They are multi-modal distributions and reveal a clear shift
to the right of DWI values over the period 2004–2014 (the solid line with the highest mode
is for 2004). We are interested in tracking inequality over this period.

A density plot of the DWI data constructed from Table 3 and a plot of some of the
quantiles over time, where the quantiles are relative to what the corresponding quantile
was in 2004 (e.g. the 0.9 quantiles for 2006, 2008 etc. are each divided by the 2004 0.9
quantile) for clearer interpretation, are provided in Figure 3. The latter plot suggests
increasing disparity in DWI up until 2010 between the wealthier and poorer sections of
the population.

To understand how we constructed the plots in Figure 3, we need to examine Table 3 in
some detail. For a partition of 29 classes of dollar incomes listed in the left-most column,
and selected financial years, each table entry gives the estimated number of DWIs (in



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 698

Table 3: Equivalized disposable weekly income (DWI) in Australian dollars, adjusted for
inflation to 2013-2014 dollars, for selected financial years. The tabled entries represent
thousands of persons. Source: [1, Table 1.3], downloaded 27 July, 2017.

2003–2004 2005–2006 2008–2010 2011–2012 2013–2014
[0, 0] 1 87.3 73.7 89.0 87.4 86.4
[1, 49] 94.1 90.1 95.8 81.6 95.3
[50, 99] 49.7 63.1 61.3 85.3 78.9
[100, 149] 94.0 66.2 84.0 92.3 47.6
[150, 199] 129.9 108.6 125.1 107.3 134.9
[200, 249] 273.7 219.6 164.7 185.6 151.1
[250, 299] 657.6 443.7 351.5 335.0 373.4
[300, 349] 1385.5 1152.0 596.3 373.9 397.9
[350, 399] 1301.8 1187.5 1195.8 913.3 636.7
[400, 449] 1231.7 1111.8 1172.4 1184.1 1135.2
[450, 499] 1093.7 1052.3 933.4 1044.7 1175.2
[500, 549] 1043.0 1097.4 991.3 1019.7 1171.7
[550, 599] 1092.2 1057.0 1009.7 980.8 1093.0
[600, 649] 1087.5 1016.2 1046.4 926.3 956.6
[650, 699] 1083.5 1066.9 987.0 1021.9 972.7
[700, 749] 1092.8 1023.3 996.9 999.2 938.9
[750, 799] 959.9 834.1 1037.1 1038.1 1009.6
[800, 849] 878.1 940.4 829.3 989.4 1013.4
[850, 899] 718.3 828.5 806.5 959.7 1099.5
[900, 949] 612.2 746.6 793.0 896.4 826.2
[950, 999] 631.8 731.9 757.8 714.9 885.6
[1000, 1049] 506.8 547.5 630.3 690.1 692.6
[1050, 1099] 492.3 515.3 730.8 803.1 695.8
[1100, 1199] 750.3 933.9 1118.5 1245.7 1379.5
[1200, 1299] 529.4 674.2 906.1 985.3 1027.2
[1300, 1499] 706.4 863.9 1400.8 1499.2 1447.8
[1500, 1699] 387.9 469.6 889.7 995.4 938.5
[1700, 1999] 263.2 427.0 682.8 850.2 862.3
[2000,+∞) 2 371.9 588.4 1106.3 1082.9 1355.6
Total 19,606.5 19,930.7 21,589.6 22,188.8 22,679.1
1 Some DWIs are negative, but these values have been rounded up to zero.
2 An upper bound on DWIs greater than $2000 is not reported.

thousands). All amounts have been converted to 2013-2014 dollars using the Consumer
Price Index. The financial year in Australia ranges from 1 July of one year to 30 June of
the next; for simplicity we hereafter write ‘2004’ for ‘2003-2004’, and similarly for other
years.. For the 2004 data the first entry tells us that there were (approximately) 87,300
zero DWIs, the second entry that there were 94,100 DWIs between 1 and 49 dollars, and
for the last class 371,900 DWIs of 2000 dollars or more. The last class has lower bound
xq = 2000, where q = 1− 379.1/19606.5 = 0.981.

Lacking the individual data, we created a population to take samples from, which
allowed for the ambiguity of missing data in the right hand tail, which is often modelled
by the Pareto distribution. We did this by generating 873 zeros, 941 random uniform



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 699

0 500 1000 1500 2000 2500

Distribution of DWI when a =  4

Disposable weekly income

D
en

si
ty

2004
2006
2010
2012
2014

2004 2006 2008 2010 2012 2014

1.
00

1.
10

1.
20

1.
30

Relative quantiles over time

Year
Q

ua
nt

ile
s 

ov
er

 ti
m

e 
re

la
tiv

e 
to

 2
00

4 
qu

an
til

e

0.9 quantile
0.8 quantile
0.2 quantile
0.1 quantile

Figure 3: Kernel density estimates and relative quantile plots over time for the five populations generated in
Section 3.1, truncated to [0,2500]. The small positive mass on 0 is smoothed out by these density estimates.

values from 1 to 49, 497 uniform values from 55 to 99, and so on. For the last category we
generated 3,719 random Pareto(a, λ) values as follows: first, for a given shape parameter
a > 0, we computed the scale parameter λ = xq/{(1−q)−1/a−1} = 2000/{(1−0.981)−1/a−
1}; secondly, we generated 3,719 uniform values ui from [0.981,1]; and thirdly, we applied
the quantile function to these values Qa,λ(ui) = λ{(1 − ui)−1/a − 1}. As we will see, the
choice of a does not affect the QRI much, although it would significantly affect most
inequality measures.

Standard kernel density plots for the five populations were generated in this way using
the default density command on R [7], one for each of the selected years, and these are
shown in Figure 3 when the choice of Pareto shape parameter a = 4. They are truncated
at income 2500, although their maxima can be much larger, as shown in Table 4. It is
evident that the distributions are moving to the right. In fact the distribution of 1996
DWI (not shown) is unimodal with mode near 500, while these populations have two or
more modes. The 2014 density plot is very similar to Graph 2 of the section ‘Household
Income and Wealth Distribution’ [1].

The ABS also provides the relative standard error (RSE), defined as standard error
of estimate divided by the estimate, for most of their results. For example, in the same
source [1, Table 1.3] from which our Table 3 was excerpted, they give the RSE = 11.8%
for the first table entry 87.3 (thousands of persons with zero DWI in 2004); that is,
the standard error is nearly 10.3 thousand. For later years where the sample number of
households was much higher, such as in 2014, the RSE’s are in the range 5–8 %. The main
point is that even with the best of survey methods, the resulting summary data listed in
the ABS Table 1.3 only approximately describes the exact populations of DWI. Our five
populations, truncated at $2500 to fit in Figure 3, are also approximations.



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 700

Table 4: Percentiles for five distributions of DWI depicted in Figure 3, and values of Î rounded to two places.
The minimum value for each distribution was 0.

P05 P10 P20 P25 P50 P75 P80 P90 P95 max Î

2004 269 320 394 433 658 928 1008 1255 1521 33520 0.51
2006 292 340 426 472 707 1003 1096 1383 1714 39441 0.51
2010 309 374 470 526 793 1163 1273 1615 2024 39666 0.52
2012 317 396 497 552 831 1188 1298 1642 1989 34226 0.52
2014 321 411 509 558 843 1196 1309 1688 2179 32970 0.52

The percentiles of our five populations are listed in Table 4. They are in good agreement
with the percentile estimates of [1, Table 1.1]. For example, for the year 2004 they obtain
P10=324, P20=395, P50=657, P80=1,008 and P90=1,255. And for 2014 they obtain
P10=415, P20=511, P50=844, P80=1308 and P90=1688.

The maximum values in the second-last column of Table 4 could have been quite
different, (and much larger for a near 1). However, extremely large incomes (outliers)
do not affect the quantile ratio index estimator Î very much. Note that Î, also listed in
Table 4, does not appear to be changing over the years 2004–2014, indicating a stable
level of inequality for this period. Changing the Pareto tail shape a from 4 to 1 greatly
increases the maximum values in this table, but has no effect on the other quantiles and
little effect on Î . This is because a small percentage of extreme outliers do not affect the
index, because their magnitudes contribute little after transformation to the [0,1] scale
(e.g, note that the ratio of a small quantile and an a very large quantile due to extreme
outliers is approximately zero, and increasing the large quantile only makes this closer to
zero). Robustness properties of the QRI are discussed in more detail in [14].

Quartile partition estimates for DWI.

Next consider the ‘quartile partition’ with members A1 = [0, 0.25] ∪ [0.75, 1] and A2 =
[0.25, 0.75]. Incomes in A2 can be considered as belonging to the ‘middle class’. Here Îk is
defined by (6) for k = 1, 2. These estimates, as well as those for I were based on samples
of size n = 10, 000 for each of the five populations generated with a = 4 and are listed in
Table 5.

Table 5: Estimates of I and Ik for the quartile partition of the five distributions of Figure 4, based on samples

of size n = 10, 000. In parentheses are the values of
√
n ŜE[Îk].

2004 2006 2010 2012 2014

Î1 0.721(0.27) 0.722(0.27) 0.742(0.27) 0.733(0.28) 0.736(0.28)

Î2 0.285(0.34) 0.289(0.33) 0.297(0.34) 0.285(0.34) 0.287(0.34)

Î 0.503(0.27) 0.506(0.26) 0.520(0.27) 0.509(0.27) 0.512(0.27)

The bottom row shows the estimates of I are only increasing slightly over time. We
have already commented upon this stability earlier for the right-most column of Table 4.
However, a two-sided level 0.05 test for a difference between the 2004 and 2014 values of
Î is just significant. We can now find the source of this increasing inequality.



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 701

The middle row of Table 5 shows that inequality in the middle class, measured by
Î2, has not significantly changed over this period. But the inequality for partition A1,
containing the lower quartile and upper quartiles, has increased from 0.721 to 0.736 which
is significant at the 0.05 level . Further, inequality for A1 peaks at 0.742 in 2010, which was
suggested in the righthand plot of Figure 3 where disparity grew at an increasing rate up
until this time. It is interesting that the standard errors of Î1 and Î are roughly the same,
while those of Î2 for the middle class are about 1/4 larger. Similar results were obtained
for samples based on n = 4000 observations, but for n = 1000 changes in inequality over
this time period were not quite statistically significant at the 0.05 level.

3.2. Example 2: Australian wealth data

In its explanatory notes of Manual 6523.0, the ABS defines household wealth by ‘Net
worth, often is the value of a household’s assets less the value of its liabilities.’ and then
goes on to explain what it means by assets and liabilities.

Table 6 is obtained from [1, Table 2.3]. Because of the large numbers involved, we
have written the dollar classes in thousands of dollars. For example, the second entry in
the column labelled 2004 is 1098.9, which means that there were an estimated 1,098,900
households in that year whose net wealth was between 0 and $50,000 in 2004.

0 500 1000 1500 2000

Distribution of Net Household Wealth

Net Household Wealth ('000)

D
en

si
ty

2004
2006
2010
2012
2014

2004 2006 2008 2010 2012 2014

1.
00

1.
10

1.
20

1.
30

Relative quantiles over time

Year

Q
ua

nt
ile

s 
ov

er
 ti

m
e 

re
la

tiv
e 

to
 2

00
4 

qu
an

til
e

0.9 quantile
0.8 quantile
0.7 quantile
0.6 quantile
median
0.4 quantile
0.3 quantile
0.2 quantile
0.1 quantile

Figure 4: Kernel density estimates and relative quantile plots over time for the five populations of wealth data.
For the densities, the solid line with the single, highest mode is the graph for 2004; it crosses the graph for
2014, also in solid line, which depicts a much more dispersed population of incomes that is bimodal.

We used the methods of Example 1 to generate a population that reflects the infor-
mation in Table 6, modulo the shape parameter for a Pareto tail for the last unbounded
dollar class. Density plots for the five populations and plots of the relative quantiles over
time are shown in Figure 4. The density graph for 2004 is unimodal, while for subsequent



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 702

Table 6: Net Household Wealth (NHW) in thousands of Australian dollars, adjusted for
inflation to 2013-2014 dollars, for all available financial years. The tabled entries represent
thousands of households. Source: [1, Table 2.3], downloaded 27 July, 2017.

2003–2004 2005–2006 2009–2010 2011–2012 2013–2014
(−∞, 0) 1 56.6 75.6 77.3 113.7 93.8
[0, 49] 1098.9 1044.6 1058.2 1075.2 1052.7
[50, 99] 547.0 583.4 577.4 617.1 667.3
[100, 149] 364.3 374.1 408.5 441.9 433.3
[150, 199] 365.4 308.0 311.5 368.5 334.1
[200, 249] 372.7 305.4 294.6 337.7 360.1
[250, 299] 393.3 354.3 309.1 319.8 363.1
[300, 349] 372.8 356.5 348.6 306.4 329.5
[350, 399] 397.5 397.0 331.0 335.9 343.3
[400, 449] 353.5 351.6 332.1 360.7 329.3
[450, 499] 335.5 361.8 348.2 333.0 342.4
[500, 599] 574.9 601.4 621.8 554.7 570.4
[600, 699] 402.6 492.9 508.6 499.4 451.0
[700, 799] 365.9 400.8 425.8 410.4 420.2
[800, 899] 295.6 252.9 344.0 377.3 325.8
[900, 999] 211.1 220.5 283.1 267.5 298.0
[1000, 1099] 179.5 189.5 235.5 234.0 261.9
[1100, 1199] 147.3 161.9 185.0 203.9 202.4
[1200, 1399] 233.7 241.9 314.7 323.6 346.4
[1400, 1599] 138.3 196.4 213.1 228.6 242.9
[1600, 1799] 92.8 122.6 161.8 158.6 171.5
[1800, 1999] 69.9 83.7 118.9 153.3 149.0
[2000, 2199] 64.9 77.4 75.6 94.7 100.8
[2200, 2399] 50.9 53.8 72.3 68.0 78.8
[2400, 2599] 30.1 44.4 62.3 55.6 62.0
[2600, 2999] 55.8 73.6 98.4 87.6 91.4
[3000, 3999] 83.6 90.1 111.1 126.3 149.2
[4000, 4999] 19.4 41.2 61.8 68.7 70.5
[5000, 6999] 34.8 36.3 51.7 55.3 63.9
[7000, 9999] 11.7 14.5 25.4 33.1 31.8
[10000,+∞) 2 15.5 18.3 31.4 19.9 29.5
Total 7,735.8 7,926.4 8,398.8 8,630.4 8,766.3
1 The unknown NHWs less than 0 will be assigned 0 in our analysis.
2 An upper bound on NHWs greater than $10,000,000.00 is not reported.

years a clear shift to bimodality is apparent. The decrease in relative quantiles for the the
less wealthy half of the population between 2010 and 2012 is not seen for the wealthier
half.

Empirical percentiles for these data in Table 7 reveal that while the lower percentiles
are not changing over the decade, the median P50 and larger percentiles appear to be
steadily increasing. The estimated inequality of wealth Î appears to be increasing only
slightly over the decade. Nevertheless Î values near 0.7 are certainly much higher than
those for disposable income, which was near 0.5, see the last column of Table 5. We now



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 703

Table 7: Percentiles for five distributions of NHW depicted in Figure 4, and values of Î. The minimum value
for each distribution was 0.

P05 P10 P20 P25 P50 P75 P80 P90 P95 max Î

2004 15 32 86 132 388 747 867 1319 1984 43203 0.71
2006 15 34 89 137 424 787 942 1477 2182 42635 0.71
2010 16 35 97 147 474 909 1070 1670 2567 62912 0.72
2012 15 34 93 140 458 913 1085 1697 2526 40650 0.73
2014 16 36 95 143 463 961 1129 1770 2717 41328 0.73

examine the possible change in inequality over the Australian population of households
and certain sub-populations of these wealth data.

3.2.1. Quartile partition estimates for NHW.

Next consider the quartile partition with members A1 = [0, 0.25] ∪ [0.75, 1] and A2 =
[0.25, 0.75]. Incomes in the population image of A2 can be considered as belonging to the
‘middle class’. Here Îk is defined by (6) for k = 1, 2. These estimates are found for each of
the five populations generated (as was done for the DWI data) starting with the grouped
data listed in Table 6. They are based on 10,000 observations from each of the respective
populations.

Table 8: Estimates of Ik for the quartile partition of the five distributions of Figure 4 based on 10,000 obser-

vations selected at random from each of them. In parentheses are the values of
√
n ŜE[Îk].

2004 2006 2010 2012 2014

Î1 0.949(0.16) 0.948(0.16) 0.952(0.16) 0.956(0.15) 0.955(0.14)

Î2 0.480(0.54) 0.466(0.55) 0.473(0.56) 0.500(0.53) 0.498(0.53)

Î 0.714(0.33) 0.707(0.33) 0.712(0.34) 0.726(0.32) 0.726(0.31)

Estimates for the quartile partition are provided in Table 8. A level 0.05 test for a
difference between the 2004 and 2014 values of Î, namely |0.726 − 0.714| = 0.012 would
reject for n = 10, 000 because then the standard error of the difference between them is
SE =

√
(0.332 + 0.312)/n = 0.004. A similar test for significant change in inequality for

the middle class over the same range of time is just significant at the 0.05 level, because the
difference is 0.018 and the standard error of the difference is 0.008. The standard errors
for the estimates of inequality in the outer quartiles Î1 are much smaller, so while the
difference between the 2014 and 2004 results is only 0.955-0.949 =0.006 but its standard
error is also much smaller at 0.002, leading to a statistically significant result. Thus most
of the change in wealth inequality over this period is due to the change in the lower and
upper quartiles. Note that such tests are correlated.

Decile partition estimates for NHW.

Next we look at a finer partition, the decile partition, to further pinpoint where the wealth
inequality is changing most.



L. Prendergast, R. Staudte / Eur. J. Pure Appl. Math, 12 (3) (2019), 689-708 704

Table 9: Estimates of Ik for the decile partitioning of the five distributions deicted in Figure 4. As in Table 8,
estimates are based on 10,000 observations selected at random from each of them, and values in parentheses

are 100× ŜE[Îk].

2004 2006 2010 2012 2014

Î1 0.991(0.04) 0.991(0.04) 0.992(0.03) 0.993(0.03) 0.993(0.03)

Î2 0.944(0.19) 0.945(0.19) 0.949(0.18) 0.949(0.18) 0.951(0.16)

Î3 0.812(0.51) 0.811(0.52) 0.827(0.48) 0.832(0.44) 0.842(0.42)

Î4 0.565(0.69) 0.557(0.69) 0.584(0.68) 0.606(0.69) 0.618(0.66)

Î5 0.223(0.51) 0.218(0.49) 0.232(0.52) 0.235(0.54) 0.229(0.55)

Î 0.707(0.33) 0.705(0.33) 0.717(0.32) 0.723(0.32) 0.729(0.31)

Estimates and their standard errors are given in Table 9. Note that the standard errors
of the estimates can vary from 0.0003 to 0.0069.

Using (3) the results in the second column show that for 2004 the overall inequality
can be broken down into Î = 0.707 = 0.198 + 0.189 + 0.162 + 0.113 + 0.045, so the first
three partition members (outer six deciles) contribute 0.549/0.707 or almost 80% to the
overall QRI and the fifth partition (central two deciles) only 0.045/0.707, or 6%.

Comparing the estimates of I5 in 2004 and 2014 for the central partition shows a less
than one standard error of increase. However, for every other partition member the QRI
has increased by more than two standard errors over this time period. We can conclude
that wealth inequality is becoming more unequal except within the central 1/5 of the
population.

4. Shiny applications

For convenience, we have created two Shiny [3] applications that some readers may
find useful. These applications can be found at

https://lukeprendergast.shinyapps.io/Decomp/

and

https://lukeprendergast.shinyapps.io/QRIestimation/.

The first calculates the QRI for several distributions considered within this manuscript.
This application also calculates the quartile and quintile decompositions for the QRI if
requested. The second allows the user to upload a csv data file for which to estimate the
QRI and its quartile and quintile decompositions. The standard errors and large sample
confidence interval estimators for the QRI and its decompositions are included in the
tabulated output. We will continue to improve these applications and are grateful for any
feedback.

5. Conclusion

We have learned that for Australian data, inequality of DWI as measured by the QRI
is almost steady at about Î = 0.5 over the years 2004 to 2014, although for a large enough



REFERENCES 705

sample it has increased by a statistically significant amount. Samples of size 1000 would
not detect such an increase. Further, by examining the QRI estimates for the quartile
partition, we found that all the change in inequality of incomes over the time period 2004
to 2014 can be attributed to the outer classes, with incomes inequality in the middle class
remaining stagnant. The inequality in wealth NHW for Australian households over this
same period was much higher Î ≈ 0.7 and very significantly increases from 2004 to 2014.
Moreover, hardly any of this increase is due to the middle two deciles.

These examples illustrate the simple utility of measuring inequality with the QRI. Not
only can we find reliable confidence intervals for the QRI of a population using relatively
small samples in the hundreds, we can also find them for symmetric partitions of quantiles,
and use them to identify those which contribute most to the population QRI and how
much them contribute. Finally we can use these results to detect changes over time.
Given income or wealth data from various countries, it would be straightforward, using
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symmetric sub-populations. One could also study income data for those in the top 10%
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Testing for changes in the QRI over time and for several components of the decom-
position means that it would be wise to take the usual care when dealing with multiple
testing. While the Bonferroni correction is often seen as too conservative, methods for
controlling the false discovery rate for dependent tests may be appealing.

Given that the QRI is a simple average of quantile inequality measures of the form
R(p) = 1−Q(p/2)/Q(1− p/2), one for each 0 < p < 1, it should be possible to introduce
weights or otherwise extend it to other partitions where meaningful decompositions arise.

Acknowledgements

The authors would like to thank the three referees and the Editor for taking their time
to review the paper and for the suggestions made.

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APPENDIX 707

Appendix

A. Derivation of the indefinite integral (4)

Starting with Rσ(p) = exp{2σzp/2} and making the change of variable y = 2σzp/2∫ r

0
Rσ(p) dp =

1

σ

∫ 2σzr/2

−∞
ey ϕ

( y
2σ

)
dy ,

where ϕ(z) = e−z
2/2/
√

2π is the standard normal density. Completing the square within
the exponential of the integrand exp{y − (y/2σ)2/2} leads to

8σ2y − y2

8σ2
=

16σ4 − (y − 4σ2)2

8σ2
, so∫ r

0
Rσ(p) dp = 2 exp{2σ2}

∫ 2σzr/2

−∞

1

2σ
ϕ

(
y − 4σ2

2σ

)
dy = 2 exp{2σ2}Φ(Φ−1(r/2)− 2σ) .

B. Exact decomposition formula for Î

Taking the weighted average of the Î
(J)
k s defined in (6) does not guarantee that it will

be exactly equal to the estimator of I found by Î(J) = {
∑J

j=1[1− R̂(pj)}/J . However, [14]
showed the estimates are stable for moderate to large choices of J so that the resulting

weighted average of the Î
(J)
k s is expected to be very close to Î(J). However, if n0 = np0 = 0,

n1 = np1, n2 = np2 . . . , nK−1 = npK−1 and nK = n/2 are all distinct integers, then it is
possible to define estimates of the Îks such that their weighted average is equal to a simple
estimator of I.

Given ordered incomes 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn, where n ≥ 2 and the frequency of 0’s
is less than 0.5, let k = bn/2c. It is shown in [14, Equation 3] that an exact estimate of I
is given by

In = I(Fn) =
2

n

k∑
j=1

(
1− xj

xn−j+1

)
. (8)

Given a symmetric K-partition {A1, . . . , AK} of the unit interval determined by 0 =
p0 < p1 < · · · < pK = 1/2, we want to decompose Î into a weighted average of individual
estimates Îk of Ik. To this end assume n0, n2, . . . as above and assume that they are distinct
integers. Define the disjoint sets Bk = {nk−1 + 1, nk−1 + 2, . . . , nk}, for k = 1, . . . ,K. The
length of Bk is mk = nk − nk−1 and

∑K
k=1mk = nK = n/2. We estimate Ik by

Îk =
1

mk

∑
j∈Bk

(
1− xj

xn−j+1

)
.

Then a simple estimator of Î based on the order statistics can be written

Î =
2

n

n/2∑
j=1

(
1− xj

xn−j+1

)



APPENDIX 708

=
2

n

K∑
k=1

∑
j∈Bk

(
1− xj

xn−j+1

)

=
2

n

K∑
k=1

mk Îk , (9)

where mk = nk − nk−1. Note that the sum of the weights
∑

k 2mk/n = 1.