The Bonferroni curve and Generalised Entropy
Inequality Measures
Effat Dalir
Mehdi Yaghoobi Awal Riyabi
Ali reza Hosseinzadeh
Department of Statistics
Department of Statistics
Department of Statistics
Islamic Azad University, Mashhad Branch
Islamic Azad University, Gonabad Branch
Islamic Azad University, Gonabad Branch
Mashhad, Iran
Gonabad, Iran
Gonabad, Iran
e_dalir_e@yahoo.com
m_yaghoobiawal@yahoo.com
hoseinzade1@yahoo.com
Abstract-Lorenz curve and the Generalised Entropy
inequality measures are popular tools for the analysis of
income inequality which are commonly used in
Thus we have:
𝑏
𝑞𝑖 =
∑𝑘𝑖 =1 𝑥𝑘(𝑖)
𝑖
𝑗
𝑝𝑖 =
,
∑𝑘=1 𝑥𝑘
𝑏𝑖
𝑗
empirical studies in parallel to each other.Another
Generalised Intropy measures size have the following
income inequality analysis tools, is the Bonferroni
general form:
Curve that was introduced by Bonferroni (1930).
𝐺𝐸(𝛼) =
This paper first introduces the Generalised Entropy
1
1
𝑞
𝛼
[ ∑𝑛𝑖=1 ( 𝑖 ) − 1] (1)
𝛼 2 −𝛼 𝑛
𝑝𝑖
inequality measures and explains their relationship with
Where 𝛼 is a nonnegative sensitivity parameter.
Lorenz curves. Then by introducing the Bonferroni
Taking the limit of equation (1), as 𝛼 → 0,1 gives
curve and expressing its relationship with Lorenz curve,
Theil̕s L and T inequality measures respectively
we derive the Generalised Intropy index by using the
Bonferroni curve. Finally, we calculate these indices for
a number of Bonferroni curve function forms.
Key words: Lorenz curve -Bonferroni Curve -Generalized
Intropy measures- Theil index.
𝑛
𝑝𝑖
𝐿 = ∑ 𝑝𝑖 ln ( )
𝑞𝑖
(2)
𝑖=1
𝑛
𝑞𝑖
𝑇 = ∑ 𝑞𝑖 ln ( )
𝑝𝑖
(3)
𝑖=1
Half the square of the coefficient of variation is as the
Ι. Introduction
result of 𝛼 = 2, and the Atkinson measure is
calculated as follows:
𝑛
Consider a population which includes j individual
and all has nonnegative income.Also, suppose to
1
1
𝑞𝑖
𝐴 = 1 − [ ∑( )1−𝜀 ]1−𝜀
𝑛
𝑝𝑖
(4)
𝑖=1
show k- th individual income with 𝑥𝑘 . If the
Where 𝜀 is an inequality aversion parameter and lies
population is formed arbitrarily, as have n groups,
between zero and infinity.
then i-th group that consisting of (𝑏𝑖 )individual and is
a share of income(𝑞𝑖 ) and a population share(𝑝𝑖 ).
II. LORENZ CURVES AND INCOME SHARES
Lorenz curves are one of the important indicators of
how wealth is distributed. These curves relate the
cumulative percentage of income with the cumulative
population percent and the first time was proposed by
Max Lorenz (1905).
function and the partial mean value,for 𝑋 ≤ 𝑥
Definition 1.2. Suppose 𝑋 is a nonnegative random
,respectively.By putting 𝐹(𝑥) = 𝑝, Bonferroni curve
variable with density function 𝑓 (𝑥) and cumulative
parametric form is as follows:
distribution function 𝐹(𝑥). Then we show the related
Lorenz curve by 𝐿(𝑝)and define it as follows:
𝐹1 [𝐹 −1 (𝑝)]
1 𝑃 −1
=
∫ 𝐹 (𝑡)𝑑𝑡 , 0 ≤ 𝑃 ≤ 1
𝑝
𝑝𝜇 0
𝐵(𝑝) =
(11)
0
As 𝑝 → 0, 𝐵(𝑝)Takes the form , Hence the
0
𝐿(𝑝) =
𝐹−1 (𝑝)
1
∫
𝐸(𝑋) 0
𝑥𝑓(𝑥)𝑑𝑥 , 𝑝 ∈ [0,1] (5)
Bonferroni curve does not always start from the
origin of coordinates, since
Rohde (2008) using the definition of definite
𝜕𝐵
𝜕𝑝
> 0,the graph of 𝐵(𝑝)
is strictly increasing but nothing can be said about the
integrals with total Riemann showed that:
sign of its second derivative.Hence ,Bonferroni Curve
𝑛
1
1 𝑞𝑖
𝐿 = lim − ∑ ln( ) = − ∫ 𝑙𝑛𝐿′ (𝑝)𝑑𝑝
𝑛→∞
𝑛 𝑝𝑖
0
(6)
𝑖=1
could be convex in some parts and concave in some
others(Olemedo,2009). Comparing the relations (5)
𝑛
1 𝑞𝑖
𝑞𝑖
𝑇 = lim ∑ ( ) ln( )
𝑛→∞
𝑛 𝑝𝑖
𝑝𝑖
and (11) we have:
𝐵: [0,1] → [0,1] → 𝐵(𝑝)
𝑖=1
1
= ∫ 𝐿′ (𝑝) ln(𝐿′ (𝑝)) 𝑑𝑝
𝐿(𝑝)
={ 𝑝
0
(7)
0
𝑛
1 2
1
1 𝑞𝑖
𝐶𝑉 = lim ∑ (( )2 − 1)
𝑛→∞ 2
2
𝑛 𝑝𝑖
0<𝑝<1
(12)
𝑝=0
For some Bonferroni curve properties and its
𝑖=1
1 1
= ∫ (𝐿′ (𝑝)2 − 1)𝑑𝑝
2 0
relationship with Lorenz curves we can refer to
(8)
Pundir (2005).
1
𝑛
1 𝑞𝑖 1−𝜀 1−𝜀
𝐴 = lim 1 − [∑ ( ) ]
𝑛→∞
𝑛 𝑝𝑖
IV. Bonferroni curves and Generalised Intropy
measures
𝑖=1
1
= 1 − [∫ 𝐿′ (𝑝)1−𝜀 𝑑𝑝]
1
1−𝜀
According to previous sections and the Bonferroni
(9)
0
III. BONFERRONI CURVE AND ITS
RELASHIONSHIP WITH THE LORENZ CURVE
curves’ relationships and Lorenz curve with each
other we can extend equations obtained for GE
inequality indices by Lorenz curves to the Bonferroni
For continuous and nonnegative random variable𝑋,
curve.
with the cumulative distribution function 𝐹(𝑥)that is
According to the formula (12) we have:
𝐿′ (𝑝) = 𝐵(𝑝) + 𝑝𝐵′ (𝑝)
derived from at least two times, Bonferroni curve as
(13)
follows coordinates on the page is defined in the unit
Placement this formula in relations with (6) to (9), we
square.
have:
𝐵(𝐹(𝑥)) =
𝜇𝑥 𝐹1 (𝑥)
=
𝜇
𝐹(𝑥)
1
(10)
Where 𝜇, 𝜇𝑥 , and 𝐹1 (𝑥) are the first moment of
𝑋about zero,i.e. ,the first moment distribution
𝐿 = − ∫ ln( 𝐵(𝑝) + 𝑝𝐵′ (𝑝))𝑑𝑝
0
(14)
1
as follows:
𝑇 = ∫ (𝐵(𝑝) + 𝑝𝐵′ (𝑝)) ln(𝐵(𝑝)
0
𝐹(𝑥) =
+ 𝑝𝐵′ (𝑝)) 𝑑𝑝
(15)
1 2 1 1
𝐶𝑉 = ∫ ((𝐵(𝑝) + 𝑝𝐵′ (𝑝))2 − 1)𝑑𝑝
2
2 0
𝐴=1−
1
[∫0 (𝐵(𝑝)
+
1
𝑝𝐵′ (𝑝))1−𝜀 ]1−𝜀
(16)
1
, 𝑥 > 0, 𝜎 > 0,0 < 𝜀 < 1
1
𝜇𝜎 1−𝜀
( )
𝑥
With some mathematical calculations, we get:
𝑥
(17)
5. Calculation of GE inequality measures based on
Bonferroni curves
𝜇𝑥 =
1
∫0 𝑡 1−𝜀 𝑑𝑡
1
𝑥
−1
∫0 𝑡 1−𝜀
=
𝑑𝑡
1
𝜎𝑝1−𝜀
𝑥 → 𝐵(𝑝) =
2−𝜀
2−𝜀
𝑈𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠 (14) 𝑡𝑜 (17) 𝑤𝑒 ℎ𝑎𝑣𝑒:
𝐿 = 1 − 𝜀 − 𝑙𝑛(𝜎)
, 𝑇
In this section,at first we calculate Bonferroni curve
= 𝜎(−1 + 𝜀 + 𝑙𝑛𝜎.
for the Pareto distribution type 1, and then we deduce
the size of GE inequality by the relations (14) to (17).
Then for some major distributions, calculation results
in
a
tabular
summary
are
1 2
1
1
𝐶𝑉 =
𝜎2 −
2
6 − 4𝜀
2
type 1, the probability distribution function is defined
1
)
2−𝜀
,
1
1
=1−(
𝜎 1−𝜀 )1−𝜀
2
(1 − 𝜀) + 1
presented.
If the 𝑋 random variable has the Pareto distribution
(18)
For some other specific distribution, Generalised
Intropy measures are calculated based on the
Bonferroni Curves.The summary results are given in
Table 1.
Table 1: Generalised Intropy measures from Bonferroni curve for the particular multiple
TABLE 1
Function
L
Rectangular
𝑥 − (1 − 𝜃)𝜇
2𝜃𝜇
𝐵(𝑝) = (1 − 𝜃) + 𝜃𝑝
𝐹(𝑥) =
(1 − 𝜃)1−𝜃
1
(ln (
)
(1 + 𝜃)1+𝜃
2𝜃
+ 𝜃 + 1)
T
ln(𝜃 + 1) (
(𝜃 + 1)2
)
4𝜃
1 + 𝜃2
−(
)
4𝜃
(1 − 𝜃)2
−
(ln(1 − 𝜃)
4𝜃
− 1)
Power
𝐹(𝑥) = 𝑥 𝛼
𝛼 > 1,0 ≤ 𝑥 ≤ 1
𝐵(𝑝) = 𝑝𝛼
1
− (ln(𝑝) − 1)
𝛼
1
− ln ( + 1)
𝛼
ln (
1+𝛼
1
)−
𝛼
𝛼+1
1 2
𝐶𝑉
2
1
6
𝜃2
A
1
−
1
((𝜃 + 1)2−𝜀
2𝜃(2 − 𝜀)
1
− (1 − 𝜃)2−𝜀 )1−𝜀
1 (𝛼 + 1)2
(
− 1)
2 𝛼(2 + 𝛼)
(1+𝛼)1−𝜀 𝛼
1
1-(𝛼1−𝜀(1−𝜀+𝛼))1−𝜀
Uniform
𝑥−𝑎
F(x)= 𝑏−𝑎
a<x<b
2𝑎 + (𝑏 − 𝑎)𝑝
𝐵(𝑝) =
𝑎+𝑏
Pareto (Ι)
1
𝜇𝜎 1
( 𝑥 )1−𝜀
𝑥 > 0, 𝜎 > 0,0 < 𝜀
<1
𝜎𝑝1−𝜀
𝐵(𝑝) =
2−𝜀
𝐹(𝑥) =
−ln(
2𝑏
)
𝑎+𝑏
1 − 𝜀 − ln(𝜎)
2𝑏
2𝑏𝑙𝑛(𝑎 + 𝑏)
𝑎+𝑏
𝜎 (−1 + 𝜀
1
+ 𝑙𝑛𝜎.
)
2−𝜀
2𝑏2
1
−
𝑎+𝑏 2
1
1
𝜎2 −
6 − 4𝜀
2
1 − ((
1−(
2𝑏 1−𝜀 1
) )1−𝜀
𝑎+𝑏
1
1
𝜎 1−𝜀 )1−𝜀
(1 − 𝜀)2 + 1
REFERENCES
[1] C.E. Bonferroni, Elementi di statistica general, Libereria Seber,
Firenze, 1930.
[2] L.J. Imedio Olemedo, E. Barcena Martin, and E.M. Parrado
Gallardo, A wide class
of inequality measures based on the Bonferroni curve, 2009.
[3] M.O. Lorenz, Methods of measuring the concentration of
wealth, Publication of American Statistical Association, 1905.
[4] S. Pundir, S. Arora, and K. Jain, Bonferroni curve and the
related statistical inference,Statistics and Probability Letters, 75
(2005), 140-150.
[5] N. Rohd, Lorenz Curves and Generalised Entropy Inequality
Measures, Modeling
Income Distributions and Lorenz Curves, Springer, 2008.