In a recent paper Ali and Son (2007) suggested measuring the concept of "inclusive growth" via the use of what they called a "social opportunity function". The latter was assumed to depend on the average opportunities available in the population and to give greater weight to the opportunities enjoyed by the poor. On the basis of this approach Ali and Son (2007) then defined an "opportunity index" and an "opportunity curve". The present paper derives the link which exists between these concepts of "opportunity index" and "opportunity curve" and what is known in the literature as the Bonferroni index and the Bonferroni curve. It also defines what could be called a Bonferroni concentration index, a Bonferroni concentration curve, a Generalized Bonferroni curve and a Generalized Bonferroni concentration curve.Bonferroni index – concentration curve - concentration index – generalized Lorenz curve – human opportunity – opportunity index – opportunity curve

Hyun Son

Jacques Silber

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 1 
  
On the link between the Bonferroni index and the measurement of inclusive 
growth.  
  
 
 
Jacques Silber  
Bar-Ilan University 
Hyun Son  
Asian Development Bank
Abstract 
In a recent paper Ali and Son (2007) suggested measuring the concept of "inclusive growth" via the use of what they 
called a "social opportunity function". The latter was assumed to depend on the average opportunities available in the 
population and to give greater weight to the opportunities enjoyed by the poor. On the basis of this approach Ali and 
Son (2007) then defined an "opportunity index" and an "opportunity curve". The present paper derives the link which 
exists between these concepts of "opportunity index" and "opportunity curve" and what is known in the literature as 
the Bonferroni index and the Bonferroni curve. It also defines what could be called a Bonferroni concentration index, 
a Bonferroni concentration curve, a Generalized Bonferroni curve and a Generalized Bonferroni concentration curve.
Jacques Silber gratefully acknowledges the financial support of the Asian Development Bank and of the Adar Foundation of the Department of 
Economics at Bar-Ilan University. 
Citation: Jacques Silber and Hyun Son, (2010) ''On the link between the Bonferroni index and the measurement of inclusive growth. '', 
Economics Bulletin, Vol. 30 no.1 pp. 421-428. 
Submitted: Dec 23 2009.   Published: February 02, 2010. 
 
     1.  Introduction 
 
In a recent paper Ali and Son (2007) suggested measuring the concept of "inclusive 
growth" via the use of what they called a "social opportunity function". The latter was 
defined as a function of both the average opportunities available in the population and 
of  the  way  opportunities  are  shared  in  the  population.  More  precisely  the  social 
opportunity function was assumed to give greater weight to the opportunities enjoyed 
by the poor: the poorer an individual is, the greater the weight given to this individual. 
On the basis of this approach Ali and Son (2007) then defined an "opportunity index" 
and an "opportunity curve". 
The  purpose  of  the  present  paper  is  to  show  the  link  which  exists  between  the 
concepts of "opportunity index" and "opportunity curve" and what is known in the 
literature as the Bonferroni index and the Bonferroni curve. The paper is organized as 
follows. Section 2 below recalls the definition of the Bonferroni index and curve. 
Section 3 then shows how these two concepts may be extended to derive a Bonferroni 
concentration  index,  a  Bonferroni  concentration  curve,  a  Generalized  Bonferroni 
curve and a Generalized Bonferroni concentration curve. Section 4 finally shows how 
to apply these concepts to measure inequality in human opportunities and indicates 
their link with what Ali and Son (2007) called "opportunity index" and "opportunity 
curve". Section 5 concludes. 
 
2.  The Bonferroni Index 
 
This index was originally proposed by Bonferroni (1930) who derived also what is 
called the Bonferroni curve. This curve is defined as follows. Assume  nindividuals 
whose shares in total incomes are defined as { } n i s s s ,..., ,..., 1  with  n i s s s ≤ ≤ ≤ ≤ ... ... 1 , 
where n is the number of individuals. On the horizontal axis plot, like for the Lorenz 
curve, the cumulative population shares  } 1 ), / ) 1 ),...(( / ( ),..., / 2 ( ), / 1 {( n n n i n n − . On the 
vertical axis however do not plot the cumulative income shares (as in the case of the 
Lorenz  curve)  but  the  ratio  of  the  cumulative  income  shares  over  the  cumulative 
population shares. In other words plot the following values: 
  
)} 1
1
1
) / (
) ... ... (
( ),...,
) / (
) ... (
( ),...,
) / 2 (
) (
( ),
) / 1 (
{(
1 2 1 2 1 1 = =
+ + + + + + + +
n n
s s s
n i
s s s
n
s s
n
s n i i  
 
Note that if  i x  is the income of individual i, with  ) / ( x n x s i i = ,  x being the average 
income in the total population, the ratio  )
) / (
) ... (
(
2 1
n i
s s s i + + +
 may be also expressed as 
)
) / ) ... ((
(
2 1
x
i x x x i + + +
, that is, as the ratio of a conditional mean (the mean income 
of the first iindividuals, ranked by increasing income) over the mean  x in the whole 
population. 
The Bonferroni index is then defined as the area lying between the Bonferroni curve 
and the horizontal line at height 1 (see Figure 1). The Bonferroni index  B I  is hence 
defined as   2
))]
) / (
) ...
( 1 )( / 1 [( ))]
) / ) 1 ((
) ...
( 1 )( / 1 [(
... ))]
) / (
) ... (
( 1 )( / 1 ...[( ))]
) / 2 (
) (
( 1 )( / 1 [( ))]
) / 1 (
( 1 )( / 1 [(
1 1 1 1
1 2 1 1
n n
s s s
n
n n
s s
n
n i
s s
n
n
s s
n
n
s
n I
n n n
i
B
+ + +
− +
−
+ +
− +
+
+ +
− +
+
− + − =
− −
(1) 
                     
From this definition of the Bonferroni index various algorithms have in fact been 
proposed  in  the  literature  to  compute  it  (see,  for  example,  Tarsitano,  1990, 
Chakravarty, 2007, and Bárcena and Imedio, 2008)  
 
Figure 1:
The Bonferroni Curve
Cumulative 
values of
1
A
O
Cumulative values of 
(1/n)
)) / /( (
1
n i s
i
j
j ∑
=
1
 
 
3.  Extensions of the Bonferroni index and curve 
 
Whereas the Bonferroni index and the Bonferroni curve which have just been defined 
may be used to measure income inequality, it is also possible to apply the concepts of 
Bonferroni index and Bonferroni curve to derive what could be called a Bonferroni 
concentration index and a Bonferroni concentration curve, in the same way as the 
Gini concentration index and the concentration curve were derived (see, Kakwani, 
1980) from the Gini index. 
Let us, for example, assume that we want to take what has been called a "bivariate 
approach to health inequality measurement" (see, O'Donnell et al., 2008) in order to 
analyze  the  link  between  health  and  income.  We  can  then  derive  a  "Bonferroni 
concentration index" B C  by computing the ratios of the "conditional means" of the 
health variable over the mean value of the health variable in the whole population, 
that is, the expressions  )
) / ) ... ((
(
2 1
h
i h h h i + + +
where the health variable  i h (e.g. the 
body mass index) and its mean  h  replace respectively the income variable  i x  in the 
definition of the conditional mean and the average income  x  in the definition of the   3
index. Note however that here we rank the health variable  i h  by increasing income  i x  
rather than by increasing values of the health variable itself. We can in fact derive 
such  a  "Bonferroni  concentration  index"  B C   from  a  graph  that  could be  called a 
"Bonferroni  concentration  curve".  Such  a  curve  is  constructed  like  a  regular 
Bonferroni curve, the only difference being that the health variable  i h  is ranked by 
increasing income  i x  .  Note that this "Bonferroni concentration curve" may at time 
lie above the equality line (horizontal line at height 1) and in such a case it can be 
shown that the area above such an equality line will be given a negative sign. 
 
Figure 2:
The Bonferroni Concentration Curve
Cumulative 
values, 
ranked by 
increasing 
income, of
1 A
O
Cumulative values of 
(1/n)
1
B
C
D
)) / /( ) (
1
n i s
i
j
j ∑
=
 
Finally a third graphical tool may be derived from the Bonferroni index. We know 
that in the case of income inequality analysis the Lorenz curve is obtained by plotting 
on the horizontal axis the cumulative population shares and on the vertical axis the 
cumulative income shares. If, on the vertical axis, we multiply the product of the 
cumulative income shares by the average income, we obtain what has been called a 
Generalized Lorenz curve (see, Shorrocks, 1983). This curve will therefore start at 
point (0,0) and end at point  ) , 1 ( x where  x is the average income.  
We can similarly derive a Generalized Bonferroni curve. On the horizontal axis plot, 
as previously, the cumulative population shares and on the vertical axis we now plot 
the  cumulative  values  )} / ) ... (( ),..., 2 / ) (( ), {( 2 1 2 1 1 n x x x x x x n + + + + .  Such  a 
Generalized Bonferroni curve, like the Generalized Lorenz curve, will start at point 
(0,1) and end at point  ) , 1 ( x . Since the Bonferroni index  B I  is equal to the area lying 
above the Bonferroni curve, the area lying above the Generalized Bonferroni curve 
will be equal to  x IB , where  B I is the Bonferroni index, and therefore the area lying 
below the Generalized Bonferroni curve will be equal to  ) 1 ( B B EB I x x I x x − = − = .  
EB x   can  therefore  be  considered  a  measure  of  welfare  similar  to  the  index 
) 1 ( G EG I x x − =  defined by Sen (1974).   4
One can naturally apply the concepts of Generalized Lorenz or Bonferroni curves to 
measure the welfare derived from some health attainment. Such welfare measures 
EG x  and  EB x  would in fact give a greater weight to an individual, the lower the level 
of his health. 
One  may  however  think  of  an  alternative  approach,  one  where  the  weight  of  an 
individual, when measuring health related welfare, would be higher, not the lower the 
level of his/her health, but the lower his/her income. This is in fact the approach taken 
by  Wagstaff  (2002)  in  his  definition  of  health  achievement.  We  can  call  such  an 
approach the pro-poor approach to the measurement of health achievements. 
Using  the  concept  of  "Bonferroni  concentration  index" B C   which  was  defined 
previously, we may therefore define the level of health achievement  B A as 
 
) 1 ( B B C h A − =                   (2) 
 
In  fact,  in  the  same  way  as  we  derived  previously  the  concept  of  Generalized 
Bonferroni  curve,  we  can  now  derive  the  concepts  of  Generalized  Bonferroni 
Concentration  curve.  We  simply  have  to  order  the  vertical  coordinates  of  the 
Generalized Bonferroni curve not by increasing values of the health variable, but by 
increasing income. It is then easy to derive that the area under such a Generalized 
Bonferroni concentration curve will be equal to half the product  B B A C x = − ) 1 ( .  
 
4.  The Bonferroni index and the measurement of inequality in human 
  opportunities 
 
Let  i p  be the probability for group i of having access to some public service (e.g. a 
hospital) and let  p be the average probability for an individual in the population to 
have access to this service. Let  i w  be the weight of population subgroup i in the total 
population. The weights  i w  may therefore represent the "prior" probability of having 
access to the service, while the expression  )) / ( ( p p w i i would represent the "posterior" 
probability of having access to the service. 
If one wishes to use the Bonferroni index to measure the degree of inequality in the 
opportunity to have access to this service (hospital), we may apply expression (1) but 
the  "a  priori"  weights  i w   will  replace  the  population  shares  ) / 1 ( n   and  the  "a 
posteriori"  weights  )) / ( ( p p w i i i = σ   will  replace  the  income  shares  i s   .  In  other 
words the Bonferroni index of inequality of opportunity would in such a case be 
defined as  
 
))]
) ... (
) ...
( 1 )( [( ))]
) ... (
) ...
( 1 )( [(
... ))]
) ... (
) ... (
( 1 )( ...[( ))]
) (
) (
( 1 )( [( ))]
) (
( 1 )( [(
1 1
1 1
1 1
1 1
1
1
1
2 1
2 1
2
1
1
1
m m
m m
m
m
m
m
i
i
i B
w w w
w
w w
w
w w
w
w w
w
w
w I
+ + +
+ + +
− +
+ +
+ +
− +
+
+ +
+ +
− +
+
+
− + − =
−
−
−
−
−
σ σ σ σ σ
σ σ σ σ σ
   (3) 
                       
where m  is the number of population subgroups.  
   5
The following graphical interpretation may be given to this approach. Let us plot on 
the  horizontal  axis  the  cumulative  values  of  the  "a  priori"  probabilities 
) 1 ,... ,..., , ( 2 1 i w w w and  on  the  vertical  axis  the  cumulative  values  of  the  ratios 
]},... ... /[ ))] / ( ( ... )) / ( {[( ]},..., / )) / ( {[( 1 1 1 1 1 1 i i i w w p p w p p w w p p w + + + + .  If  we 
multiply these vertical coordinates by the average probability  p , we will obtain new 
coordinates  which  will  be  expressed  as 
]}. ... /[ )] ( ... ) {[( ...,
]},... ... /[ )] ( ... ) {[( ]},..., / )) ( {[(
1 1 1
1 1 1 1 1 1
n n n
i i i
w w p w p w
w w p w p w w p w
+ + + +
+ + + +
  
Since  p p w p w n n = + + )] ( ... ) [( 1 1  and  1 ] ... [ 1 = + + n w w , it is then clear that we end up 
with a curve which starts also at point (0,0) at which ends at point  ) , 1 ( p . As was done 
previously  when  we  analyzed  health  achievements,  we  will  call  this  curve  a 
Generalized Bonferroni curve.  It is easy to derive that the area lying under such a 
curve is in fact equal to  ) 1 ( B I p − , an index which can be called the "Bonferroni-
related Human Opportunity Index" B HOI . 
Note that since the Bonferroni index (like the Gini index) gives a higher weight to the 
categories who have a lower probability of accessing the service, we can call the 
"Bonferroni-related Human Opportunity Index  B HOI " a Human Opportunity Index 
which is "welfare-related", in the sense that it favors groups with low probabilities of 
accessing the service. 
The approach which has just been described is based on a univariate approach to the 
measurement  of  inequality  in  opportunities.  In  other  words  we  measured  the 
inequality in the access to a hospital, no matter what the socioeconomic background 
of the individual is.  Let us however assume that we want to analyze the link which 
exists between this access to a hospital and the socioeconomic background. In such a 
case we would classify in (3) the "a priori" probabilities  i w  and the "a posteriori" 
probabilities  i σ , not by increasing ratios  ) / ( / )] / ( [ ) / ( p p w p p w w i i i i i i = = σ , but by 
increasing socioeconomic background. As a consequence we would not compute the 
Bonferroni index but the Bonferroni Concentration index  B C and derive a "Human 
Opportunity Index" on the basis of the Bonferroni concentration index, rather than of 
the Bonferroni inequality index. We will call such an index a "Bonferroni-related Pro-
Poor Human Opportunity Index"  pp B O ,  , and it will evidently be expressed as 
 
) 1 ( , B pp B C p O − =                   (4) 
 
The graphical interpretation of such an index is very simple. We rank the cumulative 
values that are plotted on the horizontal and vertical axes to derive a Generalized 
Bonferroni curve, not by increasing values of the probabilities  i p  of accessing the 
service, but by increasing values of the socioeconomic background of the individuals. 
Such a curve is clearly a "Generalized Bonferroni Concentration curve". It is easy to 
prove that the area lying under such a curve is to equal the product  pp B B O C p , ) 1 ( = − .  
It is interesting to note that what we have just called the "Bonferroni-related Pro-Poor 
Human Opportunity Index" is in fact what Ali and Son (2007) called "Opportunity 
Index  ",  while  what  we  called  "Generalized  Bonferroni  Concentration  curve"  is 
identical to what Ali and Son (2007) called "Opportunity Curve". The proof is simple. 
Ali  and  Son  (2007)  defined  their  opportunity  curve  (or  generalized  concentration   6
curve  of  opportunity)  as  follows.  In  terms  of  the  notations  previously  used,  their 
approach amounts in fact to plotting on the horizontal axis the cumulative values of 
the "a priori" population shares  i w , these probabilities being ranked by increasing 
socioeconomic background of the individuals, and on the vertical axis the cumulative 
values of the conditional means of the variable analyzed which in our case are the 
ratios: 
))] /( ) (( ...,
)), ... /( ) ... (( )),..., /( ) (( ), / ) [(( 2 1 2 2 1 1 2 1 2 2 1 1 1 1 1
∑ ∑
+ + + + + + + +
i
i
i
i i
i i i
w p w
w w w p w p w p w w w p w p w w p w
Ali and Son (2007) then define their opportunity index  * y as the area lying below this 
opportunity curve and this index is clearly identical to what we previously called the 
"Bonferroni-related Pro-Poor Human Opportunity Index"  pp B O , . 
 
5.  Conclusion 
 
Ali and Son (2007) derived their opportunity index and opportunity curve using the 
framework of a general social opportunity function, implicitly arguing that society 
should  focus  on  expanding  or  maximizing  this  social  opportunity  function.  The 
opportunity  index  provides  an  operational  way  to  implement  policies  that  would 
maximize the social opportunity function. Moreover, Ali and Son introduced the idea 
of  equity  of  opportunity  index,  which  measures  how  equitably  or  inequitably 
opportunities  are  distributed  across  the  population.  The  two  indices  are  not  only 
useful in assessing average access to the public services available to the people, but 
also in evaluating the equity of access to such services across different income groups. 
More importantly, their study has demonstrated that while the analysis can be done at 
a point of time, it is also possible to assess the changes in access to and equity of 
opportunities over time.  
The Bonferroni index was proposed in 1930 and was not mentioned very often until 
recently. Ali and Son’s paper demonstrates how useful Bonferroni’s approach may be 
to measure and analyze inclusive growth. Our paper linking the Bonferroni index and 
Ali  and  Son’s  opportunity  index  contributes  thus  to  the  revival  of  Bonferroni’s 
important contribution. 
 
References 
 
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absolute Bonferroni index of inequality" Journal of Economic Inequality 5  
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Kakwani, N. C. (1980) Income Inequality and Poverty: Methods of Estimation and 
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A deprivation-based axiomatic characterization of the absolute Bonferroni index of inequality&quot;

Analyzing Health Equity Using Household Survey Data.

Elementi di statistica generale,

Imedio (2008) &quot;The Bonferroni, Gini and De Vergottini indices.

Income Inequality and Poverty: Methods of Estimation and Policy Applications,

Inequality Aversion, Health Inequalities and Health Achievement&quot;

Information Bases of Alternative Welfare Approaches&quot;

Measuring Inclusive Growth&quot;

Ranking Income Distributions&quot;

The Bonferroni Index of Income Inequality&quot;

welfare and deprivation in the European Union in 2000&quot;

On the link between the Bonferroni index and the measurement of inclusive growth.

http://www.accessecon.com/Pubs/EB/2010/Volume30/EB-10-V30-I1-P37.pdf

On the link between the Bonferroni index and the measurement of inclusive growth.

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