Gini coefficient for ordinal categorical data Rodrigo Peñaloza∗ Department of Economics, University of Brasília September, 2016
Abstract We generalize the Gini coefficient to ordinal categorical variables. In particular, our generalized Gini coefficient provides an alternative way to measure concentration when only stratified data are available. JEL Classification: D30, D63, I31. Key words: ordinal variables, income distribution, relative Gini coefficient.
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Introduction The Gini coefficient [Gini (1912)] is the most common measure of distributional
concentration for income data. Based entirely on the relative position of the Lorenz curve of a certain income distribution with respect to both an equal-income distribution and a concentrated distribution, it has a natural geometric interpretation as a relative area. One important feature of the Gini coefficient is that it applies to a variable which is measurable on a ratio scale [Stevens (1946)], that is, one for which there is a natural zero and that is unique up to a multiplication by a positive constant, of which income is the most obvious example. Many social and economic variables, however, are of ordinal nature. When income data is stratified or society is stratified into classes, we could also see income as ordinal, but this ordinality has not been theoretically explored. There has been some attempts though, in the Statistics literature, to cope with the ordinal nature of many important social data, such as the measurement of quality of ∗
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academic teaching [Cerchiello et al. (2010)] and of self-reported health [van Doorslaer & Jones (2003)]. When dealing with categorical data, the standard measures of distributional variability that use the mean as a reference point, such as the variance, are not appropriate [Allison & Foster (2004) and Maaden 2010)]. The main reason for that is the dependence of the inequality measure with respect to the arbitrary scale applied to the categories of an ordinal variable. An obvious choice of reference point is the median. Another one is the cumulative proportion of population in each category. Strangely, the later approach, though used and proposed independently by many important authors, such as Gini himself, has been practically unknown in the applied Economics literature. One instance where such approach is of much help is the measurement of perceived wellbeing. Another approach is the transformation of ordinal variables into cardinal ones and to use these numbers in standard inequality measures, such as the use of interval regression [van Doorslaer & Jones (2003)] and Maden (2010)’s generalized entropy index. A very interesting alternative approach is the extension of the Lorenz curve to ordinal data and the subsequent application of the Gini index, which is the ranks-based Gini measure of Giudici & Raffinetti (2011). With regard to ordinal variables, the Gini coefficient has then had few counterparts. In this paper we contribute to fill out this gap by defining a Gini coefficient for ordinal data that reduces to the usual Gini coefficient in the particular case of variables measured on a ratio scale, for instance, income. Ordinal variables have proper measures of position based on the relative frequences of their categories. One of this measures show up naturally in a very specific alternative expression of the Gini index, one that says that the Gini index is the weight that turns the measure of position of the income distribution a convex combination between the measures of position associated with two extrema, the equalincome distribution and the concentrated distribution. This representation allows us to define a relative Gini index, one that relaxes the distributional characteristics of these extrema and to define a Gini coefficient for ordinal categorical variables. In section 2 we briefly review the Gini coefficient, so the reader may skip this section at will. In section 3 we present a formal definition of expectation for ordinal variables and a numerical measure of position that differs from the median and the mode, the most widely used measures of position for ordinal variables. This measure of position can be interpreted as a kind of weighted mode. In section 4 we define a relative Gini index and 2
apply it to income data by focusing on its ordinal aspect alone. We come up then with a representation that lets the door ajar for a generalization to ordinal variables. In section 5 we define the Gini coefficient for ordinal variables, show some of its properties and give some examples. Section 6 finally concludes the paper.
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Gini coefficient In this section we briefly review the construction of the Gini coefficient for the sake
of exposition only. The reader acquainted with it can move on to the next section. Let = (1 ) be a vector of nonnegative real numbers such that 1 ≤ · · · ≤ . P The number is the income of an individual at some period. Let = 1 =1 be the
mean income of the population. Let = and Φ =
=1 =1
be the cumulated proportion of population
be the cumulated proportion of income with respect to total income, P 1 ∀ = 1 . Set = 0 and Φ = 0, ∀ = 1 . Notice that Φ = =1 and
= Φ = 1. The Lorenz curve is the poligonal line L() generated by the points {( Φ ) (1 Φ1 ) ( Φ )} ⊂ [0 1] × [0 1].
Let = (1 ) be the concentrated vector of incomes, that is, the distribution P of income for which there exists some individual ∗ such that ∗ = =1 and = 0,
∀ 6= ∗ . Let L() be the Lorenz curve of . Notice that it is irrelevant who the individual ∗ is. The poligonal line L() is the Lorenz curve of the most unequal income distribution. Let = (1 ) be the vector of equal incomes, that is, = , ∀ = 1 . Let L()
be the Lorenz curve of . Clearly, L() coincides with the diagonal line of the unit square [0 1] × [0 1] L() is the graph of the function Φ[] : [0 1] → [0 1] given by Φ[] () = and L() is the graph of the function Φ[] : [0 1] → [0 1] given by: ⎧ ⎨ 0 if ∈ [0 1 − 1 ) Φ[] () = ⎩ 1 − + if ∈ [1 − 1 1]
that is, Φ[] () = (1 − + )[1− 1 1] (), where: ⎧ ⎨ 0 if ∈ () = ⎩ 1 if ∈
is the indicator function of a (measurable) set ⊂ [0 1] If = 3
R1 0
(Φ[] () − Φ[] ()),
then: =
Z
0
1
Φ[] () −
1 1 1 − × ×1 2 2 1 1 = (1 − ) 2
Z
1
Φ[] ()
0
=
Therefore, the area between the concentrated Lorenz curve L() and the equal-income
Lorenz curve L() is 12 (1− 1 ). This is the maximal inequality when the standard reference
for equanimity is the equal distribution of income and the standard reference for inequality is the concentrated distribution. Given a vector = (1 ), the Gini coefficient G() is given by the area between the Lorenz curve L() and the diagonal line of the unit square [0 1] × [0 1] relative to
= 12 (1 − 1 ), that is, relative to both the equal-income and the concentrated Lorenz curves L() and L(). In other words: R1 G() = R01 0
or G() =
1
R1 0
(Φ[] () − Φ[] ())
(Φ[] () − Φ[] ())
(Φ[] () − Φ[] ()). There are many alternative formulae for the Gini
coefficient, all obviously equivalent, but the following will be suitable for our purposes: P G() = 2 2 =1 −(1+ 1 ). We will however give another alternative representation, one that will give us a hint as to how to generalize the Gini coefficient to ordinal categorical variables.
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Ordinal variables and ordinal expectation An ordinal variable is a question for which there exists a finite set of categorical
answers 1 , ranked ordinally from lowest to highest, that is, 1 ≺ · · · ≺ , where ≺ is a transitive, complete, and strict binary relation. In other words, ≺ means that is categorically strictly inferior to . Suppose, for instance, that is the question, How is your access to public health?, for wich the only possible answers are 1 = none, 2 = difficult, 3 = acceptable, and 4 = easy. Notice that the answers are subject to a natural order of the type 1 ≺ 2 ≺ 3 ≺ 4 , where ≺ orders the ease of accessibility to 4
health care. Other examples are, What is your level of education?, or, What is your level of income? When we specify the categories of , we will write = {1 }. For ease of notation, an ordinal variable will be sometimes coupled with its binary relation ≺, so that ( ≺) will also denote an ordinal variable. In this sense, = {1 } is an ordered finite abstract set. If = {1 } is the set of incomes with 0 ≤ 1 ≤ · · · ≤ , then the order is and ( ) actually is a variable in the ratio scale, hence in particular an ordinal variable. The number of categories is finite and they are exhaustive in the sense that, if 1 are propositions regarding the levels of some variable in the ordinal scale, then 1 ∨· · ·∨ is tautological and ∧ is false, ∀ 6= Thus it is trivial to extend ≺ to a reflexive transitive and complete order 4 in such a way that ∼ if and only if = .
In order to define the first moment of an ordinal variable ( ≺), which we call ordinal
expectation, E≺ (), we have to rely on relative frequencies1 . Let be the relative frequency of respondents who falled into category and let = (1 ) be the vector P of relative frequencies. Let O ( ≺) = { ∈ [0 1] : =1 = 1} be the ( − 1)-
dimensional unit simplex. Given an ordinal variable ( ≺), an element ∈ O ( ≺) will be called an observation.2
The median is the most common measure of position for ordinal variables. We will instead consider a different measure based on post-cumulated frequencies [Souza (1977, P 1988)]. Let = = be the post-cumulated frequency of category . It measures
the total mass of respondents supported by category Obviously, 1 = 1 ≥ 2 ≥ · · · ≥
= . Given an ordinal variable ( ≺) and an observation ∈ O ( ≺), define P = 1 =1 , the average weight supported by the categories of . P A simple calculation shows that = =1 . Hence scales ordinal categories
according to their respective position on the ascending ordinal scale determined by the binary relation ≺. It is easy to show that
1
≤ ≤ 1, but the most important property
of this measure is the following: if the distribution = (1 ) is concentrated on category , that is, if = 1 and = 0, for any 6= , then = 1 = 2 = · · · = = 1 and +1 = +2 1 2
.
Indeed, in this case, P = · · · = = 0, so that = 1 =1 = .
This is a standard procedure in the theoretical literature. The meaning of this should be straightforward. An observation is just the collected array of observed
relative frequencies of answers to question .
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Therefore, acompanies the predominant position of the data on the scale. Notice that, when the distribution = (1 ) is concentrated on category , we have that = , that is, pins down exactly the ordinal position of category , which is then called the central category. The central category of the ordinal variable ( ≺) is obtained in the following way. P Given , where = =1 , take its nearest integer , that is: ⎧ ¥ ¦ ¥ ¦ ¥ ¦ ⎪ if − + 1 − ⎪ ⎨ ¥ ¦ ¥ ¦ ¥ ¦ ¥ ¦ = { + 1} if − = + 1 − ⎪ ⎪ ¥ ¦ ¥ ¦ ⎩ ¥ ¦ + 1 if − + 1 −
where bc is the integer part of a real number . Notice that if the fractional part is 05, then we take both adjacent integers. The central category is . The ordinal expectation
of ( ≺) is defined by E≺ () = . Clearly, the ordinal expectation is better than the median as a measure of position for ordinal categorical variables. Suppose, for instance, that = ( 12 0 0 0 12 ) for some ordinal variable ( ≺) with five categories 1 ≺ 2 ≺ 3 ≺ 4 ≺ 5 . Then every category
is median, that is, () = {1 2 3 4 5 }.3 On the other hand, the ordinal
expectaton is E≺ () = 3 4 . P The value = =1 gives the mean numerical position of on the unit interval
[0 1]. The ordinal expectation E≺ () gives the mean categorical position of with
respect to the whole range of categories. Both work well as measures of position, the difference being only a matter of focus, either on a numerical index or on a categorical selection. The ordinal expectation does not have to satisfy the same properties that the usual expectation for random variables does. For instance, the additive property of the expectation operator E over the space of integrable functions L1 ( ) on a measure space (Ω F ), that is to say, E( + ) = E() + E( ), where ∈ L1 ( ) and ∈ R,
only makes sense insofar as the counterdomain of the random variable : Ω → is P This will be the case if the median is defined as the category for which = ≥ 05 and P P =1 ≤ 05. In the strict sense, the median category is given by for which =+1 ≥ 05 and P−1 =1 ≤ 05. In this case, () = {2 3 4 }. Anyway, whenever the median is comprised of 3
many categories, our measure selects at most the same categories. 4 The mode is even worse, since it does not take into account the ordinality of the variable.
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a vector space endowed with the operations of addition of vectors and multiplication by scalars5 . However, in an ordered finite abstract set = {1 }, addition of categories have no meaning. Therefore, when dealing with ordinal variables from a categorical perspective, we have to rely on a more fundamental and primitive idea of the concept of expectation, the mere fact that expectation is a measure of position, or center. Let N = {1 } be the set of positive integers from 1 to and let N = 2N∗ be the set of nonempty subsets of N . Let be the correspondence : [1 ] → N defined by:
⎧ ⎪ if − bc ∈ [0 12 ) ⎪ ⎨ bc () = bc + 1 if − bc ∈ ( 12 1) ⎪ ⎪ ⎩ {bc bc + 1} if − bc = 12
Definition 1 Let ( ≺) be an ordinal variable and let ∈ O ( ≺) be an observation. Then: (a) the ordinal index of ( ≺) is I( ≺ ) =
P
=1
(b) the ordinal expectation of ( ≺) is E≺ () = (I(≺ )) . Lemma 2 Let ( ≺) be an ordinal variable. Then, for any observation ∈ O ( ≺), the following inequality holds:
1
≤ I( ≺ ) ≤ 1
Proof. Consider the vectors = (1 ), = (1 ), = (1 1) and = (0 0), all in R Now consider the following minimization primal problem: ⎧ P ⎪ min ⎪ =1 ⎨ P =1 = 1 ⎪ ⎪ ⎩ ≥ 0, for = 1 Then the pairing of primal and dual programs in matrix form is given by: ⎧ ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎨ min ⎨ max (1 ) 5 (1 ) 0 = 1 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ free =
where ∈ R is the Lagrange multiplier associated with the primal constraint and 5 is
the componentwise vector inequality on R . It is easy to show that the dual solution 5
Most commonly, is the real line R, but can also be any Banach Space.
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is ∗ = 1. Indeed, by the dual constraints, ≤ , for all = 1 , so the maximum ∗ value of satisfying the dual constraints is ∗ = 1. Let ∗ = (1∗ ) be the primal
solution. By the fundamental theorem of linear programming, we have that ∗ = 0 ∗ , P P ∗ that is, 1 = =1 . Therefore, 1 ≤ =1 , for any observation ∈ O ( ≺). P Since I( ≺ ∗ ) = =1 ∗ , then 1 ≤ I( ≺ ). Similarly, consider the following
primal-dual pair of problems: ⎧ 0 ⎪ ⎪ ⎨ max (2 ) 0 = 1 ⎪ ⎪ ⎩ =
⎧ ⎪ ⎪ ⎨ min (2 ) = ⎪ ⎪ ⎩ free
where ∈ R is the Lagrange multiplier associated with the primal constraint. The dual
constraint is ≥ , for all = 1 , so the dual solution is clearly ∗ = . Therefore, P ∗ ∗ ≥ 0 ∗ , that is, ≥ =1 , so I( ≺ ) ≤ 1, for any observation ∈ O ( ≺), which proves the result.
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Relative Gini index As any other measure of concentration, the Gini coefficient tells us how distant the ac-
tual income distribution is from the equal-income situation. From the social and economic point of view, however, we loose much understanding of its meaning by the mere fact that, even under a situation of equal opportunities in any level of the society, equal-income is likely not to be the final result of human action. People possess different capabilities, skills, and goals, so it is natural to have always some sort of unequal distribution of income. It is common sense among statisticians and economists alike that the measurement of income inequality should take an ideal Lorenz curve whose curvature resulted from natural differences among people under an environment of fairness. We do not want to get into the discussion of the meaning of fairness, since it will not in any way affect our reasoning. What really matters for our purposes is the concept of relative Gini coefficient, one that takes such ideal Lorenz curve as the standard of achievement, not necessarily the equal-income situation. Definition 3 Let Φ[] Φ[] : [0 1] → [0 1] be two different Lorenz curves such that
Φ[] () Φ[] (), ∀ ∈ (0 1). For any vector ∈ R of income data such that Φ[] () ≤ 8
Φ[] () ≤ Φ[] (), ∀ ∈ [0 1], we define the relative Gini [ ]-measure as: R1 (Φ[] () − Φ[] ()) [] () = R01 (Φ[] () − Φ[] ()) 0
Proposition 4 The relative Gini [ ]-measure [] () satisfies decomposition into strata, symmetry, the Pigou-Dalton transfers principle and the Dalton population principle. In addition, [] () =
G()−G() G()−G()
Proof. This follows straightforwardly from the fact that the relative Gini [ ]measure is a positive affine transformation of the standard Gini coefficient. Indeed, 1 G()−G()
[] () = G() − , where =
0 and =
G() G()−G()
= G() 0.
The coefficients and obviously do not depend on . Therefore, since it is a positive affine transformation of the Gini index, [] () shares all the nice properties satisfied by the Gini coefficient, for instance, decomposition into strata, symmetry, the PigouDalton transfers principle and the Dalton population principle [Chakravarty (2009, p.17)]. R1 R1 G()−G() To show that [] () = G()−G() , notice that 0 (Φ[] () − Φ[] ()) = 0 (Φ[] () − R1 R1 Φ[] ()) + 0 (Φ[] () − Φ[] ()), hence 0 (Φ[] () − Φ[] ()) = −G() + G(), R1 R1 that is, 0 (Φ[] () − Φ[] ()) = (G() − G()). Similarly, 0 (Φ[] () − Φ[] ()) = (G() − G()). Therefore [] () =
G()−G() , G()−G()
as claimed6 .
Proposition 5 Let = {1 } be the set of incomes with 1 ≤ · · · ≤ and consider the ordinal variable ( ). Let = be the participation of income =1 P in the total income =1 Likewise, define = and = where =1
=1
= (1 ) and = ( 1 ) are income arrays for which the respective Lorenz
curves Φ[] Φ[] satisfy Φ[] () Φ[] (), ∀ ∈ (0 1), and such that either = = or
1
≈ 0. Then I( ) = (1 − [] ())I( ) + [] ()I( )
P − (1 + 1 ). Similarly, G() = 22 =1 − P (1+ 1 ), where is the dimension of vector and = 1 =1 is its arithmetic mean. Proof. Recall that G() =
6
2 2
P
=1
Let be the set of all nondecreasing convex Lebesgue-measurable functions Φ on [0 1] such that
Φ(0) = 0 and Φ(1) = 1. Let L1 [0 1] be the space of all Lebesgue-integrable functions on [0 1], endowed with the weak topology. Then clearly ∀x ∈ R , Φ[x] ∈ and ⊂ L1 [0 1] is a weakly closed convex subset of L1 [0 1] If || · ||1 denotes the L1 -metric on L1 [0 1], then [] () =
||Φ[] −Φ[] ||1 ||Φ[] −Φ[] ||1 ,
because of the
fact that Φ[] and Φ[] are the supremum and the infimum of according to the order ≤, respectively,
so that Φ[] () − Φ[x] () ≥ 0, ∀ ∈ [0 1].
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Analogously for G(). We can consider three cases. First, the vetor = (1 ) (with 1 ≤ · · · ≤ ) is an array of incomes of a whole population of households in an economy, assuming, for the time being, that 1 · · · . In this case, the dimension is so large that we can assume that
1
≈ 0, that is, we can neglect the term 1 . Second, the
vector = (1 ) might refer to a sample. In this case, since we are dealing with countries, states or cities, the sample size should big enough to allow us to assume that 1
≈ 0. Third, the size might refer to the number of income strata. In this case, we can
assume that = = . Therefore, without loss of generality,
1
−
1
=
1
−
1
= 0.
It is easy to show that [] () depends only on the vectors of relative frequencies, not on their dimensions or arithmetic means. Indeed: P P 2 1 2 1 =1 − (1 + ) − 2 =1 + (1 + ) 2 P P [] () = 2 1 2 1 =1 − (1 + ) − 2 =1 + (1 + ) 2 ´ ³ P P 2 2 1 1 − 2 2 =1 =1 + − ´ ³ = P P 2 2 1 1 − + − 2 2 =1 =1 P P 1 1 =1 − 2 =1 2 P P = 1 1 =1 − 2 =1 2
Now notice that:
Let =
X 1 X 1 P = 2 =1 =1 =1 ¶ µ X P = =1 =1
P be the participation of income in the total income =1 Likewise, P P and = Then 1 2 =1 = =1 . Therefore:
=1
define =
=1
=1
[] () =
P =1 − P P=1 =1 − =1 P
I( ) − I( ) I( ) − I( ) I( ) − I( ) = I( ) − I( )
=
where = (1 ), = (1 ) and = (1 ), and is the usual transitive complete strict order on the real line restricted to . Consequently, the Gini [ ]10
measure can be defined by the equation: I( ) = (1 − [] ())I( ) + [] ()I( ) as was to be shown. In other words, the Gini [ ]-measure is the weight that turns I( ) into a
convex combination of the indices I( ) and I( ) When the proposition above is applied to the standard extreme distributions, that is, the equal-income and the concentrated distributions, then the Gini [ ]-measure [] () reduces to the standard Gini coefficient, as the following corollary shows. Corollary 6 Let = {1 } be the set of incomes with 1 ≤ · · · ≤ . Suppose that
= refers to the equal-income distribution and that is any concentrated distribution. Then [] () = G()
Proof. IIf = refers to the equal-income distribution, then = 1 , ∀, in which ³ ´ case, I( ) = 12 1 + 1 . If refers to a concentrated income distribution, then
1 = · · · = −1 = 0 and = 1, hence I( ) = 1. Therefore, I( ) = ³ ´ 2I( )−(1+ 1 ) 1 1 (1 − [] ()) 2 1 + + [] (), so that [] () = . Assuming that (1− 1 ) 1
≈
1
≈ 0, since we are dealing with the size of populations, we have [] () = 2I(
) − 1. On the other hand: 1 2 X − (1 + ) G() = 2 =1
= 2
X − 1 =1
= 2I( ) − 1 Therefore, [] () = G(). This is the particular case of the Gini coefficient.
It is the representation I( ) = (1 − [] ())I( ) + [] ()I( )
that will allow us to generalize the Gini coefficient to ordinal categorical data.
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Relative Gini coefficient for ordinal variables Instead of taking a vector = (1 ) of incomes such that 1 ≤ · · · ≤ ,
suppose that we take an ordinal variable ( ≺) with ordered categories 1 ≺ · · · ≺ . 11
In particular, the variable can be ordered strata of incomes, but can also be any other ordinal variable. We want to propose an ordinal version of the Gini coefficient for ordinal variables. Regarding the ordinal variable , let = (1 ) be the vector of relative frequencies considered to be the best or ideal distribution. Likewise, let = (1 )
be the vector of relative frequencies considered the worst. We call them, respectively, the superior and inferior distributions. When an ordinal variable ( ≺) is paired with its superior and inferior distributions, we call it a Lorenz model and write h ≺ i. Definition 7 Let ( ≺) be an ordinal variable and let ∈ O ( ≺) be an observation
for . Suppose that and are the superior and inferior distributions, respectively. The ordinal Gini measure of is given by the weight b ∈ [0 1] such that: bI( ≺ ) I( ≺ ) = (1 − b)I( ≺ ) +
We define the Gini categorical expectation as E≺ () = ((1− ))
The interpretation of the Gini categorical expectation is straightforward. The higher b the b, the higher the category ((1− )) in the ordinal scale. This is the category that
selects as the category that represents the distance from the inferior category. To simplify matters, assume we are talking about standard Gini coeffiicient for income data. If the income variable were stratified into an ascending order of income strata, then the Gini categorical expectation for a distribution with very low Gini coefficient is a stratum high in the ordered set of strata. Of course E≺ () is meaningless without the information given by E≺ (). Indeed, two distributions may be equal-income, but one with very high income and the other with very low income. The following proposition establishes the relation between ordinal expactation and Gini categorical expectation. Proposition 8 Let h ≺ i be a Lorenz model. For any observation ∈ O ( ≺),
let E≺ () and E≺ () be its Gini expectation and ordinal expectation, respectively. Suppose, for simplicity, that both (I( ≺ )) and ((1 − b)) are single-valued. Define
the following intervals = ( 2−1 2+1 ), for = 1 − 1. The following hold: 2 2
(a) If there exist ∈ {1 − 1} and ≥ such that I( ≺ ) ∈ and b ∈ − , then E≺ () 4 E≺ (); in particular, if = , then E≺ () ∼ E≺ (). 12
(b) If there exist ∈ {1 − 1} and such that I( ≺ ) ∈ and b ∈ − , then E≺ () ≺ E≺ ().
(c) in either case, E≺ () is the category and E≺ () is the category. Proof.
(a) Recall that E≺ () = (I(≺ )) and E≺ () = ((1− )) .
Then
E≺ () ∼ E≺ () if, and only if, (I( ≺ )) = ((1 − b)). If denotes such integer,
1 3 then I( ≺ ) (1−b ) ∈ (− 12 + 12 ). If = 1, then I( ≺ ) (1−b ) ∈ ( 2 2 )=
1 3 1 , that is, I( ≺ ) ∈ ( 2 2 ) = 1 and b ∈ ( 2−3 2−1 ) = −1 . By an analogous 2 2 3 5 reasoning, if = 2, then I( ≺ ) ∈ ( 2 2 ) = 2 and b ∈ ( 2−5 2−3 ) = −2 . 2 2
Thus, if there exist ∈ {1 − 1} such that I( ≺ ) ∈ and b ∈ − , then
E≺ () ∼ E≺ () = . Now if , let = − , where obviously 1 ≤ ≤ − 2 Then
I( ≺ ) lies on the interval from left to right and b lies on the ( +) interval from
right to left, which means that 1 − b lies on the ( + ) interval from left to right. In other words, I( ≺ ) ∈ and 1 − b ∈ + . Therefore, E≺ () = ≺ + = E≺ ().
(b) This case is the complement of part (a). (c) The result follows by construction.
Example 9 (stratified income data7 ): Suppose that income data is stratified into eleven categories , = 1 11. The first category 1 refers to the income interval [0 100), so there is no loss of generality if we write 1 = [0 100). Then, = [100( − 1) 100), for = 1 10, and 11 = [1000 ∞), so = 11. Suppose that in an ideal country, in the country with worst distribution, and in an intermediate country, the distribution vectors 7
In this case, the same results regarding decomposition and range within which the true Gini index is
given, for stratified data, holds true. Suppose we partition the population into strata N1 N , that
is, N1 ∪ · · · ∪N = {1 } and N ∩ N = ∅, for 6= , where N = { : ∈ [−1 )}, [−1 ) is an income stratum, for 1 ≤ ≤ . Let G be the Gini index within stratum N , =
1 |N |
be the relative
be the participation of income in the total populational proportion of stratum N , and = =1 P P income =1 Define ∗ = ∈N . Let G () be the Gini coefficient between strata, each stratum
N taken with its mean income and relative weight . Then it is easy to see that trivially the usual P decomposition into strata holds, that is, G() = G () + =1 ∗ G . Let = max{ : ∈ N } P be the maximum individual income in the stratum, and = |N1 | ∈N be the mean income of the stratum N . We know that the maximum Gini index within stratum N is G = the last stratum is of the type [−1 ∞), then G = 1 −
clearly G () ≤ G() ≤ G().
13
−1 .
Define G() = G +
( −−1 )( − ) . If ( − ) P ∗−1 =1 G . Then
of income are: = (0 0 0 0 0 01 01 03 04 0 01) = (01 02 03 0 0 0 0 0 0 03 01) = (0 0 01 0 02 0 0 02 02 03 0) respectively. Define ≺ 0 if, and only if, sup = inf 0 . Then I( ≺ ) = 076364,
I( ≺ ) = 05, and I( ≺ ) = 07. The ordinal Gini measure is the value b for
which 07 = 076364(1 − b ) + 05b , that is, b = 024139. Now, (I( ≺ )) = 8.
Then the ordinal expectation is given the eigth stratum, 8 = [700 800), that is, E≺ () = [700 800). Since ((1 − b)) = 8, then the Gini expectation is also the eigth stratum,
E≺ () = [700 800).
Example 10 (ordinal qualitative data): Suppose that the ordinal variable refers to the qualitative degrees of education. From worst to best, the categories are 1 = no education, 2 = some but less than 9 years of education, 3 = up to 12 years of education but no high school degree, 4 = high school degree, 5 = more education than high school but less than a bachelor degree, 6 = bachelor degree, 7 = master degree or higher. Consider the following populational distributions of the best, worst and intermediate countries: = (0 0 0 0 03 05 02) = (01 02 02 02 01 02 0) = (0 0 01 02 01 05 01) Then I( ≺ ) = 084286, I( ≺ ) = 051429, and I( ≺ ) = 075714. Hence
the ordinal Gini measure is b = 026089. In addition, E≺ () = E≺ () = 5 .
Example 11 (standard Gini index for income data): Income data are collected from an array of 10 people and their income levels are ordered increasingly as shown in the vector 1 2 3 4 5 8 12 15 20 30 = (1 2 3 4 5 8 12 15 20 30). Then = ( 100 100 100 100 100 100 100 100 100 100 ), so
that I( ) = 0787 In addition, E≺ () = 8 , that is, the eigth individual. Since ¡ ¢ 1 I( ) = 1 and I( ) = 12 1 + 10 , we have that the Gini index is = 11 20
b = 047 The Gini expectation is E≺ () = 5 , that is, the fifth individual. 14
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Conclusion There has been a trend in the past decade or so towards the use of multidimensional
indices of poverty and inequality. Besides monetary data, it is hard to find, however, other relevant variables measurable in the ratio scale. Inevitably we end up with a set of categorical variables, either nominal or ordinal. Therefore, our philosophy is to deal with ordinal variables exactly how they are. With this in mind, we do not have to go far beyond techniques of exploratory Statistics. In this paper we built an ordinal version of the Gini coefficient, which can in particular be applied to stratified income data, but can also be applied to any ordinal categorical variable. Its extension to multidimensional inequality measures is a natural next step. Peñaloza (2012) uses ordinal variables in the construction of multidimensional measurement of education based on Sen’s capability approach. References 1. Allison, R. & J. Foster (2004): "Measuring health inequality using qualitative data". Journal of Health Economics, 23: 505-524. 2. Cerchiello, P., E. de Quarti, P. Giudici, C. Magni (2010): "Scorecard models to evaluate perceived quality of academic teaching". Statistica & Applicazioni, 8(2): 145-155. 3. Chakravarty, S. (2009): Inequality, Polarization and Poverty: Advances in Distributional Analysis. Springer-Verlag, New York. 4. Gini, C. (1912): Variabilità e Mutabilità: Contributo allo Studio delle Distribuzioni e delle Relazioni Statistiche. C. Cuppini, Bologna, 156 pages. 5. Giudici, O. & E. Raffinetti (2011): "Multivariate ranks-based concordance indexes", in Statistical Methods for the Analysis of Large Data-Sets, Springer-Verlag, New York. 6. Madden, D. (2010): "Ordinal and cardinal measures of health inequality: an empirical comparison". Health Economics Letters, 19: 243-250.
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7. Peñaloza, R. (2012): "Measurement of Education as an ordinal variable: towards a measure of human development based on the capability approach". Discussion Paper, Department of Economics, University of Brasilia (UnB). 8. Stevens, S. (1946): "On the theory of scales of measurement". Science, 103 (No. 2684): 677-680. 9. van Doorslaer, E. & A. Jones (2003): "Inequalities in self-reported health: validation of a new approach to measurement". Journal of Health Economics, 22: 61-87.
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