Measuring relative equality of concentration between different
income/wealth distributions
Quentin L Burrell
Isle of Man International Business School
The Nunnery
Old Castletown Road
Douglas
Isle of Man IM2 1QB
via United Kingdom
Email: [email protected]
Submitted to the Organising Committee of the “International Conference to
commemorate Gini and Lorenz”, University of Siena, Italy, 23-26 May, 2005.
Abstract
In a recent paper Burrell (2005a) introduced two new measures - both based on the Gini
mean difference - for measuring the similarity of concentration of productivity between
different informetric distributions. The first was derived from Dagum’s (1987) notion of
relative economic affluence (REA); the second – in some ways analogous to the
correlation coefficient – is a new approach giving the so-called co-concentration
coefficient (C-CC). Models and methods adopted in the field of informetrics – very
roughly, the “metrics” aspects of “information systems” - are very often based upon, or
have direct analogues with, or at very least are inspired by, ones from econometrics and it
is the purpose of this paper to suggest ways in which these new measures of similarity
could be useful in, for instance, studies of income distributions (a) between different
countries and (b) over different periods of time. The measures are illustrated using
exponential, Pareto, Weibull and Singh-Maddala distributions.
2
1. Introduction
One of the most intuitively reasonable requirements of a measure of concentration of
income/wealth within a population is that it should be invariant under scale
transformations – the degree of inequality should be the same if incomes are measured in
€ or US$. The situation is rather different if we are comparing inequalities between
populations. For instance, suppose we have two populations whose unit income
distributions are identical, but with one measured in €, the other in US$. Then their
“within population” concentration measures will be the same and yet there is clearly a
difference “between populations” since if both were expressed in the same units, we
would have two populations with different degrees of affluence.
Dagum (1987) sought to address this problem by introducing a measure of relative
economic affluence (REA) based upon the Gini mean difference, defined as the average
absolute difference between incomes of (randomly chosen) members of the two
populations. It turns out that the REAs of two income distributions are the same if and
only if the means of the two distributions are the same. The first of the new measures
proposed by Burrell (2005a) is a simple adaptation of the REA; the second incorporates
the Gini mean difference and the Gini coefficients of the two populations separately to
give a normalised measure – somewhat analogous to the correlation coefficient – lying
between 0 and 1, with the upper value being achieved if and only if the two population
income distributions (measured in the same units) are identical. Although we will talk in
terms of income distributions, the notions clearly extend to other fields.
3
2. Basic definitions
We imagine a population of individuals and let X denote the income of a randomly
chosen individual. Suppose that the distribution of X in the population is given by the
(absolutely) continuous probability density function (pdf) f X ( x ) defined on the nonnegative real line. (Restricting attention to the absolutely continuous case is done purely
for simplicity of presentation.)
Notation.
∞
(i) µ X = E[X] = mean of X =
∫ xf
X
( x )dx
0
∞
(ii) Tail distribution function of X = Φ X ( x ) = P(X ≥ x ) = ∫ f X ( y)dy
x
At this early stage, let us recall that for a continuous non-negative random variable X we
have
∞
E[X] = ∫ Φ X ( x )dx = µ X
(1)
0
(see, for instance, Stirzaker (1994, p238)). Without further comment we will always
assume that the mean is finite. There are many different approaches to the measurement
of concentration/inequality of (income) distributions and we refer the reader to Lambert
(2001) and Kleiber & Kotz (2003) for further discussion. See also Egghe & Rousseau
(1990). Notwithstanding the opinion that “… the overemphasis – bordering on obsession
– on the Gini coefficient as the measure of income inequality … is an unhealthy and
possibly misleading development” (Kleiber & Kotz, 2003, p30), the Gini coefficient is
the cornerstone of our analysis.
Definition 1. The Gini coefficient/index/ratio for X is defined formally as
4
γX =
E[| X 1 − X 2 |]
, where X1 and X2 are independent copies of X.
2µ X
(See the previous references, among many others, as well as the original presentation by
Gini (1914).)
The idea behind the definition is that we look at each pair of individuals within the
population in turn, find the absolute difference between their incomes and then average
out over all possible pairs. For purposes of calculation, the above definition is not very
convenient. Of the many others available (see Yitzhaki, 1998), the one that best suits our
purposes is given in the following:
Proposition 1.
∞
γX = 1−
∫Φ
X
( x ) 2 dx
0
µX
(2)
According to Kleiber & Kotz (2003), this is originally due to Arnold & Laguna (1977)
“at least in the non-Italian literature”. It was independently rediscovered by Dorfman
(1979) in economics and by Burrell (1991, 1992a) in informetrics.
The Gini coefficient is usually held to be one of the, if not the, best inequality measures
in that it obeys all seven of the “desirable” properties proposed by Dalton (1920) for such
a measure, see Dagum (1983). (But note the previously quoted counter opinion of Kleiber
& Kotz.) One of these properties is that it is invariant under scale, or is independent of
the unit of measurement. This is clearly almost a necessary property in measuring
inequality within a population. However, this property should not necessarily carry over
to comparative studies of inequality between populations if different units of income are
used. As an example, suppose that the income in two populations each follows an
5
exponential distribution, measured in the same units but with different means. Then
clearly the general level of income is greater in the population having the greater mean.
On the other hand, since the exponential is a scale-parameter family, the Gini coefficient
for each will be the same. (Indeed, the Gini coefficient for an exponential distribution is
½, see, e.g. Burrell (1992a).) Hence reliance on standard measurements of inequality such
as the Gini coefficient is inappropriate for measuring relative inequality between
populations. Instead, we follow Dagum (1987) to extend the Gini coefficient to become a
measure of the overall inequality of income between two populations.
Aside. Within the field of income/wealth distributions, the standard graphical
representation of inequality is via the Lorenz curve where one arranges individuals in
increasing order of income. In informetrics the focus is usually upon the “most
productive sources” (= “richest individuals”) in which case it is natural to arrange
individuals in decreasing order of “income” so that what is plotted is the tail distribution
function against the tail-moment distribution function. In the informetrics literature this
graphical representation is sometimes known as the Leimkuhler curve, see Burrell (1991,
1992a). The geometric relationship with the Lorenz curve is immediate, as is the
geometric interpretation of the Gini coefficient via the Leimkuhler curve, see Burrell
(1991).
Let us denote the income of a randomly chosen individual from each population by X, Y,
respectively. The mean and tail distribution function are defined as before and denoted
µ X , Φ X respectively, and similarly for the Y population. The idea behind the
6
construction of the Gini ratio between the two populations is exactly analogous to that of
the Gini coefficient for a single population, namely we look at pairs of individuals, but
now one from each population, find the absolute difference between their incomes,
measured in the same monetary units, and average this difference over all possible pairs.
Thus we have the following:
Definition 2. The Gini ratio between the two populations, denoted by G (X, Y) , is given
by G (X, Y) =
E[| X − Y |]
, where X, Y are independent.
µX + µY
(The numerator of the above expression is what is referred to as the Gini mean difference,
Dagum (1987).)
The analogy with Definition 1 is clear. Indeed, the Gini coefficient becomes a special
case since if X and Y have the same distribution then G (X, Y) = G (X, X) = γ X so that the
(comparative) Gini ratio becomes the (single population) Gini coefficient. Note also that
0 ≤ G (X, Y) < 1 , where we can get G → 1 as a limiting case, see Dagum (1987).
Again, for purposes of calculation, the defining formula for the Gini ratio is not very
convenient so that we make use of the following:
Theorem 1. With the above notation:
∞
G ( X, Y ) = 1 −
2∫ Φ X ( x )Φ Y ( x )dx
0
µX + µY
(3)
Proof. See Burrell (2005a), reproduced here in Appendix 1.
7
3. Normalized measures.
I: The relative concentration coefficient
In the paper in which he introduced the Gini ratio, Dagum (1987) proposed the notion of
relative economic affluence. We briefly recap Dagum’s approach, but modifying some of
his notation and terminology.
As described earlier, the Gini ratio is derived from the average absolute difference in
income between the X-population and the Y-population. Dagum’s relative measure
results from splitting this difference into two components: the average excess income of
members of the X-population over less affluent Y-sources, which we denote by p(X,Y),
or p1 in Dagum’s notation; and the average excess income of Y-sources over less affluent
X-sources, denoted p(Y,X), or Dagum’s d 1 . (These two components are in fact those
considered in the proof of Theorem 1 in the Appendix.)
In Burrell (2005a) we suggested a (relative) concentration coefficient between the X and
Y populations defined as D(X,Y) = p(X,Y)/p(Y,X) assuming, without loss of generality,
that E[X] ≤ E[Y]. This is just one minus the relative economic affluence defined by
Dagum (1987, Definition 7). An alternative representation of D(X, Y) is given in the
following:
Proposition 2. Assuming, without loss of generality, that E[Y] ≥ E[X]
D( X, Y ) =
µX − ∫ ΦXΦY
µY − ∫ ΦXΦY
=
∫ Φ (1 − Φ )
∫ (1 − Φ )Φ
X
Y
X
Y
Proof. See Appendix 2.
8
Corollary.
0 < D( X, Y ) ≤
µX
≤ 1 and D(X,Y) = 1 if and only if E[X] = E[Y].
µY
The proof is immediate, but see Burrell (2005a) for details.
Thus D(X, Y) is normalized in that it lies between 0 and 1 and the upper bound is
achieved if and only if the two means are the same. However, if the means are not the
same then the upper bound is given by the ratio of the means and this leads us to make
the following
Definition 3. The (modified) relative concentration coefficient is
D * ( X, Y ) =
1−
µY
D( X, Y ) =
µX
1−
(∫ Φ
(∫ Φ
X
)
)/ µ
ΦY / µX
XΦ Y
(4)
Y
where wlog E[Y] > E[X].
II: The coefficient of co-concentration
Note that although the Gini ratio already gives some sort of measure of the degree of
similarity/dissimilarity between two income distributions so far as their
concentration/inequality is concerned, it is not very informative on its own. One problem
is that the ratio is minimised when the two distributions are the same whereas we would
like a comparative measure to be maximised in this situation. This is easily resolved if we
make the following:
9
Definition 4.
∞
θX = 1− γ X =
∫Φ
X
( x ) 2 dx
0
(5)
µX
= coefficient of equality within the distribution of X
and
∞
H ( X, Y ) = 1 − G ( X, Y ) =
2 ∫ Φ X ( x )Φ Y ( x )dx
0
(6)
µX + µY
= equality ratio between the distributions of X and Y.
Note that both θ X and H(X,Y) lie between 0 and 1 and that θ X = 1 corresponds to the case
where all individuals have the same income. If all individuals across both populations
have equal income, then H(X,Y) = 1. Also, zero values can only be achieved via a
limiting process so that in practice both may be taken as being strictly greater than zero.
We can now construct a new measure that focuses on the degree of equality rather than
inequality of concentration between the two populations.
Definition 5. The coefficient of co-concentration or co-concentration coefficient
(C-CC) is given by Q(X, Y) =
=
H(X, Y)
(1 − G(X, Y))
=
(1 − γ X )(1 − γ Y )
θ X θY
2∫ Φ X Φ Y
µXµY
µX + µY
( ∫ Φ X )( ∫ Φ Y )
2
2
(7)
(8)
where the representation (8) follows from (7) together with (5) and (6).
The following shows that this is a standardised measure and is (joint) scale invariant:
10
Theorem 2.
(i) 0 < Q(X, Y) ≤ 1
(ii) Q(X,Y) = 1 if and only if the two distributions are the same.
(iii) Q(kX, kY) = Q(X, Y) for any constant k.
Proof. See Appendix 3.
Note. The equality of distributions required for the co-concentration coefficient to
achieve its upper bound is a much stronger condition than the equality of means required
in the case of the relative concentration coefficient and, we would argue, a more natural
requirement.
4. Some theoretical examples
In Burrell (2005a), simple examples for Exponential and Pareto distributions were
considered. Here we look at two rather more substantial cases.
(a) Weibull distribution
Suppose that X ~ Wei(α, β), i.e. X has a Weibull distribution with index α and scale
parameter β, so that the tail distribution function of X is given by
Φ X ( x ) = P(X > x ) = exp[−( x / β) α ]
 1
The mean of the distribution is well known to be µ X = E[X] = βΓ1 +  .
 α
This results from, or can be viewed as providing, the useful identity
11
∞
∫ exp[−(x / β)
0
α
1

]dx = βΓ1 + 
 α
(9)
[
]
[
]
Noting that Φ X ( x ) 2 = exp − 2( x / β) α = exp − ( x / λ ) α , where λ = β / 21 / α , we can use
the identity (9) to straight away write
∞
∫Φ
X
0
(
)
 1
( x ) 2 dx = β / 21 / α Γ1 +  .
 α
It then follows that the Gini coefficient, using (2), is γ X = 1 − 2 −1 / α .
This is, of course, a well-known result; see e.g. Kleiber& Kotz (2003, p177) for an
alternative derivation.
Similarly, if X ~ Wei(α, β1) and Y ~ Wei(α, β2) then

 1
1
Φ X ( x )Φ Y ( x ) = exp − ( x / β1 ) α − ( x / β 2 ) α = exp − x α  α + α
β2

 β1
[
where now λ =
(β
]
β1β 2
α
1
+ β2
∞
∫Φ
X
( x )Φ Y ( x )dx =
0
(β
α
1
+ β2
[
]
. It then follows from the identity (9) that
)
α 1/ α
β1β 2

 = exp − ( x / λ) α ,


)
α 1/ α
 1
Γ1 + 
 α
Of course, the above derivation can be much simplified if we recall that the Weibull
parameter β is a scale parameter and that the measures we are considering are (joint)
scale invariant, see Theorem 2(iii). For instance, notice how λ in the above depends only
on the ratio of the two β-values. Hence there is no loss in assuming that, say, β1 = β and
β2 = 1 throughout. With this assumption we find the equality ratio (Definition 4) as
∞
H ( X, Y ) =
2∫ Φ X ( x )Φ Y ( x )dx
0
µX + µY
β
 1
Γ1 + 
α 1/ α
2β
(1 + β )
 α =
=
(1 + β)(1 + β α )1 / α
 1
(1 + β)Γ1 + 
 α
2
12
Also, the product of the coefficients of equality is θ X θ Y = 2 −2 / α so that the coconcentration coefficient is
H ( X, Y )
21+1 / α β
2β  2

=
=
Q( X, Y ) =
α 1/ α
(1 + β)  1 + β α
(1 + β)(1 + β )
θX θY



1/ α
Note the particular case where α = 1 gives the C-CC for the exponential distribution as
Q( X, Y ) =
4β
, see Burrell (2005a).
(1 + β) 2
Remark. In practice we have found that the graph of Q(X, Y) is fairly flat and so we
recommend using its square for both illustrative and analytic purposes. (This is analogous
to using the R2 measure, or coefficient of variation, rather than the basic correlation
coefficient in correlation studies.)
The graph of Q2 as a function of the scaling ratio β is given in Figure 1 for various values
of α. Note how in each case the peak, where Q2 = 1, occurs when the scale ratio β = 1,
which corresponds to the two distributions coinciding.
******************* Insert Figure 1 about here ****************************
For the modified relative concentration coefficient, using the formula (4) and the results
above, routine algebra gives
(1 + β α )1 / α − 1
D * ( X, Y ) =
if β ≤ 1
(1 + β α )1 / α − β
=
(1 + β α )1 / α − β
if β > 1.
(1 + β α )1 / α − 1
13
This is illustrated in Figure 2 over the same range and for the same values of α. Note the
severely peaked nature of the graphs around β = 1 for α > 1. The differences between the
general forms of the graphs reflect the different emphases of the two measures in
assessing differences/similarities in concentrations.
************************* Insert Figure 2 about here *********************
(b) Singh-Maddala distribution
The Singh-Maddala (1975, 1976) distribution is a very flexible three-parameter income
distribution model. (See Kleiber & Kotz (2003) for a concise treatment of its various
attributes.) For our purposes, an attractive feature is the simple form of its tail distribution
function. Indeed, adopting the Kleiber & Kotz notation, if X ~ SM(a, b, q) then
  x a 
Φ X ( x ) = 1 +   
  b  
−q
and hence, as before,
−q
  x a 
bΓ(1 + 1 / a )Γ(q − 1 / a )
µ X = E[X] = ∫ Φ X ( x )dx = ∫ 1 +    dx =
Γ (q )
0
0
  b  
∞
∞
(10)
The final equality of (10) provides our “useful identity”. (Note that we have merely
quoted the expression for the mean of the distribution, see Kleiber & Kotz (2003, p201).)
Clearly, then
  x a 
(
x
)
dx
Φ
=
∫0 X
∫0 1 +  b  


∞
∞
2
−2 q
dx =
bΓ(1 + 1 / a )Γ(2q − 1 / a )
Γ ( 2q )
using (10), so that the coefficient of equality is
θX =
Γ(q)Γ(2q − 1 / a )
Γ(q − 1 / a )Γ(2q )
14
Also if X ~ SM(a, b, q) and Y ~ SM(a, b, p) then, using the same “identity” in (10)
  x a 
∫0 Φ X (x )Φ Y (x )dx = ∫0 1 +  b  


∞
∞
−(q+p)
dx =
bΓ(1 + 1 / a )Γ(q + p − 1 / a )
Γ (q + p )
and the equality ratio is then
∞
H ( X, Y ) =
2∫ Φ X ( x )Φ Y ( x )dx
0
µX + µY
2Γ ( q + p − 1 / a )  Γ ( q − 1 / a ) Γ ( p − 1 / a 


=
+
Γ (q + p)
Γ(p) 
 Γ (q )
−1
Note that both the coefficient of equality and the equality ratio do not involve the
parameter b, as should be expected since it is a scale parameter for the SM distribution.
From the above, it is clearly straightforward to derive a general expression for the C-CC
although it is rather cumbersome and not too enlightening. However, certain special cases
simplify matters greatly. For instance, if we take a = 1 then we find after a little algebra
that
Q( X, Y ) =
(q − 1)(2q − 1)(p − 1)(2p − 1)
(q + p − 1)(q + p − 2)
It is now straightforward to plot this as a function of p > 1 for any value of q > 1 (to
ensure finite means). This is illustrated in Figure 3, again using the squared form of the
function. Notice that here we get Q2 = 1 when p = q, again corresponding to the two
distributions coinciding.
***************************** Insert Figure 3 about here *********************
15
For the modified relative concentration coefficient, rather than give the general form let
us just stay with the special case where a = 1 as considered above. Routine calculation
leads, for a given value of q, to
D* = p/q if p < q, D* = q/p if p > q.
Thus the graph of D* is linear in p for p < q and is proportional to 1/p for p > q. See
Figure 4 and again compare with the corresponding Figure 3 for the Q2 measure. Our
view is that, once again, D*is rather harsh in distinguishing between the distributions,
which is not surprising given that it hinges on the mean rather than the overall
distribution.
****************** Insert Figure 4 about here *******************************
Remark. The reason that the Singh-Maddala example works so neatly in the calculation
of the C-C coefficient above is that the tail distribution function is of the form
Φ ( x ) = g( x ) α where α is the sole parameter of interest. This then gives the identity
∫Φ = ∫g
α
= µ(α) , say, and then ∫ Φ 2 = ∫ g 2 α = µ(2α) . Also if X and Y belong to the
same parametric family, with parameter values α, β respectively, then
∫Φ
X
Φ Y = ∫ g α +β = µ(α + β)
Substituting into (8) then gives the C-C coefficient as
Q( X , Y ) =
2µ(α + β)
µ(α )µ(β)
µ(α) + µ(β) µ(2α )µ(2β)
Use of this formula allows us to straight away write down the C-CC for such as the
exponential and Pareto distributions as well as the Singh-Maddala considered here.
16
5. Concluding remarks.
In this paper we have merely defined and given some simple examples of the Q2 and D*
measures and have not made any investigation of their statistical properties, although we
hope to have convinced the reader of the superiority of Q2 as the more subtle measure.
Nor have we considered empirical applications, though there are several possibilities,
including:
•
Within informetrics, examples of comparative studies over several data sets have
been given in Burrell (2005b), leading to a so-called co-concentration matrix. The
analogous treatments of income/wealth distributions for different countries or for
the same country during different years, maybe in “real” terms, are obvious
applications.
•
It would seem that it could also be used in investigative studies to assess the
effects of (proposed) taxation changes or degrees of inflation.
•
Again from informetrics, much use is made of time-dependent stochastic models
in which case the entire distributional shape changes as the length of the time
period increases. This means that concentration, as measured by the Gini index or
illustrated via the Lorenz curve, also changes, see Burrell (1992a, b). An
investigation of the behaviour of the Q2 measure in such circumstances is
currently in hand (Burrell, 2005c). Are there similar models appropriate for
income/wealth distributions? After all, if we double the period of observation, we
(roughly) double the average income so how does the income distribution
change?
17
Any conclusions regarding the efficacy of the Q2 measure must be tentative at this stage –
anything definitive requires further experience of its application and interpretation - but
we are hopeful that it might be a useful additional tool in comparative studies.
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20
Appendix
1. Proof of Theorem 1.
Although it is straightforward to give a general proof, either using Lebesgue-Stieltjes
integration or via the expectation operator (see Dagum, 1987), we prefer to use
elementary methods and restrict attention to the (absolutely) continuous case. Suppose
that X and Y are independent copies of the variables. Then
E[| X − Y |] = ∫∫ | x − y | f X ( x )f Y ( y)dxdy
Splitting the region of integration into {x>y} and {y>x}, the former yields, let us say
x

p(X,Y) = ∫∫ ( x − y)f X ( x )f Y ( y)dxdy = ∫  ∫ ( x − y)f Y ( y)dy f X ( x )dx
x>y
0 0

∞
x
∞
 x

= ∫  x ∫ f Y ( y)dy − ∫ yf Y ( y)dy f X ( x )dx
0  0
0

x



x

= ∫  xFY ( x ) −  yFY ( y) | 0 − ∫ FY ( y)dy f X ( x )dx

0 
0


∞
∞ x


= ∫  ∫ FY ( y)dy f x ( x )dx
0 0

x

= ∫  ∫ f X ( x )FY ( y)dy dx
0 0

∞
∞ ∞

= ∫  ∫ f X ( x )FY ( y)dx dy

0 
y
∞
∞
0
0
= ∫ Φ X ( y)FY ( y)dy = ∫ Φ X ( y)(1 − Φ Y ( y) )dy
Interchanging the roles of x and y in the above leads straight to
21
p(Y,X) =
∞
∞
0
0
∫∫ ( y − x )f X ( x )f Y ( y)dxdy = ∫ Φ Y ( y)FX ( y)dy = ∫ Φ Y ( y)(1 − Φ X ( y))dy
y>x
Adding these two expressions gives
E[| X − Y |] = p(X, Y ) + p(Y, X ) = ∫∫ | x − y | f X ( x )f Y ( y)dxdy
∞
∞
0
0
= ∫ Φ X ( y)(1 − Φ Y ( y) )dy + ∫ Φ Y ( y)(1 − Φ X ( y) )dy
∞
= ∫ (Φ X ( y) + Φ Y ( y) − 2Φ X ( y)Φ Y ( y) )dy
0
= ∫ Φ X + ∫ Φ Y − 2∫ Φ X Φ Y
= µ X + µ Y − 2∫ Φ X Φ Y
(A1)
and the result follows.
2. Proof of Proposition 5.
Eliminating p(X,Y) from (6) and (7), and rearranging, gives
p(Y, X) = (E[| X − Y |] + µ Y − µ X ) / 2
and similarly
p(X, Y) = (E[| X − Y |] − (µ Y − µ X )) / 2
Hence if µ Y > µ X ,
D( X, Y ) =
E[| X − Y |] − (µ Y − µ X )
E[| X − Y |] + (µ Y − µ X )
Now substitute from (A1) and then dividing numerator and denominator by µ X + µ Y
gives the result.
3. Proof of Theorem 2.
22
∞
(i) H(X, Y) = 1 − G (X, Y) =
 2 ΦXΦY
∫
2
H ( X, Y ) = 
 µ +µ
Y
 X
2∫ Φ X ( x )Φ Y ( x )dx
0
µX + µY
(
2
so that
)
2

 = 4 ∫ ΦXΦY

(µ X + µ Y ) 2

(A2)
Now
(∫ Φ
X
ΦY
) ≤ (∫ Φ )(∫ Φ )
2
2
2
X
(A3)
Y
by the Cauchy-Schwarz inequality, variants of which can be found in most introductory
texts on analysis, see also Stuart & Ord (1987, p. 65). Also
(µ X + µ Y ) 2 = (µ X − µ Y ) 2 + 4µ X µ Y ≥ 4µ X µ Y
(A4)
Combining (A3) and (A4) then gives, from (A2)
H ( X, Y ) =
2
(
4 ∫ ΦXΦY
) (∫ Φ )(∫ Φ )
2
(µ X + µ Y ) 2
2
≤
X
µXµY
2
Y
= θXθY
and the result follows.
(ii) For Q(X,Y) = 1, both of the above inequalities (A3) and (A4) must be equalities. For
the second, trivially the equality holds if and only if µ X = µ Y .
The Cauchy-Schwarz inequality reduces to an equality if and only if there is a constant c
such that, for all x, Φ X ( x ) = cΦ Y ( x ) . Then using the note at the end of the end of the
Proof of Theorem 1 above, this leads to
µ X = ∫ Φ X = c ∫ Φ Y = cµ Y . Having established the requirement that the two means must
be the same, this implies that c = 1 and hence the two distributions are the same.
23
Figure 1 : Q-squared for the Weibull distribution
1.2
1
Q-squared
0.8
alpha = 1/2
alpha = 2
alpha = 5
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Scale ratio, beta
Figure 2. D-star for the Weibull distribution
1.2
1
D-star
0.8
0.6
alpha = 1/2
alpha = 2
alpha = 5
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-0.2
Scale ratio, beta
24
Figure 3: Q-squared for the Singh-Maddala distribution
1.2
1
Q-squared
0.8
q = 1.5
q=2
0.6
q=3
0.4
0.2
0
1
2
3
4
5
6
7
8
9
Parameter value, p
Figure 4: D-star for the Singh-Maddala distribution
1.2
1
D-star
0.8
q = 1.5
0.6
q=2
q=3
0.4
0.2
0
1
2
3
4
5
6
7
8
9
Parameter value, p
25