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A new class of inequality measures based on a ratio of L-statistics
by
Agostino Tarsitano
Summary.
The choice of a suitable weight function is a feasible and desirable direction for inequality
comparison. This paper proposes a new class of indices involving a ratio of two L-statistics in
which the scores at the numerator are proportionate to the expected values of order statistics
from a specified random variable. A large_scale Monte Carlo study has been conducted to assess
their sampling and asymptotic properties. The simulations show that the new class competes
well with the traditional measures of income inequality and industrial concentration. In particular, if the income-weighting scheme is “leftist”, the corresponding index has less bias, lower
relative variability, and faster convergence to the normal distribution.
Key Words: order statistics, economic inequality, simulations.
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1. Introduction
Linear function of order statistics (L-statistics) have wide applicability in the study of income
distribution both to investigate asymptotic behavior of inequality indices and analyze their relative sensitivity to transfers at different points of the distributions. In fact, Stigler (1974) states
his principal motivation for making research on L-statistics has been that “...these estimators
exhibit desirable robustness, particularly to heavy-tailed distributions or outlying distributions”.
Here we describe a class of indices constructed as a ratio of L-statistics. What is new is that the
weights are the expected values of a given probability distribution. The new class has great
flexibility, optimal sampling properties, and provides a fresh interpretation of some classical
measures of income inequality
The contents of the various sections are as follows. Section 2 reviews the general formulation of L-statistics highlighting their more salient features. In the third section, the results of
section 2 are applied to show that a ratio of L-statistics yields valid measures of economic
inequality. Section 4 reports the results of a simulation experiment designed to compare the
finite sampling properties of alternative measures of income inequality and to compare them
with the asymptotic results. Finally, there is a conclusion in which the major merits of the new
class are reviewed.
2. L-Statistics and inequality measures
Let {X1, X2,…, Xn} be a random sample of size n from a distribution function F with E(|X|)<∞
and let {X1:n, X2:n,…, Xn:n} denote the associated order statistics. For a fixed sequence of weight
functions Jn and Kn our interest will center on statistics of the type
n
i 
n −1 ∑ Jn 
Xi:n

n + 1
i
=
1
Ln =
;
n
i 

−1
n ∑ Kn
X
 n + 1  i: n
i =1
(1)
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where the denominator does not vanish. Furthermore we assume that the weight functions verify
the following constraints
n
i 
n −1 ∑ Jn 
= 0;
 n + 1
i =1
n
i 
n −1 ∑ Kn 
=1
 n + 1
i =1
(2)
The class Ln is particularly valuable in studies connected with income distributions because of
its homogeneity of degree zero; moreover, it is computationally simple, exhibits desirable robustness and is asymptotically efficient given the proper choice of the weight functions. In the
present paper Jn(t) allows weight to be put on all observations cutting off the results obtained for
trimmed L-statistics which appear inappropriate in income studies where all observations are
relevant for an accurate measurement of inequality. Table 1 presents some indices falling in
class (1). In particular, column J(t) reports real valued functions defined on (0,1) where Jn(t)
converges uniformly for J(t). The coefficients in the various numerators are obtained by setting
t=i/(n+1). Fn(x) is the proportion of observed values which are less than or equal to x; [x] is the
largest integer less than or equal x; ε(x) denotes the function which is one if x is true and zero
otherwise. The Pietra-Ricci is defined as one-half the relative mean deviation; the Gini/2, proposed by Gini in 1932, is given by the absolute mean deviation from the median divided by the
arithmetic mean x . The Goldie index is a measure of cumulative concentration as the area
below the Lorenz curve: G=(1+R)/2 where R is the Gini index. The Gini (R), Mehran (M) and
Piesh (P) are special cases of the measures proposed by Mehran (1976). Note that R=(2P+M)/3.
The standardized range is equal to (Xn:n-X1:n)/ x whereas the normalized range is (Xn:n-X1:n)/
Xn:n. Finally, the top 100p% represents the total worth of the richest 100p% of the population
and the Bowley’s interquartile measure is defined by (Q3-Q1/(Q3+Q1).
The multiplier in the denominators is Kn(t)≡1 for most entries with the exception of De
Vergottini (K n= ∑ i -1-1), normalized range with K n(t)= ε (t=1), ninth-first decile ratio:
Kn(t)=ε(t=0.1) and Bowley: Kn(t)=ε(t=0.75)+ ε(t=0.25).
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Table 1: inequality measures belonging to class Ln.
Index
Gini
Pietra − Ricci
Gini2
Mehran
Piesch
J n (t )
J (z)
 n + 1 (2t − 1)
2z − 1
 n − 1
Fn ( x ) − ε [t < Fn ( x )]
F(µ ) − ε [ z < F(µ )]
sgn(t − 0.5)
sgn( z − 0.5)
−1
2
1 − 3(1 − z )2
(n − 1) (2n + 1)(3t − 1) − 3(n + 1)t
2
2  2 nt + 2t − 1
3(n + 1)
− 3 − 4n2 / 4
 2n + 2 
3z 2 − 1 / 2
2
n − 3n + 1
(
)
(
n  n(1− t )
1 
1− ∑


n −1
j =1 n − j + 1
 n + 1 t
Goldie
 n 
Eltetö − Frigyes
[t > Fn ( x )]
n + 1
S. and N . Range 
(2t − 1)ε (t = 0 ∪ t = 1)
 n − 1
Top 100p%
ε (t > p )
Bowley
ε (t = 0.75) − ε (t = 0.25)
 nt 
1
De Vergottini
−1
∑
j =1 n − j + 1
Lorenz statistics ε (t ≤ p)
Ninth − first d .r. ε (t = 0.9)
Bonferroni
)
(
)
1 + Ln( z )
z
ε [ z > F(µ )]
(2 z − 1)ε ( z = 0 ∪ z = 1)
ε ( z > p)
ε ( z = 0.75) − ε ( z = 0.25)
−[1 + Ln(1 − z )]
ε ( z ≤ p)
ε ( z = 0.9)
The asymptotic behavior of L-statistics has been examined under various sets of assumptions.
Serfling (1980, Ch.8), Shorack and Wellner (1986, Ch. 19) give notable surveys of asymptotic
results on L-statistics and several different methodological approaches are outlined. Cicchitelli
(1976), Sendler (1979), Palmitesta et al (1999) extended the results to ratios of L-Statistics. The
selection of Jn and Kn is somewhat arbitrary and optimality depends on the aims of the investigators because each approach involves a trade-off between requirements on the weight functions and requirements on the distribution function. Stronger conditions on F allow more general weight functions and assumptions on the weight functions must be strengthened in order to
put less severe restrictions on F. For example, if zero or negligible weights are attached to
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extreme order statistics, quite general scores are permitted on remaining observations and the
distribution need not satisfy any smoothness condition (F may be discrete or may not have a
finite variance and the Xi may be non identically distributed). On the other hand, if nonzero
weights are placed on the extremes then Ln is sensitive to outliers and the distribution function
must satisfy higher moment conditions. Furthermore, if Jn is sectionally constant over (0,1) then
Ln fails the Pigou-Dalton principle of transfers (such is the case for the Pietra-Ricci, Gini/2,
Eltetö-Frygies, relative range, top p%, Lorenz statistics, ninth-first decile ratio).
The indices of Table 1 share the characteristic that, as n→∞, Jn(t)→J(t) uniformly for
0<t<1. If the denominator converges in probability to a finite positive constant, then the asymptotic properties of Ln can be established by ascertaining the convergence in distribution of its
numerator and then by applying the Slutsky’s theorem. Two important theorems obtain the following results
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a) Lim µn = µ = ∫ J (t ) F −1 (t )dt;
n →∞
b) Lim nσ
n →∞
c)
0
2
n
= σ 2 = 2∫
−1
_1
∫ J (t ) J (s)t (1 − s)dF (t )dF (s) < ∞
0 < t < s <1
J − µ d
n n
→ N (0,1) as n → ∞; if σ > 0

σ
 n 
(3)
Theorem S1. Stigler (1974).
Assumption 1)
i) Jn(t) is bounded and continuous.
ii) J(t) is continuous except possibly at a finite number of points at which F-1(t) is continuous.
iii) Jn(t)→J(t) uniformly in an open neighborhood of t as n→∞
iv) J (t2 ) − J (t1 ) ≤ M t2 − t1 , for t1 , t2 ∈ [λt,1 − λ (1 − t )], 0 < λ < 1, M > 0, α > 0.5
α
v) ∫J(t)dt=0
Assumption 2)
∫ F( x )[1 − F( x )]dx < ∞
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Theorem S2. Shorack -Wellner (1986, pp. 662-665).
Assumption 1)
For some M>0 and B(t ) = Mt − b1 (1 − t )− b2 fot 0<t<1 with max(b1,b2)<1.
i) |Jn(t)| ≤ B(t) .
ii) J(t) is continuous except possibly at a finite number of points at which F-1(t) is continuous.
iii) Jn(t)→J(t) uniformly in some small neighborhood of t as n→∞.
iv.1) |J(t)| ≤ B(t); iv.2) J’(t) exists on (0,1) and J' (t ) ≤ B(t )[t (1 − t )]−1 for 0 < t < 1
v) ∫J(t)dt=0.
Assumption 2)
F −1 (t ) < Mt − d1 (1 − t )− d2 for 0 < t < 1, max{(b1 + b2 ), ( d1 + d2 )} < 0.5
The conditions of Theorem S1 are fulfilled by the index of Gini, Goldie, and Piesch (the weight
function of the Mehran index fails condition 1.iv so that µn cannot be replaced by µ in 3c). Since
it is not required the existence of J’, Theorem S1 is also applicable to the Pietra-Ricci, Gini2,
relative range, Eltetö-Frygies, top p%, Lorenz statistics provided that J(t) and F-1(t) possess no
common discontinuities. Theorem S2 extends the results (3) to the Bonferroni whose weight
function is not bounded, but verifies conditions iv).
Assumption 2 is cryptic in both theorems although the more familiar requirement of
E(|X|2+d)<∞, d>0 suffices in order that F should verify this assumption both in S1 (Singh,
1981) and in S2 (Shorack, 1972). However, the moment condition is more demanding since it
may be difficult to satisfy for the skewed and highly leptokurtic distributions frequently encountered in the analysis of income data. Similar results can be generalized to indices expressed as
linear combinations of function of order statistics (Chernoff et al., 1967; Shorach, 1972; Sendler,
1972).
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3. A new class of inequality measures
Inequality comparisons are always subject to arbitrary specifications of weights. For the purpose of large-sample theory J should have a reasonably smooth behavior. It is convenient to
specify this by means of a probability distribution. Let G(z) be an absolutely continuous distribution function with associated density g(z) continuous and strictly positive for all z and assume
that E(Z)=m and σ(Z)=τ<∞. The model G and its parameters m and τ are known in advance.
The random variable Z can also take negative values, but the mean is always positive: m>0.
If Zi:n i=1,2,…,n is the i-th order statistics of a random sample {Z1, Z2,…, Zn} of size n
from G, then its expected value is E(Zi:n)=mi:n, i=1,2,…,n. Let {X1,X2,…,Xn} be a random sample
of incomes from F with E(X)=µ and 0<µ<∞ and consider the scatterplot with coordinates
 Xi:n − x   mi:n − m 
,

x   bnτ 2 


;

i = 1, 2,…, n;
(4)
Where bn>0 is a constant depending only on n and G. It the regression is linear, then the least
squares estimate of the slope
n
Qn =
bn ∑ ( min − m) Xi:n
i =1
n
∑ Xi:n
(5)
i =1
can measure the inequality in a sample distribution of incomes. Of course, Qn≥0 since both the
Xi:n’s and the mi:n’s are arranged in increasing order of numerical magnitude. Furthermore, Qn is
scale invariant with respect to X. By definition (5) obeys the Pigou-Dalton principle since the
effect of a neutral transfer of d>0 from the unit j to the unit i, with i<j, is bnd(mi:n-mj:n)<0. The
lower limit Qn=0 corresponds to the situation in which everyone receives the same income:
_
Xi:n= x . This condition is independent of the model G. However, Qn,→0 can also be interpreted as a departure from linearity which is indicative of lack of fit. The upper limit of Qn, as
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the number of individuals diverges while only one of them gets all the income is bn(mn:n-m).
Hence Qn can be normalized by bn=b=(m∞:∞-m)-1 provided that m∞:∞<∞. The denominator of
(5) converges in probability to µ=E(X). Thus, the behavior of Qn as n goes to infinity can be
assessed following the approach outlined in Section 2. In particular, to determine the limit weight
function corresponding to Jn(t) we can recall that m(n+1)t:n is continuous for 0<t<1 and converges to G-1(t) as n→∞. Thus, for n sufficiently large
[
Jn (t ) → J (t ) = b G −1(t ) − m
]
with bn → b > 0
(6)
Moreover, by construction G-1(t) and dG-1(t)/dt exist and are continuous on (0,1). If Jn(t) is
uniformly bounded, then the asymptotic properties of Qn follow from Theorem S1 provided that
J(t) satisfies the Lipschitz condition iv). If this is not the case, the properties (3) can still be
obtained under the less convenient assumptions of Theorem S2 on condition that
[
]
t b1 +1(1 − t )b2 +1 < M g G −1(t )
for 0 < t < 1
(7)
The rationale behind (4) and /5) is that to every G there exists a corresponding income weighting scheme which measures a different facet of economic inequality.
Example 1: Let G(z)=z be the uniform distribution over the unit interval. Then mi:n=i/(n+1)
and bn=2(n+1)/(n-1) yields the Gini index which can now be regarded as a measure of the
average feeling in a situation in which the real distribution of income is unknown and everyone
supposes that incomes are dropped at random over a fixed interval.
Example 2: Let G be the unit exponential distribution G(z)=1-e-z, z>0. It follows that
i
1
;
j =1 n − j + 1
mi:n = ∑
(
)
Jn (t ) = bn 1 − m( n +1)(1− t ):n ;
bn = 1 − n −1
(8)
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which yields the Bonferroni index. The log-rank scores appearing in Jn throw interesting sidelights on the index. For example, the i-th weight of Bonferroni represents the selection differential (in terms of mean units) obtained by comparing the mean of the bottom i ordered incomes
with the total mean.

−1
Example 3: G( z ) = 1 − k (1 − z )

[
1
1
α2
]
 α1
−1
 ,1 − k ≤ z ≤ 1, k = α1 B α1 , α 2 + 1

[(
α

  i  α 1  2 
X
∑ 1 − kn 1 −
 n + 1   i : n
i =1




Tn (α1, α 2 ) = 
; kn =
n
∑ Xi:n
)]
−1
n
i =1
(
The limit weight function J (t;α1, α 2 ) = 1 − k 1 − t α 1
)
α2
n
  i  α1 
∑ 1 −
 n + 1 
i =1

n
a2
→k
(9)
permits a broad range of shape. For ex-
ample, Tn will be relatively insensitive to outlying incomes if J assigns most of its mass to the
central part of the unit interval. The Mehran index is a special case of (10) for α1=1, α2=2. If
α1 =1 then T coincides with the generalization of the Gini index proposed by Donaldson and
Weymark (1980) and analyzed by Zidikis and Gastwirth (2002). For 0≤α1≤1 and α2≥1, T fulfills the Kolm principle of diminishing transfers that is, an inequality-reducing transfer at a
lower income level has not greater impact than at a higher income level. In addition, T satisfies
the assumptions of theorem S2 (or also S1 if J is bounded) provided that X possesses a finite
absolute moment of order r>2.
Example 4: The power-function distribution has G( z ) = z
1
α,
0 ≤ z ≤ 1, α > 0 . The correspond-
ing weight function is
J (t ; α ) =
(α + 1)tα − 1
α
(10)
If α=2 then (10) becomes the Piesch index with J(t)=(3t2-1)/2. Note that the Piesch index
violates the Kolm principle because J”(t;2)>0. The Kolm principle is always satisfied for 0<α<1.
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Example 5: For the generalized lambda distribution G −1(t ) = t λ1 − (1 − t )λ 2 , 0 ≤ t ≤ 1, index (5)
takes the form
n  B(λ , n + 1)

B(λ2 , n + 1)
1
−
− m Xi:n
∑
B(λ1, i )
B(λ2 , n + 1 − i )
(λ2 − λ1 )
i =1

; m=
λn (λ1, λ2 ) = 
n
(λ1 + 1)(λ2 + 1)
(1 − m) ∑ Xi:n
(11)
i =1
The parameters λ1 and λ2 determine the shape of the score functions. In particular, if λ1=λ2 then
Jn(t) and J(t) are symmetric about zero which means that a similar weight (of opposite sign) is
attached to ordered incomes at equal depths from the extremes. For 0 ≤ λ1≤1, 1≤λ2<∞, not only
G is a valid distribution, but the limit weight function J(t)=[G-1(t)-m]/(1-m) satisfies the Kolm
principle of diminishing transfers. Furthermore, if X has a finite absolute moment of order r>2,
then Theorem S1 becomes applicable to λn.
Example 6: Kadane (1971) showed that mi:n=fsi-1, i=1,2,…,n; s≥1,f>0 are the expectations of
order statistics for a sample from some positive random variable possessing a finite mean. Let
s=α1/(n+1), α>1, f=s. Then

 i
 α n +1 − m 
Xi:n
∑
α −m 
i =1


kn (α ) =  n
∑ Xi:n
n
n
with
m=
∑
i
n
+
α 1
i =1
n
→
α −1
Ln(α )
(12)
i =1
is a family of normalized inequality measures indexed by α. Specifically, kn(α) is “rightist” in
the sense that transfers at the upper end of the distribution are weighted more heavily than
transfers at the lower end. The larger is α the greater is concern for those on higher incomes.
Since J(t)=(αt-m)/(α-m) is convex, Theorem S1 applies to kn(α) for each α>1.
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4. A Monte Carlo experiment
To achieve valid conclusions in measuring income inequality on a sample basis one needs to
know the exact sampling distribution of the statistics used. However, the resulting distributions
tend to be quite complicated making it prohibitively difficult to determine standard errors, confidence intervals and test statistics. Moreover, the exact nature of F is usually unknown so that
one must appeal to asymptotic distribution theory and extensive computer simulations.
Simulated data
Giorgi and Pallini (1987, 1990), Palmitesta et al. (1999) performed various experiments to assess the asymptotic behavior of Gini, Mehran and Piesch index in samples drawn from a Burr/3
distribution. A question of both theoretical and practical interest is how the inequality measures
would vary their performance as the parent population undergoes change. In this sense our
Monte Carlo experiment involves two special cases of the generalized Beta (GB) distribution
introduced by McDonald and Xu (1985)
q −1
a

x 
a x 1 − (1 − c)  
 b 

h( x; a, b, c, p, q ) =
a p+q

x 

ap
b B( p, q )1 + c
 b  

ap −1
for 0 < x a <
ba
; 0 ≤ c < 1, p, q ≥ 0
1− c
(13)
The first GB model is the Burr 3 (alias Dagum type 1) with density given by a=−γ-1, b=1, c→1,
p=1, q=δ-1 (γ coincides with the Gini’s ratio when δ=1). For δγ>1 the density function is
twisted L-shaped. The Monte Carlo study involved simulations for the parameter values (δ, γ)
C1:(1,0.15), C2:(1,0.20), C3:(2,0.20), C4:(2,0.25), C5: (3,0.25), C6:(3,0.30), C7:(4,0.30). Figure 1a illustrates the Burr/3 densities employed in the simulation plan. The curves C1-C7 are
arranged according to an ascending level of inequality (the same ranking could be obtained for
a diminishing mean income).
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Figure 1a: Burr/3 distributions used in simulations.
The other model is the Beta density corresponding to (13) with a=1,b=1,c=0, p=δ, q=γ.
which has a compact support. In this case x should be interpreted as a normalized value x=(yminymax)/(ymin-ymax) where ymin and ymax denote the lower and upper limit (known in advance) of
income over the population considered. The closer the values of δ and γ are, the more symmetrical the density. The condition δ<γ suffices to ensure that the Beta density is positively
skewed; in particular, for δ<1, the density has a singularity at x=0 and it is L-shaped. Large
values of δ and γ imply that the Beta is better approximated by a Normal distribution. For
decreasing values of the ratio δ/γ the Beta model yields progressively narrowing densities characterized by an increasing probability of obtaining values near the extremes. The parameter
combinations chosen for the simulations are ( δ,γ)=C1:(6.946,7443), C2:(4.073,5.102),
C3:(2.586,3.746), C4:(1.719,2.843), C5:(1.179,2.069), C6: (0.851,1.711), C7:(0.633,1.354). The
corresponding curves are shown in Figure 1b where the densities are ordered for increasing
level of inequality (the same ordering could be obtained for a decreasing mean income or a
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decreasing ratio δ/γ).
Figure 1b. The Beta distributions used in simulations.
It can be easily checked that for the selection of parameters considered in Figure 1a and
Figure 1b, the distribution functions possess a finite moment of order 3. Moreover, the models
have the same value of the Gini index: C1=0.15000,C2=0.2000, C3=0.25373, C4=0.31103,
C5=0.36603, C6=0.42418, C7=0.47357. The Beta and Burr/3 have a high flexibility of shape
and provide realistic frameworks to test the inequality indices. Furthermore, the Beta is also a
good fit (Boswell and De Angelis, 1981) to another well known model: the Burr 12 (also known
as Singh-Maddala) included in (15) as GB(a>0;1,1,p>0,1).
Nine estimators were selected for inclusion in the study: Gini (R), Mehran (M), Piesch
(P), Bonferroni (B), Pietra-Ricci (D), Gini/2 (G2), Kadane k(1.1), λ(0.8,4), T(0.75,2). The last
two indices were chosen after a preliminary study (not reported here) involving 117 combinations of parameters: [0.1(0.1)0.9)x(1(0.25)4]. Both λ and T are “leftist” giving more weight to
transfers affecting lower income earners. This choice has the effort of giving less importance to
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observations in the upper tail than do other measures counteracting the fact that extreme incomes, though rare, have considerable effects on the sampling distributions. Moreover, since the
top order statistics do not contribute too much to λ and T, their convergence to the normal
distribution should be faster. In this sense, K represents essentially a planned failure.
The design consisted of generating L=25000 samples of various sizes: (n=200, 500, 1000,
2000, 4000, 8000) for the seven parameter combinations with a total 84 distinct experiments.
The exponential spacings method proposed by Lurie and Mason (1973) was first used to generate uniform order statistics Ui:n, i=1,2,…n and then the inversion method determined {Xi:n,
i=1,…,n} in a random sample from a Burr/3 distribution: Xi:n=[(Ui:n)-δ-1]-γ. To simulate the
order statistics in a random sample from a beta distribution the independent variates (X1, X2,…,
Xn) were generated according to the algorithms described in Rubinstein (1981, procedures Be-3
and Be-4) and sorted in natural order as {Xi:n, i=1,…,n} by using Hoare’s Quicksort.
Since the simulation plan analyzes sequences longer than 2x108, the fast multiple recursive generator (FMRG) proposed by Deng and Lin (2000) was applied to produce uniform
pseudorandom number on the unit interval (all the sequences were initialized from the same
seeds). The FMRG has a period of (262-1) but yields values lower than 231 so that repetitions in
the same sample are probable. This is in contrast to the hypothesis of order statistics from continuous distributions in which the probability of ties among the observations is zero. However,
because of the large discrepancy between the number of potential calls and the effective number
of values used in a single sample, this drawback should not interfere constructively with the
reliability of the simulations.
Two simple coefficients of performance have been considered for comparison
1  N θ − θ0 
Mb(θ ) =  ∑ i
;
N  i =1 θ 0 
with θ 0 = µ
−1
1
−1
∫ J (t )GB (t; a, b, c, p, q)dt
0
(14)
15
θ −θ 
∑ i

i =1 θ
N
N
Cv(θ ) =
2
N
with θ =
∑ θˆi
i =1
(15)
N
For θ ∈ (R,M,P,B,K,λ,D,G2,T). The relative mean bias Mb(θ) quantifies the average magnitude
of the estimator’s accuracy and Cv(θ) reflects the estimator’s variation from sample to sample.
The evaluation of θ0, that is the values of θ in the parent distribution, has been carried out by
standard numerical integration routines.
All the software has been written in Future Basic 3 running on a G4 (one processor)
computer using MacOS 9.2 operating system. Program codes as well as numerical results are
available from the author on request.
Assessing Performance
For reason of space only a selection of the results can be presented here. Table 2, in two parts,
shows the values of (14) and (15) for the nine measures of inequality. For comparison are also
included analogous results for the sample mean.
The general behavior of all the indices follows foreseeable lines. Consistent with asymptotic results, the absolute bias of all the indices becomes smaller and the coefficient of variation
diminishes as sample size increases although, the gain in accuracy and in low variance decays as
the number of sampling units grows up. The best performance has been obtained by Β and T
(the former tends to have lower relative bias and lower relative variation than the latter when the
sample size and/or the inequality increases) with Μ and λ close competitors. The indices G2 , R
and K are less than satisfactory. The last position was almost always occupied by P. The PietraRicci has good results under the Burr/3 model, but its performance decreases in quality under
the Beta model. A close (and surprising) resemblance between R and K is also apparent.
Table 2 reveals that the higher is the level of inequality expressed by θ the lower are Mb(θ)
and Cv(θ) for each θ (the estimated values of the population mean exhibit the reverse of this
condition). To some extent this would be expected since (14) and (15) depend on the size of the
16
Table 2a: results fo the Burr/3 model
δ
2
γ
0.20
n
200 EV
500
1000
2000
4000
8000
200 MB
500
1000
2000
4000
8000
200 CV
500
1000
2000
4000
8000
R
0.2536
0.2537
0.2537
0.2537
0.2538
0.2537
0.0152
0.0096
0.0068
0.0048
0.0034
0.0024
0.0600
0.0378
0.0269
0.0190
0.0133
0.0095
M
0.3595
0.3606
0.3609
0.3610
0.3612
0.3612
0.0189
0.0119
0.0085
0.0060
0.0042
0.0030
0.0527
0.0331
0.0235
0.0166
0.0117
0.0083
P
0.2003
0.2002
0.2001
0.2000
0.2001
0.2000
0.0138
0.0087
0.0062
0.0044
0.0031
0.0022
0.0691
0.0435
0.0310
0.0219
0.0154
0.0109
B
0.3689
0.3707
0.3712
0.3714
0.3717
0.3717
0.0182
0.0114
0.0081
0.0057
0.0040
0.0029
0.0494
0.0309
0.0219
0.0154
0.0109
0.0077
λ
0.3293
0.3314
0.3320
0.3323
0.3326
0.3326
0.0174
0.0110
0.0078
0.0055
0.0039
0.0028
0.0528
0.0331
0.0235
0.0166
0.0117
0.0083
K
0.2503
0.2504
0.2503
0.2503
0.2504
0.2503
0.0151
0.0095
0.0068
0.0048
0.0034
0.0024
0.0605
0.0380
0.0271
0.0191
0.0134
0.0095
T
0.3899
0.3918
0.3924
0.3926
0.3929
0.3929
0.0198
0.0125
0.0089
0.0063
0.0044
0.0031
0.0508
0.0318
0.0226
0.0160
0.0112
0.0080
PR
0.1766
0.1769
0.1769
0.1769
0.1770
0.1770
0.0106
0.0067
0.0047
0.0034
0.0024
0.0017
0.0601
0.0377
0.0268
0.0189
0.0133
0.0094
G2
0.3493
0.3502
0.3503
0.3504
0.3506
0.3506
0.0201
0.0126
0.0090
0.0064
0.0045
0.0032
0.0574
0.0360
0.0257
0.0181
0.0127
0.0090
µ
0.8526
0.8528
0.8526
0.8526
0.8527
0.8526
0.0294
0.0185
0.0130
0.0092
0.0066
0.0047
0.0345
0.0216
0.0153
0.0108
0.0077
0.0055
3
0.25
200 EV
500
1000
2000
4000
8000
200 MB
500
1000
2000
4000
8000
200 CV
500
1000
2000
4000
8000
0.3658
0.3660
0.3660
0.3659
0.3661
0.3660
0.0207
0.0131
0.0093
0.0066
0.0046
0.0033
0.0565
0.0357
0.0255
0.0180
0.0126
0.0090
0.5074
0.5088
0.5093
0.5094
0.5097
0.5096
0.0233
0.0147
0.0105
0.0074
0.0052
0.0037
0.0459
0.0288
0.0206
0.0145
0.0102
0.0072
0.2945
0.2944
0.2943
0.2941
0.2943
0.2942
0.0200
0.0126
0.0090
0.0064
0.0045
0.0032
0.0678
0.0429
0.0306
0.0217
0.0152
0.0108
0.5073
0.5088
0.5093
0.5095
0.5098
0.5097
0.0210
0.0132
0.0094
0.0066
0.0047
0.0033
0.0414
0.0259
0.0184
0.0130
0.0091
0.0065
0.4631
0.4659
0.4667
0.4671
0.4675
0.4675
0.0214
0.0135
0.0097
0.0068
0.0048
0.0034
0.0462
0.0290
0.0207
0.0146
0.0103
0.0073
0.3613
0.3615
0.3615
0.3614
0.3616
0.3615
0.0206
0.0130
0.0093
0.0066
0.0046
0.0033
0.0571
0.0360
0.0257
0.0182
0.0128
0.0091
0.5440
0.5464
0.5471
0.5474
0.5478
0.5478
0.0233
0.0147
0.0105
0.0074
0.0052
0.0037
0.0429
0.0269
0.0192
0.0135
0.0095
0.0067
0.2583
0.2587
0.2587
0.2588
0.2589
0.2589
0.0148
0.0093
0.0067
0.0047
0.0033
0.0023
0.0575
0.0361
0.0257
0.0182
0.0127
0.0090
0.5063
0.5075
0.5078
0.5080
0.5082
0.5082
0.0266
0.0167
0.0120
0.0084
0.0059
0.0042
0.0525
0.0329
0.0236
0.0166
0.0117
0.0083
0.6993
0.6995
0.6992
0.6992
0.6993
0.6993
0.0361
0.0227
0.0160
0.0113
0.0080
0.0057
0.0516
0.0324
0.0229
0.0162
0.0115
0.0082
4
0.30
200 EV
500
1000
2000
4000
8000
0.4731
0.4734
0.4735
0.4734
0.4736
0.4736
0.6357
0.6374
0.6379
0.6381
0.6384
0.6384
0.3912
0.3912
0.3911
0.3910
0.3912
0.3911
0.6207
0.6220
0.6224
0.6226
0.6228
0.6228
0.5790
0.5822
0.5832
0.5837
0.5841
0.5842
0.4679
0.4682
0.4683
0.4682
0.4684
0.4684
0.6722
0.6748
0.6757
0.6760
0.6764
0.6764
0.3403
0.3408
0.3409
0.3409
0.3411
0.3411
0.6531
0.6548
0.6552
0.6554
0.6557
0.6557
0.5786
0.5789
0.5786
0.5785
0.5787
0.5786
200 MB
500
1000
2000
4000
8000
0.0253
0.0161
0.0116
0.0082
0.0058
0.0041
0.0248
0.0157
0.0112
0.0079
0.0056
0.0040
0.0262
0.0167
0.0120
0.0085
0.0060
0.0043
0.0220
0.0139
0.0099
0.0070
0.0049
0.0035
0.0234
0.0148
0.0106
0.0075
0.0053
0.0038
0.0253
0.0161
0.0116
0.0082
0.0058
0.0041
0.0237
0.0150
0.0107
0.0076
0.0053
0.0038
0.0188
0.0118
0.0085
0.0060
0.0042
0.0030
0.0296
0.0187
0.0134
0.0095
0.0066
0.0047
0.0419
0.0263
0.0185
0.0131
0.0093
0.0066
200 CV
500
1000
2000
4000
8000
0.0535
0.0340
0.0244
0.0173
0.0122
0.0087
0.0390
0.0246
0.0176
0.0124
0.0087
0.0062
0.0670
0.0428
0.0308
0.0218
0.0154
0.0109
0.0354
0.0223
0.0160
0.0113
0.0079
0.0056
0.0403
0.0255
0.0182
0.0129
0.0091
0.0065
0.0542
0.0345
0.0247
0.0175
0.0124
0.0088
0.0353
0.0222
0.0158
0.0112
0.0079
0.0056
0.0552
0.0347
0.0248
0.0175
0.0123
0.0087
0.0454
0.0285
0.0204
0.0144
0.0101
0.0072
0.0724
0.0454
0.0320
0.0226
0.0161
0.0115
17
Table 2b: results fo the Beta model
δ
2.586
γ
3.746
n
200 EV
500
1000
2000
4000
8000
200 MB
500
1000
2000
4000
8000
200 CV
500
1000
2000
4000
8000
R
0.2540
0.2538
0.2538
0.2538
0.2537
0.2537
0.0383
0.0243
0.0171
0.0121
0.0085
0.0060
0.0480
0.0304
0.0214
0.0152
0.0107
0.0076
M
0.3726
0.3736
0.3739
0.3740
0.3741
0.3741
0.0367
0.0233
0.0164
0.0116
0.0082
0.0058
0.0461
0.0291
0.0205
0.0145
0.0103
0.0072
P
0.1942
0.1938
0.1936
0.1936
0.1935
0.1935
0.0411
0.0260
0.0183
0.0130
0.0091
0.0064
0.0513
0.0325
0.0229
0.0162
0.0115
0.0081
B
0.3807
0.3822
0.3827
0.3829
0.3831
0.3831
0.0339
0.0213
0.0150
0.0106
0.0075
0.0053
0.0423
0.0266
0.0187
0.0132
0.0094
0.0066
λ
0.3388
0.3407
0.3414
0.3417
0.3419
0.3419
0.0364
0.0229
0.0161
0.0113
0.0080
0.0056
0.0451
0.0285
0.0201
0.0142
0.0100
0.0071
K
0.2502
0.2500
0.2500
0.2500
0.2499
0.2499
0.0384
0.0244
0.0171
0.0121
0.0086
0.0060
0.0482
0.0305
0.0215
0.0152
0.0107
0.0076
T
0.4062
0.4081
0.4087
0.4090
0.4091
0.4092
0.0360
0.0227
0.0159
0.0113
0.0080
0.0056
0.0448
0.0283
0.0199
0.0141
0.0100
0.0070
PR
0.1828
0.1830
0.1830
0.1831
0.1831
0.1831
0.0422
0.0268
0.0188
0.0134
0.0094
0.0067
0.0531
0.0335
0.0236
0.0167
0.0118
0.0084
G2
0.3645
0.3652
0.3655
0.3656
0.3657
0.3657
0.0419
0.0266
0.0187
0.0133
0.0094
0.0067
0.0527
0.0333
0.0234
0.0166
0.0117
0.0083
µ
0.4083
0.4084
0.4084
0.4084
0.4084
0.4084
0.0250
0.0159
0.0112
0.0079
0.0056
0.0039
0.0313
0.0199
0.0140
0.0099
0.0070
0.0049
1.179
2.069
200 EV
500
1000
2000
4000
8000
200 MB
500
1000
2000
4000
8000
200 CV
500
1000
2000
4000
8000
0.3665
0.3662
0.3661
0.3661
0.3660
0.3660
0.0357
0.0227
0.0160
0.0113
0.0081
0.0057
0.0447
0.0284
0.0201
0.0142
0.0101
0.0071
0.5269
0.5281
0.5284
0.5286
0.5287
0.5288
0.0319
0.0203
0.0144
0.0103
0.0074
0.0054
0.0398
0.0253
0.0179
0.0126
0.0090
0.0063
0.2857
0.2851
0.2848
0.2847
0.2847
0.2847
0.0407
0.0259
0.0182
0.0130
0.0093
0.0067
0.0503
0.0320
0.0227
0.0160
0.0114
0.0080
0.5221
0.5232
0.5235
0.5237
0.5238
0.5239
0.0277
0.0180
0.0132
0.0101
0.0082
0.0072
0.0344
0.0217
0.0153
0.0108
0.0077
0.0054
0.4761
0.4785
0.4792
0.4796
0.4798
0.4800
0.0312
0.0195
0.0138
0.0097
0.0069
0.0049
0.0385
0.0244
0.0172
0.0122
0.0087
0.0061
0.3615
0.3611
0.3610
0.3609
0.3609
0.3609
0.0360
0.0228
0.0161
0.0114
0.0081
0.0057
0.0449
0.0286
0.0202
0.0143
0.0102
0.0071
0.5671
0.5692
0.5699
0.5703
0.5705
0.5706
0.0304
0.0192
0.0137
0.0099
0.0072
0.0054
0.0373
0.0236
0.0167
0.0118
0.0084
0.0059
0.2692
0.2695
0.2696
0.2696
0.2697
0.2697
0.0415
0.0271
0.0199
0.0151
0.0122
0.0106
0.0510
0.0325
0.0229
0.0162
0.0115
0.0081
0.5340
0.5350
0.5353
0.5355
0.5356
0.5357
0.0395
0.0257
0.0186
0.0140
0.0111
0.0094
0.0490
0.0312
0.0220
0.0155
0.0110
0.0078
0.3630
0.3630
0.3630
0.3630
0.3630
0.3630
0.0363
0.0231
0.0163
0.0115
0.0082
0.0057
0.0455
0.0289
0.0204
0.0144
0.0103
0.0072
0.633
1.354
200 EV
500
1000
2000
4000
8000
0.4743
0.4739
0.4737
0.4736
0.4736
0.4736
0.6625
0.6638
0.6642
0.6644
0.6645
0.6646
0.3796
0.3787
0.3784
0.3782
0.3781
0.3781
0.6375
0.6382
0.6384
0.6385
0.6386
0.6386
0.5948
0.5975
0.5984
0.5988
0.5990
0.5992
0.4684
0.4679
0.4677
0.4676
0.4675
0.4675
0.7021
0.7044
0.7052
0.7055
0.7057
0.7058
0.3581
0.3585
0.3586
0.3586
0.3587
0.3587
0.6989
0.7004
0.7008
0.7011
0.7012
0.7013
0.3185
0.3185
0.3185
0.3186
0.3186
0.3186
200 MB
500
1000
2000
4000
8000
200 CV
500
1000
2000
4000
8000
0.0329
0.0211
0.0149
0.0104
0.0074
0.0052
0.0411
0.0263
0.0186
0.0131
0.0093
0.0065
0.0262
0.0168
0.0118
0.0083
0.0059
0.0042
0.0329
0.0210
0.0148
0.0104
0.0074
0.0052
0.0395
0.0253
0.0178
0.0125
0.0089
0.0062
0.0492
0.0314
0.0223
0.0156
0.0111
0.0078
0.0223
0.0143
0.0100
0.0070
0.0050
0.0035
0.0280
0.0178
0.0126
0.0088
0.0063
0.0044
0.0259
0.0164
0.0115
0.0081
0.0058
0.0040
0.0320
0.0203
0.0144
0.0101
0.0072
0.0051
0.0332
0.0213
0.0150
0.0106
0.0075
0.0053
0.0415
0.0265
0.0188
0.0132
0.0094
0.0066
0.0238
0.0151
0.0106
0.0075
0.0053
0.0037
0.0296
0.0188
0.0133
0.0093
0.0067
0.0047
0.0386
0.0248
0.0175
0.0123
0.0088
0.0062
0.0484
0.0309
0.0219
0.0154
0.0110
0.0077
0.0337
0.0216
0.0152
0.0107
0.0076
0.0054
0.0422
0.0269
0.0191
0.0134
0.0096
0.0067
0.0478
0.0303
0.0215
0.0152
0.0107
0.0076
0.0600
0.0379
0.0270
0.0190
0.0135
0.0094
18
location parameter.
To obtain a mean-independent comparison the two quantities
7
Ak =
7
∑ Mbj [θ (γ ,δ )]
j =1
7
Vk =
,
∑ Cv j [θ (γ ,δ )]
j =1
k = 1, 2,…, 5
;
7
(16)
were computed and reported in Table 3.
Table 3: overall measure of bias and relarive variation
λ
n
R
M
P
B
K
T
PR
G2
µ
Burr/3 Bias
200
500
1000
2000
4000
8000
0.0472
0.0300
0.0215
0.0152
0.0107
0.0076
0.0394
0.0248
0.0177
0.0125
0.0088
0.0062
0.0557
0.0356
0.0255
0.0181
0.0127
0.0091
0.0369
0.0232
0.0165
0.0116
0.0082
0.0058
0.0404
0.0253
0.0180
0.0127
0.0089
0.0064
0.0476
0.0303
0.0217
0.0153
0.0108
0.0077
0.0376
0.0235
0.0167
0.0118
0.0083
0.0059
0.0478
0.0302
0.0215
0.0152
0.0107
0.0076
0.0435
0.0274
0.0195
0.0138
0.0097
0.0069
0.0357
0.0225
0.0159
0.0113
0.0080
0.0057
Var.
200
500
1000
2000
4000
8000
0.0600
0.0379
0.0270
0.0191
0.0134
0.0096
0.0495
0.0311
0.0222
0.0156
0.0110
0.0078
0.0711
0.0450
0.0321
0.0227
0.0160
0.0114
0.0463
0.0291
0.0206
0.0146
0.0103
0.0073
0.0502
0.0316
0.0225
0.0159
0.0112
0.0079
0.0605
0.0382
0.0273
0.0193
0.0136
0.0096
0.0469
0.0294
0.0209
0.0148
0.0104
0.0074
0.0604
0.0379
0.0270
0.0191
0.0134
0.0095
0.0547
0.0343
0.0245
0.0173
0.0121
0.0086
0.0452
0.0284
0.0200
0.0142
0.0101
0.0072
Bias
200
500
1000
2000
4000
8000
0.0368
0.0233
0.0164
0.0117
0.0082
0.0058
0.0338
0.0214
0.0151
0.0107
0.0076
0.0054
0.0407
0.0258
0.0182
0.0129
0.0091
0.0064
0.0308
0.0195
0.0138
0.0099
0.0071
0.0052
0.0334
0.0210
0.0148
0.0105
0.0074
0.0052
0.0369
0.0234
0.0165
0.0117
0.0083
0.0058
0.0326
0.0206
0.0145
0.0103
0.0073
0.0052
0.0413
0.0263
0.0187
0.0134
0.0097
0.0072
0.0397
0.0253
0.0179
0.0128
0.0092
0.0067
0.0310
0.0196
0.0139
0.0098
0.0070
0.0049
Var.
200
500
1000
2000
4000
8000
0.0460
0.0292
0.0206
0.0146
0.0103
0.0073
0.0424
0.0268
0.0189
0.0134
0.0095
0.0067
0.0507
0.0322
0.0227
0.0161
0.0114
0.0080
0.0384
0.0243
0.0171
0.0121
0.0085
0.0060
0.0414
0.0262
0.0185
0.0131
0.0092
0.0065
0.0462
0.0293
0.0207
0.0147
0.0104
0.0073
0.0405
0.0256
0.0181
0.0128
0.0091
0.0064
0.0517
0.0328
0.0231
0.0164
0.0116
0.0082
0.0497
0.0315
0.0223
0.0158
0.0111
0.0079
0.0388
0.0246
0.0174
0.0123
0.0087
0.0061
Beta
The findings in Table 3 confirm the tendencies already shown in Table 2. In fact, both the relative bias and the relative variation declines as the sample size goes to infinite. Furthermore, the
19
arrangement of the indices according to ascending values of Ak and Vk has (Β, T) in the first two
ranks, (λ, M) is the second best, and (P,PR,G2) have the last three positions, regardless the size
of the sample considered. The Gini index (and its twin, K) attains results good enough.
In scanning Table 2 and Table 3 one can note that the parent distribution has scarce or null
influence on the behavior of the inequality indices as far as mean bias and relative variation are
concerned except that Mb(θ) and Cv(θ) associated with the beta are slightly smaller than those
obtained from the Burr 3.
What is impressive about the results is that we could name several measures having similar optimum performance at least from a sample size n≥ 500. Thus, in order to choose among
competing indices we need some further criterion. As we have seen in Section 2, many L-statistics commonly employed in the measurement of income inequality are asymptotically normally
distributed. Formula (3), however, is useless for applications unless we can establish how fast
the convergence is taking place. Helmers (1977, 1981) determined a Berry-Esséen bound of
order n-0.5 by requiring that F must possess a finite third absolute moment and that J must satisfy
a Lipschitz condition of order one on (0,1). Helmers (1980) obtained a faster convergence of
order n-1 under additional constraints on F (e.g. a finite fourth moment) and that J is three times
differentiable with bounded derivatives.
When the population distribution function is fixed the rate of convergence depends only
on the limit weight function J which can be used to investigate the asymptotic properties of class
(5). Table 4a and 4b show the results of testing H0: Fn{[θ-E(θ)]/s(θ)}=N(0,1) against the alternative H1:Fn{[θ-E(θ)]/s(θ)} ≠ N(0,1) with the Chi-square goodness-of-fit test. The N repetitions
of θ have been grouped into k=40 disjoint intervals each having an expected number of frequencies of 1250. The class limits have been chosen such that all theoretical probabilities are
equal to 1/40 which gives equal importance to all partition of the Gaussian model. The column
R° indicates the level of inequality (as measured by the Gini index). The observed values of the
Chi-square greater than χ 02, 05 (37) = 52.1923 have been replaced by “*” meaning that H0 cannot
be accepted.
20
Table 4a: asymptotic normality under the Burr/3 model
λ
R°
n
R
M
P
B
K
T
0.150
200
.
*
* 47.38
*
*
*
0.200
200
.
*
*
*
*
*
*
0.253
200
. 45.44
*
*
*
* 40.01
0.311
200
.
*
* 46.45
*
*
*
0.366
200
. 34.00
* 28.93
*
*
*
0.424
200
.
*
*
*
*
* 44.40
0.473
200
.
*
*
*
*
* 45.60
PR
G2
µ
*
*
*
*
*
*
*
*
*
*
*
*
*
48.47
*
*
*
*
*
*
*
0.150
0.200
0.253
0.311
0.366
0.424
0.473
500
500
500
500
500
500
500
*
*
*
*
*
*
*
49.98
*
*
50.29
35.77
*
44.54
*
*
*
*
* 43.14
* 45.57
* 50.42
*
*
*
*
*
*
45.25
*
*
*
*
*
*
*
*
*
*
*
39.97
*
32.89
39.09
37.87
45.36
35.40
*
*
*
*
*
*
*
48.36
*
48.72
*
*
*
*
46.52
*
51.10
*
*
*
*
0.150
0.200
0.253
0.311
0.366
0.424
0.473
1000
1000
1000
1000
1000
1000
1000
*
*
*
*
*
*
*
*
*
43.49
*
36.60
49.37
36.95
*
*
*
*
* 45.29
* 45.70
* 55.87
*
*
*
*
*
*
*
*
38.71
*
*
*
*
*
*
*
*
*
*
* 39.05
* 49.40
* 40.14
*
*
51.24
*
48.28
*
*
*
*
*
47.63
45.32
51.73
47.36
25.17
30.36
28.44
35.91
30.25
*
*
0.150
0.200
0.253
0.311
0.366
0.424
0.473
2000
2000
2000
2000
2000
2000
2000
37.05
51.08
39.71
*
*
*
*
21.03
37.02
42.26
46.40
44.89
47.69
26.58
*
*
48.83
*
*
*
*
31.09
30.47
*
35.08
33.91
*
29.52
43.59
25.68
37.44
47.88
40.87
*
35.73
43.65
*
35.95
*
*
*
*
42.64
21.79
47.06
25.00
34.72
29.69
33.26
35.58
44.34
32.52
41.03
42.40
*
50.12
49.87
47.96
40.70
30.06
40.25
38.32
48.99
36.36
33.33
43.68
50.25
47.61
*
*
0.150
0.200
0.253
0.311
0.366
0.424
0.473
4000
4000
4000
4000
4000
4000
4000
45.94
*
41.31
*
48.50
*
*
31.30
46.04
43.04
43.24
42.65
38.08
38.34
46.63
*
*
*
*
*
*
23.88
35.75
44.77
36.29
45.30
51.00
51.83
32.73
41.25
*
34.85
46.22
*
48.90
37.08
*
43.16
*
48.86
*
*
40.58
32.60
29.65
46.27
33.89
34.31
31.53
30.04
46.89
43.54
*
41.33
*
48.85
32.86
*
27.58
49.46
30.31
35.49
24.54
48.80
43.28
39.17
44.73
41.01
41.00
*
0.150
0.200
0.253
0.311
0.366
0.424
0.473
8000
8000
8000
8000
8000
8000
8000
37.59
42.20
46.06
39.81
36.67
*
*
44.80
47.53
39.19
35.40
34.90
44.40
36.38
44.06
*
*
*
39.81
*
*
41.39
46.55
33.71
42.13
31.12
48.03
32.08
38.51
50.63
46.64
37.72
51.18
*
40.94
42.00
41.26
41.11
39.01
38.32
*
52.06
39.75
*
33.89
28.09
28.02
37.18
*
40.73
47.61
47.25
42.82
28.84
50.36
34.33
45.80
41.19
39.87
*
36.09
36.04
34.06
50.39
32.48
*
48.53
*
44.21
43.55
21
Table 4b: asymptotic
R°
n
R
0.150
200 36.89
0.200
200 40.65
0.253
200 32.99
0.311
200 37.15
0.366
200 46.88
0.424
200 43.83
0.473
200 30.20
normality under the beta model
λ
M
P
B
K
34.52 48.55 49.00 35.42 36.32
* 40.00 51.86 44.38 35.81
26.69
* 33.36 33.40 43.97
41.00 47.13 28.07 39.03 31.72
41.40 43.79 43.96 36.35 45.62
* 43.01
*
* 44.57
45.51 36.39
* 51.57 29.46
T
PR
G2
µ
52.22
46.43
42.59
38.95
42.42
*
*
38.22
41.17
*
39.97
*
37.34
36.97
41.23
24.25
*
35.74
*
36.56
40.79
34.44
38.62
26.42
28.49
37.33
44.72
47.93
0.150
0.200
0.253
0.311
0.366
0.424
0.473
500
500
500
500
500
500
500
43.36
*
36.39
40.03
35.16
25.88
*
49.04
32.91
27.27
44.63
38.28
*
51.62
*
37.96
27.85
29.59
39.02
28.92
46.70
50.55
35.13
33.56
49.16
42.58
48.19
*
33.09
31.45
25.96
*
39.47
*
*
42.39
*
31.74
39.46
32.69
28.38
*
48.74
36.51
27.36
41.23
40.81
44.03
*
43.77
34.50
29.20
37.02
37.80
31.07
35.73
44.61
36.52
43.16
40.01
31.39
46.44
*
35.66
42.67
*
31.88
23.55
43.66
42.93
0.150
0.200
0.253
0.311
0.366
0.424
0.473
1000
1000
1000
1000
1000
1000
1000
37.45
49.06
28.95
41.36
*
38.12
23.62
43.14
23.13
*
42.52
48.73
26.53
42.81
32.84
36.77
29.80
33.12
*
44.10
35.57
33.34
47.15
29.75
28.72
38.64
31.55
38.82
47.02
30.33
34.49
34.20
48.12
31.44
36.45
33.61
*
30.49
39.83
*
37.54
27.40
32.09
29.43
36.63
35.07
39.67
44.46
34.18
40.50
37.79
39.10
40.96
39.06
19.16
25.01
46.54
34.54
38.95
29.56
37.51
48.23
37.90
27.59
26.26
33.58
34.34
32.16
41.07
40.72
0.150
0.200
0.253
0.311
0.366
0.424
0.473
2000
2000
2000
2000
2000
2000
2000
47.56
20.73
25.15
50.31
*
27.83
40.58
35.81
36.02
45.61
33.42
23.20
25.94
23.14
29.39
28.48
34.85
30.59
37.00
33.11
45.63
27.36
25.15
39.25
37.43
42.53
30.44
24.06
31.27
32.19
41.12
32.15
29.56
32.60
17.05
50.04
26.88
25.85
*
*
26.08
35.77
45.03
39.08
44.89
26.04
24.08
34.55
26.44
41.13
*
37.39
27.76
40.20
23.64
30.02
44.53
46.86
34.27
46.36
39.63
36.25
36.31
29.52
28.25
23.53
40.00
51.90
47.72
27.39
0.150
0.200
0.253
0.311
0.366
0.424
0.473
4000
4000
4000
4000
4000
4000
4000
27.31
17.01
30.39
27.91
*
29.24
41.57
36.38
24.60
39.11
*
34.36
29.48
54.83
47.55
37.26
23.07
26.16
45.34
35.48
52.18
35.29
41.12
34.06
27.35
33.93
39.02
42.67
43.91
28.90
28.18
37.22
31.08
33.64
40.53
29.76
21.65
33.03
23.88
47.98
32.74
38.65
34.88
36.04
34.22
42.26
36.73
33.86
44.15
29.58
33.68
42.76
33.61
42.02
20.05
36.25
32.89
35.13
33.53
27.46
23.96
20.68
40.44
29.13
23.24
31.64
50.73
34.43
44.96
44.81
0.150
0.200
0.253
0.311
0.366
0.424
0.473
8000
8000
8000
8000
8000
8000
8000
27.86
26.48
23.30
30.03
51.40
27.22
41.45
29.53
47.83
30.56
27.75
38.07
42.73
38.20
49.52
36.39
39.71
32.99
42.87
26.91
42.18
26.26
36.94
30.84
40.65
48.29
32.10
35.65
29.34
32.46
29.86
36.39
48.87
38.92
38.28
33.18
26.56
26.17
29.34
*
35.25
41.00
28.09
36.39
37.06
27.77
36.26
32.96
14.18
40.82
26.50
24.88
48.58
43.69
24.84
38.22
37.61
42.99
32.74
46.39
39.51
37.16
45.63
32.93
34.26
29.04
33.96
21.49
42.13
34.93
22
It is generally observed that, with sporadic exceptions, the tested empirical distributions are
acceptably normal when the model underlying the sample values is the Beta distribution and
sample sizes of n≥2000 are available. The insufficient agreement between Fn(θ) and φ() for
smaller sized samples and low or moderate levels of inequality (curve C1-C3) should be ascribed
to the slow convergence of the standard errors in (3). The leftist measures λ, B, T, M perform
well throughout; in particular, the best values of χ c2 were attained by B and λ. The normal model
is also a plausible representation of the empirical distribution of PR and G2. The difference
between G and K tends to disappear when n becomes larger
Under the Burr/3 model the performances of most indices deteriorate appreciably and a
great variation is observed in the values of χ c2 across levels of inequality and across sample
sizes. Specifically, the N(0,1) was found to be appropriate for the empirical distribution of M, B
and λ for n≥2000. Also PR and G2 have a sampling distribution which shows an encouraging
degree of fitting to the Normal whereas the class frequencies observed for the indices R, K, and
P contradict the postulated model. For high levels of inequality (curves C4-C7) the convergence
to the normal is barely evident (which means that to achieve comparable numerical accuracy, it
is necessary to consider samples of a larger size than with curves C1-C3, but the indices have
similar properties. In particular, K is very unreliable (together with G) except for the largest
samples and P exhibits a test statistic systematically greater than the others. On the other hand,
the sampling behavior of λ, T, B, M appears to be quite close to what asymptotic theory suggests.
The ranking P<G<M for the speed of convergence found by Giorgi and Pallini (1990) is
not confirmed neither for the Beta nor for the Burr/3 model.
23
5. Concluding remarks
This article introduced the expected values of order statistics from a given random variable to
define the weights of a new class of indices for measuring the inequality in a distribution of
incomes. A Monte Carlo experiment has highlighted both the computational feasibility of these
alternative estimators and their strength and shortcomings. The general conclusions that can be
drawn, at least in the framework of the models considered in the experiment, are the following.
i) Although there appears to be a common thread among the nine indices of inequality considered their sampling properties are not invariant. Specifically, the approach to normality for λ, B,
T, and M is more rapid than for the other indices.
ii) The importance given to D and G2 in empirical work is not questionable on the ground of
their asymptotic performances which, on the contrary, seems to be very reasonable.
iii) The fashion of a “simple form” for the weight function is not defensible since the relatively
complex formulae for λ and T have often obtained better results than the more elegant and
computationally simple expressions of R, P, and M. However, λ, T, and K have the practical
inconvenient of relying on specific (and arbitrary) parametric assumptions.
iv) The Gini’s ratio does not appear to possess any special advantage. More room should be
allowed for leftist income-weighting schemes like the Bonferroni.
v) If one has a priori reason to believe that the inequality is high and that the underlying distribution is heavy-tailed there is little to choose among the indices although B or λ might be preferred
in practice because of their low bias and relative variation.
vi) The performance of an index is unlikely to be affected seriously by the underlying model
provided that the skewness of the potential distribution functions is at comparable level and the
size of the sample is 8000 or larger.
24
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