# Estimates of Inequality Indices Based on Simple Random, Ranked Set, and Systematic Sampling

Authors: Pooja Bansal; Sangeeta Arora; Kalpana K. Mahajan
Journal: ISRN Probability and Statistics
 (2013)

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## Abstract

Gini index, Bonferroni index, and Absolute Lorenz index are some popular indices of inequality showing different features of inequality measurement. In general simple random sampling procedure is commonly used to estimate the inequality indices and their related inference. The key condition that the samples must be drawn via simple random sampling procedure though makes calculations much simpler but this assumption is often violated in practice as the data does not always yield simple random sample. Nonsimple random samples like Ranked set sampling or stratified sampling are gaining popularity for estimating these indices. The purpose of the present paper is to compare the efficiency of simple random sample estimates of inequality indices with their nonsimple random counterparts. Monte Carlo simulation technique is applied to get the results for some specific distributions.


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## Body

## 1. Introduction

Lorenz curve [1] and the associated Gini index [2] are one of the most popular and frequently used tools to measure income inequality. Certain other variants of Lorenz curve, namely, Generalized Lorenz curve [3], Absolute Lorenz curve [4], Bonferroni curve [5], and Comic curves [3, 6] and associated inequality indices, that is, Generalized Lorenz index, Absolute Lorenz index, Bonferroni index, and Comic index, are some of the popular alternatives to Gini index used to study certain specialized features of inequality. In practice, mostly the simple random sampling procedure is used to derive the statistical inference or obtaining the sample estimates of these inequality measures. The key condition that the samples must be drawn via simple random sampling procedure though makes calculations much simpler but this assumption is often violated in practice as the data does not always yield simple random sample. For example, in India the socioeconomic data collected by the National Sample Survey Organization (NSSO) is not drawn through simple random sampling but follows two-stage stratified sampling techniques. Also in the United States, commonly used income and earnings data, such as the Current Population Survey (CPS) and the Panel Study of Income Dynamics (PSID), are all multistage random samples, where simple random sampling is not the only method applied at each stage. One may quote stratified sampling, cluster sampling, multistage cluster sampling, and so forth, as alternatives to simple random sampling while estimating inequality indices [7].Besides, these available traditional sampling methods, the method of Ranked set sampling (RSS), have also gained popularity in the literature for estimating parameters like population mean and variance and its associated statistical inference. This RSS method of sampling is proved to be more efficient than simple random sampling [8]. Thus, the RSS technique may also be extended to obtain the sample estimates of these inequality measures, and it will be extended to compare the efficiency of RSS with simple random and other sampling estimates as applied to inequality measures. [9] have given the comparison of ranked set sample estimate of Gini index with simple random sampling for a nonstandard Lorenz curve equation using simulation technique and are not true in general as no standard distributions are considered.The purpose of the present work is to compare the nonsimple random estimates of the inequality measures with that of simple random estimates. In particular, we have applied two nonsimple random sampling techniques—Ranked set sampling and systematic sampling methods and have obtained the sample estimates of some popular inequality indices—Gini, Bonferroni, and Absolute Lorenz index via method of Monte Carlo simulation [10] and have compared their efficiencies with those of simple random estimates. Simulated results showing the relative efficiencies have been obtained for Pareto, Exponential, and power distribution. It is interesting to note that the simple random estimates have much less efficiency as compared to their nonsimple counterparts, that is, ranked set or systematic sample estimates. We have also compared the ranked set estimates with systematic sample estimates and found that systematic sample estimates are more efficient than the ranked set sample estimates.The present paper has been organized in the following manner. In Section2, we have applied the estimates of some popular inequality indices—Gini Index, Bonferroni Index, and Absolute Lorenz Index for three sampling procedures, namely, simple random sampling, ranked set sampling, and systematic sampling—and applied the method of Monte Carlo method of simulation to obtain the estimates. A brief introduction of the inequality indices and their estimation in case of three sampling techniques is given in this section. Simulation study has been performed in Section 3, where we have computed the efficiencies of these estimates for some specific distributions, namely, Exponential, Pareto, and Power distributions. The table showing the various comparison of Mean Square Errors (MSE) under simple random sampling, ranked set sampling, and systematic sampling is given at the end of Section 3 along with the conclusion.


## 2. Monte Carlo Estimates for Different Sampling Techniques

In this section, we have discussed three sampling techniques, namely, simple random sampling, ranked set sampling, and systematic sampling procedures and obtained the Monte Carlo estimates of the three indices Gini, Bonferroni, and Absolute Lorenz.


### 2.1. Simple Random Sampling Procedure

The sample mean Monte Carlo procedure for the inequality indices using simple random sampling is explained below.


#### 2.1.1. Gini Index

Gini index is defined as(1)G=1-2∫01L(p)dp(2)=1-2∫01L(p)f(p)·f(p)dp=1-2E[L(p)f(p)]=1-2I1,where(3)L(p)=1μ∫0pF-1(t)dt,0≤p≤1
is the Lorenz curve at p ordinates [1] and
(4)F-1(p)=inf{x:F(x)≥p},I1=E[L(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(5)f(p)={1if0<p<1,0o.w.⟹I1=E[L(p)].
An unbiased estimator of I is sample mean
(6)I^1=1n∑i=1nL(pi),(Reference[10]),
where n is the sample size.Therefore, unbiased estimator ofG is
(7)G^=1-2I^1.
Using (6)
(8)G^=1-2n∑i=1nL(pi),(9)Var(G^)=var[1-2n∑i=1nL(pi)]=4n2var[∑i=1nL(pi)]=4n{E[L(p)]2-I12},from(5)=4n[∫01(L(p))2dp-(1-G2)2],from(1).
Algorithm for computing simple random estimate for Gini index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeL(pi),i=1,…,N.(iv)
ComputeG^=1-(2/n)∑i=1nL(pi).


#### 2.1.2. Bonferroni Curve

Bonferroni index is defined as(10)B=1-∫01B(p)dp=1-∫01B(p)f(p)·f(p)dp=1-E[B(p)f(p)]=1-I2,
where
(11)B(p)=L(p)p=1μ·p∫0pF-1(t)dt,0≤p≤1
is the Bonferroni curve defined at p ordinates [5]
(12)I2=E[B(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(13)⟹I2=E[B(p)].
An unbiased estimator of I is sample mean
(14)I^2=1n∑i=1nB(pi),
where n is the sample size.Therefore, unbiased estimator ofB is
(15)B^=1-I^2.
Using (14),
(16)B^=1-1n∑i=1nB(pi),(17)Var(B^)=var[1-1n∑i=1nB(pi)]=1n2var[∑i=1nB(pi)]=1n{E[B(p)]2-I22},from(13)=1n[∫01(B(p))2dp-(1-B)2],from(10).
Algorithm for computing simple random estimate for Bonferroni index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeB(pi),i=1,…,N.(iv)
ComputeB^=1-(1/n)∑i=1nB(pi).


#### 2.1.3. Absolute Lorenz Index

Absolute Lorenz index is defined as(18)A=-∫01AL(p)dp=-∫01AL(p)f(p)·f(p)dp=-E[AL(p)f(p)]=-I3,
where
(19)AL(p)=-μ2+∫01xF(x)dF(x)
is the Absolute Lorenz curve defined at p ordinates [6]
(20)I3=E[AL(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(21)⟹I3=E[AL(p)].
An unbiased estimator of I is sample mean
(22)I^3=1n∑i=1nAL(pi),
where n is the sample size.Therefore, unbiased estimator ofA is
(23)A^=-I^3
using (22)
(24)A^=-1n∑i=1nAL(pi),(25)Var(A^)=var[-1n∑i=1nAL(pi)]=1n2var[∑i=1nAL(pi)]=1n{E[AL(p)]2-I32},from(21)=1n[∫01(AL(p))2dp-A2],from(18).
Algorithm for computing simple random estimate for Absolute Lorenz index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeAL(pi),i=1,…,N.(iv)
ComputeA^=-(1/n)∑i=1nAL(pi).Simple random sampling estimates of Gini, Bonferroni, and Absolute Lorenz indices using Monte Carlo approach are given in the Table 1.Table 1
Simple random estimates of inequality indices.
Inequality indices
Simple random sampling estimates
Gini index (G)
G
^
=
1
-
2
n
∑
i
=
1
n
L
(
p
i
)
Bonferroni index (B)
B
^
=
1
-
1
n
∑
i
=
1
n
B
(
p
i
)
Absolute Lorenz index (A)
A
^
=
-
1
n
∑
i
=
1
n
AL
(
p
i
)


### 2.2. Ranked Set Sampling Procedure

General, ranked setsampling procedure involves the following steps [6].(i) Select a simple random sample ofm units from the population and subject it to ordering on attribute of interestvia some ranking process, that is,
(26)X(1)11<⋯<X(m)11.This judgment ranking can result from a variety of mechanisms, including expert’s opinion, visual comparisons, or the use of easy-to-obtain auxiliary variables, but it cannot involve actual measurements of the attribute of interest on sample units.Once this judgement ranking ofm units in our initial random sample has been accomplished, the item judged to be smallest is included as the first item in the ranked set sample and the attribute of interest is formally measured on this unit and remaining (m-1) unmeasured units in the first random sample are not considered further. 								(27)X(1)11<⋯<X(m)11⟹X[1]11.This measurement is denoted byX[1]11, the smallest orderstatistics or the smallest judgement order item. It may or may not actually have the smallest attribute measurement among the m sampled units. Note that X(2)11<⋯<X(m)11 are not considered further in the selection of our ranked set sample. The sole purpose of (m-1) units is to help in selecting an item for measurement that represents the smaller attribute values in the population.(ii) Following selection ofX[1]11, a second independent random sample of size m is selected from the population and judgement ranking without formal measurement on the attribute of interest is made and the second smallest item of m units in second random sample is selected and is included in the ranked set sample for measurement of attribute of interest; that is,
(28)X(1)21<⋯<X(m)21⟹X[2]21.
The second measured observation is denoted by X[2]21.(iii) Similarly, from a third independent random sample, we select the unit judgement ranked to be the third smallest,X[3]31, for measurement and inclusion in the rankedset sample, that is,
(29)X(1)31<⋯<X(m)31⟹X[3]31.This process is continued until we have selected the unit judgement ranked to be largest ofm units in the mth random sample, denoted by X[m]m1; that is, X(1)m1<⋯<X(m)m1⇒X[m]m1 for inclusion in our ranked set sample.This entire process is referred to as a cycle and the number of observations in each random sample,m in our example, is called set size.Thus, to complete a single ranked set cycle, we need to judge rankm independent random samples of size m involving a total of m2 sample units in order to obtain m measured observations. These m observations represent a balanced ranked set sample with set size m.In practice, the sample sizem is kept small to ease the visual ranking, the RSS literature suggested m=2,3,4,5 or 6. Therefore, in order to obtain a ranked set sample with a desired total number of measured observation n=r*m, we repeat the entire cycle process r independent times, yielding the data
(30)X[1]ij<⋯<X[m]ij,i=1,…,m;j=1,…,r.Algorithm for obtaining the RSS estimate of Gini index via method of Monte Carlo simulation is given as follows.(a)
Generate an RSSU[m]ij,i=1,…,m;j=1,…,r of size n=m×r from U(0,1).(b)
Assignp(i)j=U[m]ij,i=1,…,m;j=1,…,r.(c)
ComputeL(p(i)j).(d)
Compute the ranked set sample estimate of Gini index as(31)G^=1-2∑j=1r∑i=1mL(p(i)j)n.Algorithm for obtaining the RSS estimate of Bonferroni index via method of Monte Carlo simulation is given as following.(a)
Generate a RSSU[m]ij,i=1,…,m;j=1,…,r of size n=m×r from U(0,1).(b)
Assignp(i)j=U[m]ij,i=1,…,m;j=1,…,r.(c)
ComputeB(p(i)j).(d)
Compute the ranked set sample estimate of Bonferroni index as follows.(32)B^=1-∑j=1r∑i=1mB(p(i)j)n.Algorithm for obtaining the RSS estimate of Absolute Lorenz index via method of Monte Carlo simulation is given as follows.(a)
Generate a RSSU[m]ij,i=1,…,m;j=1,…,r of size n=m×r from U(0,1).(b)
Assignp(i)j=U[m]ij,i=1,…,m;j=1,…,r.(c)
ComputeAL(p(i)j).(d)
Compute the ranked set sample estimate of Absolute Lorenz index as(33)A^=-∑j=1r∑i=1mAL(p(i)j)n.Ranked set sampling estimates of Gini, Bonferroni, and Absolute Lorenz indices using Monte Carlo approach are given in Table 2.Table 2
Ranked set sample estimates of inequality indices.
Inequality indices
Ranked set sampling estimates
Gini index (G)
G
^
=
1
-
2
∑
j
=
1
r
∑
i
=
1
m
L
(
p
(
i
)
j
)
n
Bonferroni index (B)
B
^
=
1
-
∑
j
=
1
r
∑
i
=
1
m
B
(
p
(
i
)
j
)
n
Absolute Lorenz index (A)
A
^
=
-
∑
j
=
1
r
∑
i
=
1
m
AL
(
p
(
i
)
j
)
n


### 2.3. Systematic Sampling Procedure

Systematic sampling procedure is particular case of stratified sampling procedure where the stratas are of equal size.


#### 2.3.1. Gini Index

Gini index is defined as(34)G=1-2∫01L(p)dp=1-2I1.
Its unbiased estimator in case of stratified sampling is given as
(35)⟹G^=1-2I^1,
where
(36)I1=∫01L(p)dp=∫DL(p)dp=∫DL(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(37)g(p)=L(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m; that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j where ∅ is null set.Define(38)I1i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I1=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI1i.
Define
(39)gi(p′)={g(p)p∈Di,0o.w.,I1i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(40)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
1
i can be estimated by
(41)I^1i=PiNi∑ki=1Nig(pki′),i=1,…,m.Therefore, integralI1 is estimated by
(42)I^1=∑i=1mI^1i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(43)⟹G^=1-2I^1,G^=1-2∑i=1mPiNi∑ki=1Nig(pki′),Var(I^1)=∑i=1mPi2Ni·Var[g(pi′)]=∑i=1mPi2σi2Ni,
where
(44)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I1i2Pi2,Var(G^)=Var(1-2I^1)=4Var(I^1)=4·∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Gini index as
(45)G^=1-2N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uk,k=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(46)G^=1-2N∑i=1m∑ki=1Nig(pki).


#### 2.3.2. Bonferroni Index

Similarly, Bonferroni index is defined as(47)B=1-∫01B(p)dp=1-I2.
Its stratified estimate is given as
(48)⟹B^=1-I^2,
where
(49)I2=∫01B(p)dp=∫DB(p)dp=∫DB(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(50)g(p)=B(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m, that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j, where ∅ is null set.Define(51)I2i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I2=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI2i.
Define
(52)gi(p′)={g(p)p∈Di,0o.w.,I2i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(53)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
2
i can be estimated by
(54)I^2i=PiNi∑ki=1Nig(pki′),i=1,…,m.
Therefore, integral I2 is estimated by
(55)I^2=∑i=1mI^2i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(56)⟹B^=1-I^2,B^=1-∑i=1mPiNi∑ki=1Nig(pki′),Var(I^2)=∑i=1mPi2Ni·Var[g(pki′)]=∑i=1mPi2σi2Ni,
where
(57)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I2i2Pi2,Var(G^)=Var(1-I^2)=Var(I^2)=∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Bonferroni index as
(58)B^=1-1N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uk,k=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(59)B^=1-1N∑i=1m∑ki=1Nig(pki).


#### 2.3.3. Absolute Lorenz Index

Absolute Lorenz index is defined as(60)A=-∫01AL(p)dp=-I3.
Its stratified estimate is given as
(61)⟹A^=-I^3,
where
(62)I3=∫01AL(p)dp=∫DAL(p)dp=∫DAL(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(63)g(p)=AL(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m; that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j where ∅ is null set.Define(64)I3i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I3=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI3i.
Define
(65)gi(p′)={g(p)p∈Di,0o.w.,I3i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(66)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
3
i can be estimated by
(67)I^3i=PiNi∑ki=1Nig(pki′),i=1,…,m.
Therefore, integral I3 is estimated by
(68)I^3=∑i=1mI^3i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(69)⟹A^=-I^3,A^=-∑i=1mPiNi∑ki=1Nig(pki′),Var(I^3)=∑i=1mPi2Ni·Var[g(pi′)]=∑i=1mPi2σi2Ni,
where
(70)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I3i2Pi2,Var(A^)=Var(-I^3)=Var(I^3)=∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Absolute Lorenz index as
(71)A^=-1N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uki,ki=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(72)A^=-1N∑i=1m∑ki=1Nig(pki).Systematic random sampling estimates of Gini, Bonferroni, and Absolute Lorenz indices using Monte Carlo approach are given in Table 3.Table 3
Systematic sample estimates of inequality indices.
Inequality indices
Systematic random sampling estimates
Gini index (G)
G
^
=
1
-
2
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)
Bonferroni index (B)
B
^
=
1
-
1
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)
Absolute Lorenz index (A)
A
^
=
-
1
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)


## 2.1. Simple Random Sampling Procedure

The sample mean Monte Carlo procedure for the inequality indices using simple random sampling is explained below.


### 2.1.1. Gini Index

Gini index is defined as(1)G=1-2∫01L(p)dp(2)=1-2∫01L(p)f(p)·f(p)dp=1-2E[L(p)f(p)]=1-2I1,where(3)L(p)=1μ∫0pF-1(t)dt,0≤p≤1
is the Lorenz curve at p ordinates [1] and
(4)F-1(p)=inf{x:F(x)≥p},I1=E[L(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(5)f(p)={1if0<p<1,0o.w.⟹I1=E[L(p)].
An unbiased estimator of I is sample mean
(6)I^1=1n∑i=1nL(pi),(Reference[10]),
where n is the sample size.Therefore, unbiased estimator ofG is
(7)G^=1-2I^1.
Using (6)
(8)G^=1-2n∑i=1nL(pi),(9)Var(G^)=var[1-2n∑i=1nL(pi)]=4n2var[∑i=1nL(pi)]=4n{E[L(p)]2-I12},from(5)=4n[∫01(L(p))2dp-(1-G2)2],from(1).
Algorithm for computing simple random estimate for Gini index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeL(pi),i=1,…,N.(iv)
ComputeG^=1-(2/n)∑i=1nL(pi).


### 2.1.2. Bonferroni Curve

Bonferroni index is defined as(10)B=1-∫01B(p)dp=1-∫01B(p)f(p)·f(p)dp=1-E[B(p)f(p)]=1-I2,
where
(11)B(p)=L(p)p=1μ·p∫0pF-1(t)dt,0≤p≤1
is the Bonferroni curve defined at p ordinates [5]
(12)I2=E[B(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(13)⟹I2=E[B(p)].
An unbiased estimator of I is sample mean
(14)I^2=1n∑i=1nB(pi),
where n is the sample size.Therefore, unbiased estimator ofB is
(15)B^=1-I^2.
Using (14),
(16)B^=1-1n∑i=1nB(pi),(17)Var(B^)=var[1-1n∑i=1nB(pi)]=1n2var[∑i=1nB(pi)]=1n{E[B(p)]2-I22},from(13)=1n[∫01(B(p))2dp-(1-B)2],from(10).
Algorithm for computing simple random estimate for Bonferroni index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeB(pi),i=1,…,N.(iv)
ComputeB^=1-(1/n)∑i=1nB(pi).


### 2.1.3. Absolute Lorenz Index

Absolute Lorenz index is defined as(18)A=-∫01AL(p)dp=-∫01AL(p)f(p)·f(p)dp=-E[AL(p)f(p)]=-I3,
where
(19)AL(p)=-μ2+∫01xF(x)dF(x)
is the Absolute Lorenz curve defined at p ordinates [6]
(20)I3=E[AL(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(21)⟹I3=E[AL(p)].
An unbiased estimator of I is sample mean
(22)I^3=1n∑i=1nAL(pi),
where n is the sample size.Therefore, unbiased estimator ofA is
(23)A^=-I^3
using (22)
(24)A^=-1n∑i=1nAL(pi),(25)Var(A^)=var[-1n∑i=1nAL(pi)]=1n2var[∑i=1nAL(pi)]=1n{E[AL(p)]2-I32},from(21)=1n[∫01(AL(p))2dp-A2],from(18).
Algorithm for computing simple random estimate for Absolute Lorenz index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeAL(pi),i=1,…,N.(iv)
ComputeA^=-(1/n)∑i=1nAL(pi).Simple random sampling estimates of Gini, Bonferroni, and Absolute Lorenz indices using Monte Carlo approach are given in the Table 1.Table 1
Simple random estimates of inequality indices.
Inequality indices
Simple random sampling estimates
Gini index (G)
G
^
=
1
-
2
n
∑
i
=
1
n
L
(
p
i
)
Bonferroni index (B)
B
^
=
1
-
1
n
∑
i
=
1
n
B
(
p
i
)
Absolute Lorenz index (A)
A
^
=
-
1
n
∑
i
=
1
n
AL
(
p
i
)


## 2.1.1. Gini Index

Gini index is defined as(1)G=1-2∫01L(p)dp(2)=1-2∫01L(p)f(p)·f(p)dp=1-2E[L(p)f(p)]=1-2I1,where(3)L(p)=1μ∫0pF-1(t)dt,0≤p≤1
is the Lorenz curve at p ordinates [1] and
(4)F-1(p)=inf{x:F(x)≥p},I1=E[L(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(5)f(p)={1if0<p<1,0o.w.⟹I1=E[L(p)].
An unbiased estimator of I is sample mean
(6)I^1=1n∑i=1nL(pi),(Reference[10]),
where n is the sample size.Therefore, unbiased estimator ofG is
(7)G^=1-2I^1.
Using (6)
(8)G^=1-2n∑i=1nL(pi),(9)Var(G^)=var[1-2n∑i=1nL(pi)]=4n2var[∑i=1nL(pi)]=4n{E[L(p)]2-I12},from(5)=4n[∫01(L(p))2dp-(1-G2)2],from(1).
Algorithm for computing simple random estimate for Gini index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeL(pi),i=1,…,N.(iv)
ComputeG^=1-(2/n)∑i=1nL(pi).


## 2.1.2. Bonferroni Curve

Bonferroni index is defined as(10)B=1-∫01B(p)dp=1-∫01B(p)f(p)·f(p)dp=1-E[B(p)f(p)]=1-I2,
where
(11)B(p)=L(p)p=1μ·p∫0pF-1(t)dt,0≤p≤1
is the Bonferroni curve defined at p ordinates [5]
(12)I2=E[B(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(13)⟹I2=E[B(p)].
An unbiased estimator of I is sample mean
(14)I^2=1n∑i=1nB(pi),
where n is the sample size.Therefore, unbiased estimator ofB is
(15)B^=1-I^2.
Using (14),
(16)B^=1-1n∑i=1nB(pi),(17)Var(B^)=var[1-1n∑i=1nB(pi)]=1n2var[∑i=1nB(pi)]=1n{E[B(p)]2-I22},from(13)=1n[∫01(B(p))2dp-(1-B)2],from(10).
Algorithm for computing simple random estimate for Bonferroni index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeB(pi),i=1,…,N.(iv)
ComputeB^=1-(1/n)∑i=1nB(pi).


## 2.1.3. Absolute Lorenz Index

Absolute Lorenz index is defined as(18)A=-∫01AL(p)dp=-∫01AL(p)f(p)·f(p)dp=-E[AL(p)f(p)]=-I3,
where
(19)AL(p)=-μ2+∫01xF(x)dF(x)
is the Absolute Lorenz curve defined at p ordinates [6]
(20)I3=E[AL(p)f(p)].
Since p is distributed uniformly over [0,1], that is, p~U(0,1),
(21)⟹I3=E[AL(p)].
An unbiased estimator of I is sample mean
(22)I^3=1n∑i=1nAL(pi),
where n is the sample size.Therefore, unbiased estimator ofA is
(23)A^=-I^3
using (22)
(24)A^=-1n∑i=1nAL(pi),(25)Var(A^)=var[-1n∑i=1nAL(pi)]=1n2var[∑i=1nAL(pi)]=1n{E[AL(p)]2-I32},from(21)=1n[∫01(AL(p))2dp-A2],from(18).
Algorithm for computing simple random estimate for Absolute Lorenz index is given as follows.(i)
Generate a sequence{Xi}i=1N of N random numbers from U(0,1).(ii)
Computepi=Xi,i=1,…,N.(iii)
ComputeAL(pi),i=1,…,N.(iv)
ComputeA^=-(1/n)∑i=1nAL(pi).Simple random sampling estimates of Gini, Bonferroni, and Absolute Lorenz indices using Monte Carlo approach are given in the Table 1.Table 1
Simple random estimates of inequality indices.
Inequality indices
Simple random sampling estimates
Gini index (G)
G
^
=
1
-
2
n
∑
i
=
1
n
L
(
p
i
)
Bonferroni index (B)
B
^
=
1
-
1
n
∑
i
=
1
n
B
(
p
i
)
Absolute Lorenz index (A)
A
^
=
-
1
n
∑
i
=
1
n
AL
(
p
i
)


## 2.2. Ranked Set Sampling Procedure

General, ranked setsampling procedure involves the following steps [6].(i) Select a simple random sample ofm units from the population and subject it to ordering on attribute of interestvia some ranking process, that is,
(26)X(1)11<⋯<X(m)11.This judgment ranking can result from a variety of mechanisms, including expert’s opinion, visual comparisons, or the use of easy-to-obtain auxiliary variables, but it cannot involve actual measurements of the attribute of interest on sample units.Once this judgement ranking ofm units in our initial random sample has been accomplished, the item judged to be smallest is included as the first item in the ranked set sample and the attribute of interest is formally measured on this unit and remaining (m-1) unmeasured units in the first random sample are not considered further. 								(27)X(1)11<⋯<X(m)11⟹X[1]11.This measurement is denoted byX[1]11, the smallest orderstatistics or the smallest judgement order item. It may or may not actually have the smallest attribute measurement among the m sampled units. Note that X(2)11<⋯<X(m)11 are not considered further in the selection of our ranked set sample. The sole purpose of (m-1) units is to help in selecting an item for measurement that represents the smaller attribute values in the population.(ii) Following selection ofX[1]11, a second independent random sample of size m is selected from the population and judgement ranking without formal measurement on the attribute of interest is made and the second smallest item of m units in second random sample is selected and is included in the ranked set sample for measurement of attribute of interest; that is,
(28)X(1)21<⋯<X(m)21⟹X[2]21.
The second measured observation is denoted by X[2]21.(iii) Similarly, from a third independent random sample, we select the unit judgement ranked to be the third smallest,X[3]31, for measurement and inclusion in the rankedset sample, that is,
(29)X(1)31<⋯<X(m)31⟹X[3]31.This process is continued until we have selected the unit judgement ranked to be largest ofm units in the mth random sample, denoted by X[m]m1; that is, X(1)m1<⋯<X(m)m1⇒X[m]m1 for inclusion in our ranked set sample.This entire process is referred to as a cycle and the number of observations in each random sample,m in our example, is called set size.Thus, to complete a single ranked set cycle, we need to judge rankm independent random samples of size m involving a total of m2 sample units in order to obtain m measured observations. These m observations represent a balanced ranked set sample with set size m.In practice, the sample sizem is kept small to ease the visual ranking, the RSS literature suggested m=2,3,4,5 or 6. Therefore, in order to obtain a ranked set sample with a desired total number of measured observation n=r*m, we repeat the entire cycle process r independent times, yielding the data
(30)X[1]ij<⋯<X[m]ij,i=1,…,m;j=1,…,r.Algorithm for obtaining the RSS estimate of Gini index via method of Monte Carlo simulation is given as follows.(a)
Generate an RSSU[m]ij,i=1,…,m;j=1,…,r of size n=m×r from U(0,1).(b)
Assignp(i)j=U[m]ij,i=1,…,m;j=1,…,r.(c)
ComputeL(p(i)j).(d)
Compute the ranked set sample estimate of Gini index as(31)G^=1-2∑j=1r∑i=1mL(p(i)j)n.Algorithm for obtaining the RSS estimate of Bonferroni index via method of Monte Carlo simulation is given as following.(a)
Generate a RSSU[m]ij,i=1,…,m;j=1,…,r of size n=m×r from U(0,1).(b)
Assignp(i)j=U[m]ij,i=1,…,m;j=1,…,r.(c)
ComputeB(p(i)j).(d)
Compute the ranked set sample estimate of Bonferroni index as follows.(32)B^=1-∑j=1r∑i=1mB(p(i)j)n.Algorithm for obtaining the RSS estimate of Absolute Lorenz index via method of Monte Carlo simulation is given as follows.(a)
Generate a RSSU[m]ij,i=1,…,m;j=1,…,r of size n=m×r from U(0,1).(b)
Assignp(i)j=U[m]ij,i=1,…,m;j=1,…,r.(c)
ComputeAL(p(i)j).(d)
Compute the ranked set sample estimate of Absolute Lorenz index as(33)A^=-∑j=1r∑i=1mAL(p(i)j)n.Ranked set sampling estimates of Gini, Bonferroni, and Absolute Lorenz indices using Monte Carlo approach are given in Table 2.Table 2
Ranked set sample estimates of inequality indices.
Inequality indices
Ranked set sampling estimates
Gini index (G)
G
^
=
1
-
2
∑
j
=
1
r
∑
i
=
1
m
L
(
p
(
i
)
j
)
n
Bonferroni index (B)
B
^
=
1
-
∑
j
=
1
r
∑
i
=
1
m
B
(
p
(
i
)
j
)
n
Absolute Lorenz index (A)
A
^
=
-
∑
j
=
1
r
∑
i
=
1
m
AL
(
p
(
i
)
j
)
n


## 2.3. Systematic Sampling Procedure

Systematic sampling procedure is particular case of stratified sampling procedure where the stratas are of equal size.


### 2.3.1. Gini Index

Gini index is defined as(34)G=1-2∫01L(p)dp=1-2I1.
Its unbiased estimator in case of stratified sampling is given as
(35)⟹G^=1-2I^1,
where
(36)I1=∫01L(p)dp=∫DL(p)dp=∫DL(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(37)g(p)=L(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m; that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j where ∅ is null set.Define(38)I1i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I1=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI1i.
Define
(39)gi(p′)={g(p)p∈Di,0o.w.,I1i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(40)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
1
i can be estimated by
(41)I^1i=PiNi∑ki=1Nig(pki′),i=1,…,m.Therefore, integralI1 is estimated by
(42)I^1=∑i=1mI^1i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(43)⟹G^=1-2I^1,G^=1-2∑i=1mPiNi∑ki=1Nig(pki′),Var(I^1)=∑i=1mPi2Ni·Var[g(pi′)]=∑i=1mPi2σi2Ni,
where
(44)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I1i2Pi2,Var(G^)=Var(1-2I^1)=4Var(I^1)=4·∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Gini index as
(45)G^=1-2N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uk,k=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(46)G^=1-2N∑i=1m∑ki=1Nig(pki).


### 2.3.2. Bonferroni Index

Similarly, Bonferroni index is defined as(47)B=1-∫01B(p)dp=1-I2.
Its stratified estimate is given as
(48)⟹B^=1-I^2,
where
(49)I2=∫01B(p)dp=∫DB(p)dp=∫DB(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(50)g(p)=B(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m, that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j, where ∅ is null set.Define(51)I2i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I2=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI2i.
Define
(52)gi(p′)={g(p)p∈Di,0o.w.,I2i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(53)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
2
i can be estimated by
(54)I^2i=PiNi∑ki=1Nig(pki′),i=1,…,m.
Therefore, integral I2 is estimated by
(55)I^2=∑i=1mI^2i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(56)⟹B^=1-I^2,B^=1-∑i=1mPiNi∑ki=1Nig(pki′),Var(I^2)=∑i=1mPi2Ni·Var[g(pki′)]=∑i=1mPi2σi2Ni,
where
(57)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I2i2Pi2,Var(G^)=Var(1-I^2)=Var(I^2)=∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Bonferroni index as
(58)B^=1-1N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uk,k=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(59)B^=1-1N∑i=1m∑ki=1Nig(pki).


### 2.3.3. Absolute Lorenz Index

Absolute Lorenz index is defined as(60)A=-∫01AL(p)dp=-I3.
Its stratified estimate is given as
(61)⟹A^=-I^3,
where
(62)I3=∫01AL(p)dp=∫DAL(p)dp=∫DAL(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(63)g(p)=AL(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m; that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j where ∅ is null set.Define(64)I3i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I3=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI3i.
Define
(65)gi(p′)={g(p)p∈Di,0o.w.,I3i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(66)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
3
i can be estimated by
(67)I^3i=PiNi∑ki=1Nig(pki′),i=1,…,m.
Therefore, integral I3 is estimated by
(68)I^3=∑i=1mI^3i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(69)⟹A^=-I^3,A^=-∑i=1mPiNi∑ki=1Nig(pki′),Var(I^3)=∑i=1mPi2Ni·Var[g(pi′)]=∑i=1mPi2σi2Ni,
where
(70)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I3i2Pi2,Var(A^)=Var(-I^3)=Var(I^3)=∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Absolute Lorenz index as
(71)A^=-1N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uki,ki=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(72)A^=-1N∑i=1m∑ki=1Nig(pki).Systematic random sampling estimates of Gini, Bonferroni, and Absolute Lorenz indices using Monte Carlo approach are given in Table 3.Table 3
Systematic sample estimates of inequality indices.
Inequality indices
Systematic random sampling estimates
Gini index (G)
G
^
=
1
-
2
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)
Bonferroni index (B)
B
^
=
1
-
1
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)
Absolute Lorenz index (A)
A
^
=
-
1
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)


## 2.3.1. Gini Index

Gini index is defined as(34)G=1-2∫01L(p)dp=1-2I1.
Its unbiased estimator in case of stratified sampling is given as
(35)⟹G^=1-2I^1,
where
(36)I1=∫01L(p)dp=∫DL(p)dp=∫DL(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(37)g(p)=L(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m; that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j where ∅ is null set.Define(38)I1i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I1=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI1i.
Define
(39)gi(p′)={g(p)p∈Di,0o.w.,I1i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(40)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
1
i can be estimated by
(41)I^1i=PiNi∑ki=1Nig(pki′),i=1,…,m.Therefore, integralI1 is estimated by
(42)I^1=∑i=1mI^1i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(43)⟹G^=1-2I^1,G^=1-2∑i=1mPiNi∑ki=1Nig(pki′),Var(I^1)=∑i=1mPi2Ni·Var[g(pi′)]=∑i=1mPi2σi2Ni,
where
(44)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I1i2Pi2,Var(G^)=Var(1-2I^1)=4Var(I^1)=4·∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Gini index as
(45)G^=1-2N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uk,k=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(46)G^=1-2N∑i=1m∑ki=1Nig(pki).


## 2.3.2. Bonferroni Index

Similarly, Bonferroni index is defined as(47)B=1-∫01B(p)dp=1-I2.
Its stratified estimate is given as
(48)⟹B^=1-I^2,
where
(49)I2=∫01B(p)dp=∫DB(p)dp=∫DB(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(50)g(p)=B(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m, that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j, where ∅ is null set.Define(51)I2i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I2=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI2i.
Define
(52)gi(p′)={g(p)p∈Di,0o.w.,I2i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(53)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
2
i can be estimated by
(54)I^2i=PiNi∑ki=1Nig(pki′),i=1,…,m.
Therefore, integral I2 is estimated by
(55)I^2=∑i=1mI^2i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(56)⟹B^=1-I^2,B^=1-∑i=1mPiNi∑ki=1Nig(pki′),Var(I^2)=∑i=1mPi2Ni·Var[g(pki′)]=∑i=1mPi2σi2Ni,
where
(57)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I2i2Pi2,Var(G^)=Var(1-I^2)=Var(I^2)=∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Bonferroni index as
(58)B^=1-1N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uk,k=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(59)B^=1-1N∑i=1m∑ki=1Nig(pki).


## 2.3.3. Absolute Lorenz Index

Absolute Lorenz index is defined as(60)A=-∫01AL(p)dp=-I3.
Its stratified estimate is given as
(61)⟹A^=-I^3,
where
(62)I3=∫01AL(p)dp=∫DAL(p)dp=∫DAL(p)f(p)·f(p)dp=∫Dg(p)·f(p)dp,
where
(63)g(p)=AL(p)f(p).Let regionD be divided into m disjoint subregions Di,i=1,…,m; that is, D=⋃i=1mDi and Dk∩Dj=∅,k≠j where ∅ is null set.Define(64)I3i=∫Dig(p)·f(p)dp,Pi=∫Dif(p)dp,∑i=1mPi=1,I3=∫Dg(p)·f(p)dp=∑i=1m∫Dig(p)·f(p)dp=∑i=1mI3i.
Define
(65)gi(p′)={g(p)p∈Di,0o.w.,I3i=∫Dig(p)·f(p)dp=∫DiPi·g(p)·f(p)Pidp=Pi∫Dig(p′)·f(p)Pidp=PiE[gi(p′)],
where
(66)∫Dif(p)Pidp=1.Random variablep′ is distributed according to f(p)/Pi on Di.I
3
i can be estimated by
(67)I^3i=PiNi∑ki=1Nig(pki′),i=1,…,m.
Therefore, integral I3 is estimated by
(68)I^3=∑i=1mI^3i=∑i=1mPiNi∑ki=1Nig(pki′),
where pi′ is distributed according to f(p)/Pi on Di(69)⟹A^=-I^3,A^=-∑i=1mPiNi∑ki=1Nig(pki′),Var(I^3)=∑i=1mPi2Ni·Var[g(pi′)]=∑i=1mPi2σi2Ni,
where
(70)σi2=Var[g(pi′)]=1Pi∫Dig2(p)·f(p)dp-I3i2Pi2,Var(A^)=Var(-I^3)=Var(I^3)=∑i=1mPi2σi2Ni.
In particular, if Pi=1/m and Ni=N/m, we get the systematic sampling estimate of Absolute Lorenz index as
(71)A^=-1N∑i=1m∑ki=1Nig(pki′).
Algorithm for computing systematic random sampling estimate is given as follows.(i)
Divide the range[0,1] of the cumulative distribution into m intervals each of width 1/m.(ii)
Generate{Uki,ki=1,…,N/m;i=1,…,m} from U(0,1).(iii)
ComputeYki=(i-1+Uki)/m;ki=1,…,N/m;i=1,…,m.(iv)
Computepki=F-1(Yki).(v)
Compute(72)A^=-1N∑i=1m∑ki=1Nig(pki).Systematic random sampling estimates of Gini, Bonferroni, and Absolute Lorenz indices using Monte Carlo approach are given in Table 3.Table 3
Systematic sample estimates of inequality indices.
Inequality indices
Systematic random sampling estimates
Gini index (G)
G
^
=
1
-
2
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)
Bonferroni index (B)
B
^
=
1
-
1
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)
Absolute Lorenz index (A)
A
^
=
-
1
N
∑
i
=
1
m
∑
k
i
=
1
N
i
g
(
p
K
i
′
)


## 3. Simulation Study

In this section, we illustrate the performance of the estimators for the three inequality indices based on the previously mentioned sampling procedures. 10000 random samples each of size 6, 8, 10, 12, 15, 20, 25, 30, 60, 80, 100, 120, 300, 400, 500, 600, 800, 1000, 1200, 1500, 2000, 2500, and 3000 were generated to compare the ranked set and systematic sampling estimates with simple random sampling estimates of the Gini index from exponential, Pareto(α=3), and Power (α=3); Bonferroni index from exponential,  Pareto (α=3), Power (α=3), and Absolute Lorenz index from exponential (3), Pareto (α=3), and rectangular (θ=3) distribution. The simulated ranked set samples with set size m=3,4,5,6 and number of repetitions, r=2,5,20,200, and 500 were chosen. The mean squared errors (MSE) and efficiencies were computed for the three procedures. Simulation results are shown in Tables 7, 8, 9, 10, 11, 12, 13, 14, and 15.The expressions for Gini index, Bonferroni, and absolute Lorenz index are given in Tables4, 5, and 6. The relation between Pareto, Power, and Rectangular distribution can be noted from the following remark.Table 4
Gini index (G) for given distributions.
Distribution
cdf
Lorenz curveL(p)
Gini index (G)
Exponential
1
-
e
-
λ
x
,
λ
>
0
,
x
≥
0
p
+
(
1
-
p
)
log
(
1
-
p
)
1
2
Pareto
1
-
(
x
x
0
)
-
α
,
α
>
1
,
x
≥
x
0
>
0
1
-
(
1
-
p
)
(
α
-
1
)
/
α
1
(
2
α
-
1
)
Power
x
α
,
α
>
1
,
0
≤
x
≤
1
p
(
α
+
1
)
/
α
1
(
2
α
+
1
)Table 5
Bonferroni index (B) for given distributions.
Distribution
cdf
Bonferroni curveB(p)
Bonferroni index (B)
Exponential
1
-
e
-
λ
x
,
λ
>
0
,
x
≥
0
p
+
(
1
-
p
)
log
(
1
-
p
)
p
π
2
6
-
1
Pareto
1
-
(
x
x
0
)
-
α
,
α
>
1
,
x
≥
x
0
>
0
1
-
(
1
-
p
)
(
α
-
1
)
/
α
p
ψ
(
2
)
-
ψ
(
(
α
-
1
)
α
)
Power
x
α
,
α
>
1
,
0
≤
x
≤
1
p
1
/
α
1
(
α
+
1
)
Whereψ(x) is the digamma function.Table 6
Absolute Lorenz index (A) for given distributions.
Distribution
cdf
Absolute Lorenz curve
Absolute Lorenz index (A)
Exponential
1
-
e
-
λ
x
,
λ
>
0
,
x
≥
0
(
1
-
p
)
log
(
1
-
p
)
λ
1
4
λ
Pareto
1
-
(
x
x
0
)
-
α
,
α
>
1
,
x
≥
x
0
>
0
α
x
0
(
1
-
p
)
[
1
-
(
1
-
p
)
-
1
/
α
]
(
α
-
1
)
α
x
0
2
(
α
-
1
)
(
2
α
-
1
)
Rectangular
x
-
a
θ
,
a
≤
x
≤
a
+
θ
,
θ
>
0
,
a
≥
0
θ
p
(
p
-
1
)
2
θ
12Table 7
Relative efficiency of Gini index for exponential distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
0.02498093
0.04457217
0.00910646
1.784248
4.894566
2.743210
4
0.01530442
0.03360818
0.004001166
2.195979
8.399597
3.824990
5
0.01046073
0.02692506
0.002174010
2.573918
12.384975
4.811721
6
0.007608975
0.02218654
0.001295014
2.915837
17.132278
5.875593
5
3
0.01002222
0.01747601
0.003524347
1.743726
4.958652
2.843710
4
0.00611229
0.01333912
0.001584625
2.182345
8.417840
3.857247
5
0.004159828
0.01100573
0.0008643665
2.645717
12.732712
4.812574
6
0.003042587
0.00911744
0.0005193771
2.996608
17.554567
5.858146
20
3
0.002467123
0.00450777
0.0009057722
1.827136
4.976715
2.723779
4
0.001525546
0.003313244
0.0004123536
2.171842
8.034958
3.699606
5
0.001024757
0.002690909
0.0002184094
2.625900
12.320482
4.691909
6
0.0007222134
0.002298709
0.0001300434
3.182866
17.676476
5.553634
100
3
0.0005033155
0.0008831595
0.0001777265
1.754684
4.969205
2.831967
4
0.0003078470
0.0006709811
8.14811
e
-
05
2.179593
8.234807
3.778140
5
0.0002099437
0.0005397776
4.331869
e
-
05
2.571059
12.460617
4.846492
6
0.0001524544
0.0004400118
2.622736
e
-
05
2.886186
16.776824
5.812800
200
3
0.0002471209
0.0004459992
8.781071
e
-
05
1.804781
5.079098
2.814246
4
0.000153431
0.0003294056
4.053465
e
-
05
2.14693
8.126519
3.785181
5
0.0001031339
0.0002670593
2.214463
e
-
05
2.589443
12.059777
4.657287
6
7.600503
e
-
05
0.0002178877
1.308801
e
-
05
2.866754
16.647886
5.807226
500
3
0.0001002982
0.0001764347
3.693063
e
-
05
1.759102
4.777463
2.715854
4
5.943145
e
-
05
0.0001320398
1.595697
e
-
05
2.221716
8.274741
3.724482
5
4.208938
e
-
05
0.0001076813
8.8025
e
-
06
2.558395
12.233036
4.781526
6
3.117683
e
-
05
9.065143
e
-
05
5.230148
e
-
06
2.907654
17.332479
5.960984Table 8
Relative efficiency of Gini index for pareto (α=3) distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
0.02332234
0.04529824
0.00589787
1.942268
7.680441
3.954367
4
0.01381014
0.03470658
0.002580012
2.513122
13.452100
5.352743
5
0.009193211
0.02779215
0.001352259
3.023116
20.552387
6.798410
6
0.00659181
0.02279101
0.0008011307
3.457473
28.448554
8.228133
5
3
0.009292055
0.01775528
0.002399531
1.910802
7.399479
3.872446
4
0.005570443
0.01392200
0.001050541
2.499263
13.252220
5.302452
5
0.003769394
0.01075287
0.0005584852
2.852678
19.253635
6.749318
6
0.00268615
0.009290222
0.0003206307
3.458554
28.974836
8.377707
20
3
0.002275199
0.004430962
0.0006094044
1.947505
7.270971
3.733480
4
0.001390079
0.003490160
0.0002649288
2.510764
13.173955
5.246991
5
0.0009236206
0.002648793
0.0001380824
2.867836
19.182698
6.688909
6
0.0006634941
0.002265676
8.109345
e
-
05
3.414765
27.939075
8.181846
100
3
0.000461317
0.00090545
0.0001197988
1.962750
7.558089
3.850765
4
0.0002732137
0.0006699247
5.275955
e
-
05
2.452017
12.697695
5.178469
5
0.0001887224
0.0005433437
2.791159
e
-
05
2.879063
19.466598
6.761435
6
0.0001309059
0.0004580686
1.628860
e
-
05
3.499220
28.122036
8.036658
200
3
0.0002253251
0.0004546124
6.028802
e
-
05
2.017584
7.540676
3.737477
4
0.0001392043
0.0003422813
2.601848
e
-
05
2.458842
13.155315
5.350209
5
9.250598
e
-
05
0.0002770614
1.426628
e
-
05
2.995065
19.420718
6.484240
6
6.674519
e
-
05
0.0002295553
7.895343
e
-
06
3.439278
29.074772
8.453742
500
3
9.473933
e
-
05
0.0001802806
2.443929
e
-
05
1.902912
7.376671
3.876517
4
5.606348
e
-
05
0.0001383186
1.074361
e
-
05
2.467178
12.874499
5.218309
5
3.699474
e
-
05
0.0001102943
5.493003
e
-
06
2.981350
20.079053
6.734884
6
2.701999
e
-
05
9.018714
e
-
05
3.301841
e
-
06
3.337793
27.314198
8.183310Table 9
Relative efficiency of Gini index for power (α=3) distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
0.03002883
0.05904002
0.006465061
1.966111
9.132168
4.644787
4
0.01837878
0.04518542
0.002769992
2.458565
16.312473
6.634958
5
0.01202134
0.03688425
0.001390412
3.068232
26.527569
8.645883
6
0.008600471
0.02950916
0.000838656
3.431109
35.186250
10.255064
5
3
0.01183113
0.02385245
0.00263158
2.016075
9.063927
4.495828
4
0.007109205
0.01767874
0.001112017
2.486739
15.897904
6.393072
5
0.004782672
0.01397249
0.0005790662
2.921482
24.129348
8.259284
6
0.003415356
0.01160659
0.0003343554
3.398354
34.713332
10.214748
20
3
0.003007737
0.005933085
0.0006660064
1.972607
8.908450
4.516078
4
0.001856484
0.004438954
0.0002774993
2.391054
15.996271
6.690049
5
0.001176903
0.003597289
0.0001420534
3.056573
25.323498
8.284934
6
0.0008649264
0.00293501
8.396861
e
-
05
3.393364
34.953657
10.300592
100
3
0.0006040928
0.001167984
0.0001285668
1.933451
9.084647
4.698669
4
0.0003599259
0.0008980578
5.426289
e
-
05
2.495118
16.550128
6.633003
5
0.0002358945
0.0007081798
2.842227
e
-
05
3.002103
24.916370
8.299636
6
0.000173568
0.0005906876
1.640179
e
-
05
3.403206
36.013606
10.582260
200
3
0.0002970297
0.0005835807
6.760029
e
-
05
1.964722
8.632814
4.393912
4
0.0001840207
0.0004393907
2.766382
e
-
05
2.387724
15.883226
6.652035
5
0.0001199216
0.0003552514
1.421686
e
-
05
2.962364
24.988035
8.435168
6
8.84773
e
-
05
0.0002953183
8.21482
e
-
06
3.337786
35.949455
10.770449
500
3
0.0001217598
0.0002407522
2.666487
e
-
05
1.977272
9.028816
4.566300
4
7.261201
e
-
05
0.0001784493
1.124082
e
-
05
2.457572
15.875114
6.459672
5
4.8425
e
-
05
0.0001436364
5.589331
e
-
06
2.966163
25.698317
8.663828
6
3.399731
e
-
05
0.0001184066
3.256773
e
-
06
3.482823
36.357032
10.438956Table 10
Relative efficiency of Bonferroni index for rectangular (θ=3) distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
0.06202481
0.1235527
0.01377850
1.991989
8.967065
4.501565
4
0.0381469
0.09533465
0.005916627
2.499146
16.113007
6.447407
5
0.02535770
0.07558111
0.002995199
2.980598
25.234086
8.466115
6
0.01791203
0.06181676
0.001748279
3.451131
35.358636
10.245521
5
3
0.02490822
0.05036193
0.005549141
2.0219
9.075626
4.488662
4
0.01489808
0.03685433
0.002358322
2.473765
15.627353
6.317237
5
0.00982462
0.03053955
0.001206127
3.108471
25.320344
8.145593
6
0.007220886
0.02487922
0.0006952421
3.445452
35.784973
10.386146
20
3
0.006252341
0.01245597
0.001364699
1.992209
9.127265
4.581480
4
0.003685856
0.009230949
0.0005877362
2.504425
15.705939
6.271276
5
0.002550649
0.007501943
0.0002866664
2.941190
26.169593
8.897621
6
0.001808095
0.00619635
0.0001755058
3.427005
35.305671
10.302195
100
3
0.001249233
0.002446855
0.0002789078
1.958686
8.772989
4.479018
4
0.0007403018
0.001914464
0.0001151496
2.586059
16.625885
6.429044
5
0.000504336
0.001465721
6.01177
e
-
05
2.906238
24.380856
8.389143
6
0.0003529966
0.001248915
3.414360
e
-
05
3.538037
36.578305
10.338588
200
3
0.0006307183
0.001248089
0.0001376957
1.978838
9.064110
4.580523
4
0.0003722346
0.0009302265
5.962156
e
-
05
2.499033
15.602183
6.243289
5
0.0002493340
0.0007560325
2.945604
e
-
05
3.032207
25.666468
8.464614
6
0.0001767726
0.000618749
1.745216
e
-
05
3.500253
35.454007
10.128981
500
3
0.0002467436
0.0005050265
5.529099
e
-
05
2.046767
9.133975
4.462637
4
0.0001511701
0.0003785248
2.356102
e
-
05
2.503966
16.065722
6.416110
5
9.98555
e
-
05
0.0002998347
1.208355
e
-
05
3.002686
24.813461
8.263755
6
7.085566
e
-
05
0.000252477
6.991167
e
-
06
3.563258
36.113713
10.135026Table 11
Relative efficiency of Bonferroni index for exponential distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
0.005826023
0.01085295
0.001687301
1.86284
6.432136
3.452865
4
0.003477376
0.008182712
0.0007416163
2.353128
11.033619
4.688915
5
0.002229820
0.006484168
0.0003876997
2.907933
16.724718
5.751410
6
0.001690472
0.005454195
0.0002310937
3.226432
23.601660
7.315093
5
3
0.002319801
0.004423554
0.0006583515
1.906868
6.719137
3.523651
4
0.001337874
0.003238717
0.0002952573
2.420794
10.969134
4.531214
5
0.0009042652
0.002668140
0.0001607564
2.950617
16.597411
5.625065
6
0.0006640723
0.002151227
9.39234
e
-
05
3.239447
22.904058
7.070361
20
3
0.0005728049
0.001073308
0.0001686894
1.873776
6.362629
3.395619
4
0.0003477821
0.0008003468
7.638552
e
-
05
2.301288
10.477729
4.552985
5
0.0002317084
0.0006640685
4.120043
e
-
05
2.865966
16.117999
5.623932
6
0.0001682733
0.0005685107
2.490111
e
-
05
3.378496
22.830737
6.757663
100
3
0.0001145552
0.0002185333
3.457197
e
-
05
1.907668
6.321112
3.313528
4
7.047152
e
-
05
0.0001665346
1.634645
e
-
05
2.363148
10.187814
4.311121
5
4.76686
e
-
05
0.0001299972
9.897842
e
-
06
2.727103
13.133893
4.816060
6
3.368466
e
-
05
0.0001080593
6.489773
e
-
06
3.20796
16.650706
5.190422
200
3
5.866765
e
-
05
0.0001113661
1.836847
e
-
05
1.898254
6.062895
3.193932
4
3.637083
e
-
05
8.33706
e
-
05
9.291632
e
-
06
2.292238
8.972654
3.914364
5
2.444185
e
-
05
6.776085
e
-
05
5.651263
e
-
06
2.772329
11.990390
4.325024
6
1.846277
e
-
05
5.761897
e
-
05
4.05105
e
-
06
3.120819
14.223219
4.557527
500
3
2.478413
e
-
05
4.502048
e
-
05
8.473576
e
-
06
1.816505
5.313044
2.924873
4
1.551794
e
-
05
3.491700
e
-
05
4.598167
e
-
06
2.250106
7.593678
3.374810
5
1.109970
e
-
05
2.797701
e
-
05
3.230694
e
-
06
2.520520
8.659752
3.435701
6
8.396832
e
-
06
2.362577
e
-
05
2.635701
e
-
06
2.813653
8.963752
3.185806Table 12
Relative efficiency of Bonferroni index for pareto (α=3)  distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
0.0004945458
0.0009228851
0.0001830762
1.866127
5.040989
2.701311
4
0.0003013004
0.0007029501
8.527298
e
-
05
2.333054
8.243527
3.533363
5
0.0002011407
0.0005511385
4.790515
e
-
05
2.740064
11.504786
4.198728
6
0.0001507043
0.0004636178
2.906871
e
-
05
3.076342
15.949032
5.184417
5
3
0.0001979686
0.0003752962
7.295216
e
-
05
1.895736
5.144415
2.713677
4
0.0001211198
0.0002751505
3.327257
e
-
05
2.271723
8.269590
3.640230
5
8.27896
e
-
05
0.0002221785
1.902978
e
-
05
2.683652
11.675306
4.350528
6
5.93931
e
-
05
0.0001855068
1.188047
e
-
05
3.123373
15.614433
4.999221
20
3
4.902086
e
-
05
9.181063
e
-
05
1.821531
e
-
05
1.872889
5.040300
2.691190
4
3.027866
e
-
05
7.056706
e
-
05
8.84171
e
-
06
2.330587
7.981155
3.424525
5
2.076647
e
-
05
5.352124
e
-
05
4.825012
e
-
06
2.577291
11.092457
4.303921
6
1.512937
e
-
05
4.587691
e
-
05
2.918024
e
-
06
3.032308
15.721910
5.184800
100
3
9.94706
e
-
06
1.860485
e
-
05
3.674845
e
-
06
1.870386
5.062758
2.706797
4
6.097801
e
-
06
1.383726
e
-
05
1.745582
e
-
06
2.269221
7.927018
3.493277
5
3.963538
e
-
06
1.106979
e
-
05
9.846638
e
-
07
2.792905
11.242203
4.025270
6
2.950961
e
-
06
9.162954
e
-
06
6.031919
e
-
07
3.105075
15.190778
4.892242
200
3
4.966537
e
-
06
9.244004
e
-
06
1.813586
e
-
06
1.861257
5.097086
2.738518
4
2.995594
e
-
06
6.901277
e
-
06
8.687236
e
-
07
2.303809
7.944157
3.448271
5
2.069782
e
-
06
5.520984
e
-
06
4.823448
e
-
07
2.667423
11.446136
4.291084
6
1.466736
e
-
06
4.654902
e
-
06
2.952955
e
-
07
3.173647
15.763539
4.967011
500
3
1.986694
e
-
06
3.73913
e
-
06
7.290523
e
-
07
1.882087
5.128754
2.725036
4
1.206677
e
-
06
2.715055
e
-
06
3.486712
e
-
07
2.250026
7.786863
3.460788
5
8.166757
e
-
07
2.197725
e
-
06
1.930648
e
-
07
2.691062
11.383354
4.230060
6
5.903014
e
-
07
1.834876
e
-
06
1.190148
e
-
07
3.108371
15.417209
4.959899Table 13
Relative efficiency of absolute Lorenz index for pareto (α=3, x0=200) distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
20.70694
28.60494
10.57485
1.381418
2.704997
1.958131
4
13.5872
21.77554
5.390407
1.602651
4.039684
2.520626
5
9.815137
17.37329
3.06463
1.770051
5.668968
3.202715
6
7.333079
14.3627
1.935963
1.958618
7.418892
3.787820
5
3
8.490195
11.32587
4.191259
1.333994
2.702260
2.025691
4
5.523156
8.495127
2.138295
1.538093
3.972851
2.582972
5
3.839579
6.901874
1.21959
1.79756
5.659176
3.148254
6
2.848677
5.687296
0.7867597
1.996469
7.228759
3.620771
20
3
2.050789
2.857925
1.037954
1.393574
2.753422
1.975800
4
1.377661
2.149993
0.521389
1.560612
4.123587
2.642290
5
0.9706388
1.714577
0.301321
1.766442
5.690201
3.221278
6
0.7303821
1.413525
0.1940954
1.935322
7.282630
3.763006
100
3
0.4120384
0.5679051
0.2109544
1.378282
2.692075
1.953211
4
0.270179
0.4210431
0.1080168
1.558386
3.897941
2.501268
5
0.1956007
0.3382184
0.06211348
1.729127
5.445169
3.149086
6
0.1465505
0.2881033
0.03872126
1.965897
7.440442
3.784755
200
3
0.2098861
0.2912766
0.1067612
1.387785
2.728300
1.965940
4
0.1389657
0.2102593
0.05198104
1.513031
4.044923
2.673392
5
0.1003489
0.1699745
0.03067446
1.693835
5.541239
3.271415
6
0.07082068
0.1436694
0.01940810
2.028637
7.402548
3.649027
500
3
0.08326179
0.1144187
0.04192018
1.374205
2.729442
1.986198
4
0.0548825
0.08290751
0.02124204
1.510636
3.902992
2.583674
5
0.03854217
0.0686116
0.01242014
1.780170
5.524221
3.103199
6
0.02853415
0.05816632
0.007684862
2.038481
7.568948
3.713033Table 14
Relative efficiency of absolute Lorenz index for rectangular (θ=3) distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
0.0018881
0.002099699
0.00105751
1.11207
1.985512
1.785420
4
0.001239755
0.001571081
0.0004786396
1.267251
3.282388
2.590164
5
0.0008749366
0.001232462
0.0002404251
1.408630
5.126179
3.639123
6
0.0006767988
0.001043956
0.0001412633
1.542491
7.390143
4.791045
5
3
0.0007426104
0.000812753
0.0004213969
1.094454
1.928711
1.762259
4
0.0005023406
0.0006275718
0.0001826594
1.249295
3.435749
2.750149
5
0.0003583338
0.0005023294
9.73136
e
-
05
1.401848
5.161965
3.682258
6
0.0002722183
0.0004232076
5.552897
e
-
05
1.554662
7.621384
4.902275
20
3
0.0001900359
0.0002072712
0.0001066924
1.090695
1.942699
1.781157
4
0.0001270530
0.0001569948
4.505728
e
-
05
1.235664
3.484338
2.819811
5
9.21853
e
-
05
0.0001217164
2.413549
e
-
05
1.320345
5.043047
3.819492
6
6.948416
e
-
05
0.0001039845
1.432389
e
-
05
1.496521
7.259515
4.850928
100
3
3.739423
e
-
05
4.054800
e
-
05
2.125268
e
-
05
1.084339
1.907901
1.759507
4
2.485619
e
-
05
3.087114
e
-
05
9.368584
e
-
06
1.241990
3.295177
2.653143
5
1.785318
e
-
05
2.452328
e
-
05
4.819807
e
-
06
1.373609
5.088021
3.704128
6
1.317393
e
-
05
2.022185
e
-
05
2.820976
e
-
06
1.534990
7.168388
4.669990
200
3
1.865214
e
-
05
2.0668
e
-
05
1.051127
e
-
05
1.108077
1.966270
1.774490
4
1.261365
e
-
05
1.541276
e
-
05
4.629849
e
-
06
1.221911
3.328998
2.724419
5
9.145499
e
-
06
1.250770
e
-
05
2.469242
e
-
06
1.367635
5.065401
3.703768
6
6.61361
e
-
06
1.047759
e
-
05
1.394760
e
-
06
1.584246
7.512110
4.741755
500
3
7.542962
e
-
06
8.347197
e
-
06
4.224176
e
-
06
1.106621
1.976053
1.785665
4
5.027219
e
-
06
6.440567
e
-
06
1.884210
e
-
06
1.281139
3.418179
2.668078
5
3.551602
e
-
06
4.918697
e
-
06
9.686214
e
-
07
1.384923
5.078039
3.666657
6
2.65354
e
-
06
4.210024
e
-
06
5.679265
e
-
07
1.586569
7.412973
4.672330Table 15
Relative efficiency of absolute Lorenz index for exponential (λ=3) distribution.
r
m
MSE (RSS)
MSE (SRS)
MSE (SYS)
E
(
1
)
E
(
2
)
E
(
3
)
2
3
0.0001772438
0.0002139371
9.98993
e
-
05
1.207022
2.141528
1.774225
4
0.0001160937
0.001597166
4.811047
e
-
05
1.375756
3.319789
2.413065
5
8.333168
e
-
05
0.0001291083
2.622109
e
-
05
1.549330
4.923834
3.178040
6
6.31974
e
-
05
0.0001062572
1.622325
e
-
05
1.681354
6.549686
3.895483
5
3
7.01547
e
-
05
8.522752
e
-
05
3.934262
e
-
05
1.214851
2.166290
1.783173
4
4.750068
e
-
05
6.440414
e
-
05
1.899583
e
-
05
1.355857
3.390436
2.500585
5
3.310539
e
-
05
5.144982
e
-
05
1.046542
e
-
05
1.554122
4.916173
3.163312
6
2.532269
e
-
05
4.341858
e
-
05
6.32867
e
-
06
1.714612
6.860617
4.001266
20
3
1.813884
e
-
05
2.141139
e
-
05
1.017992
e
-
05
1.180417
2.103296
1.781825
4
1.161498
e
-
05
1.671353
e
-
05
4.712122
e
-
06
1.438963
3.546922
2.464915
5
8.348938
e
-
06
1.282813
e
-
05
2.611685
e
-
06
1.536498
4.911821
3.196763
6
6.300591
e
-
06
1.076669
e
-
05
1.601593
e
-
06
1.708839
6.722488
3.933953
100
3
3.576201
e
-
06
4.282342
e
-
06
1.949306
e
-
06
1.197455
2.196855
1.834602
4
2.350533
e
-
06
3.236420
e
-
06
9.673106
e
-
07
1.376887
3.345792
2.429967
5
1.667317
e
-
06
2.622885
e
-
06
5.363255
e
-
07
1.573117
4.890472
3.108778
6
1.220823
e
-
06
2.203291
e
-
06
3.159623
e
-
07
1.804759
6.973272
3.863825
200
3
1.776957
e
-
06
2.109820
e
-
06
1.023774
e
-
06
1.187322
2.060826
1.735693
4
1.175403
e
-
06
1.581642
e
-
06
4.78013
e
-
07
1.345617
3.308784
2.458935
5
8.277293
e
-
07
1.306533
e
-
06
2.650969
e
-
07
1.578454
4.928511
3.122365
6
6.383724
e
-
07
1.065240
e
-
06
1.576827
e
-
07
1.668682
6.755592
4.048462
500
3
7.048948
e
-
07
8.251703
e
-
07
4.060795
e
-
07
1.170629
2.032041
1.735854
4
4.695757
e
-
07
6.461275
e
-
07
1.943934
e
-
07
1.375981
3.323814
2.415595
5
3.350298
e
-
07
5.157602
e
-
07
1.047704
e
-
07
1.539446
4.922766
3.197752
6
2.468869
e
-
07
4.345264
e
-
07
6.523285
e
-
08
1.760022
6.661159
3.784702Remark
(1)
If random variableX~Pow(α,β), that is, f(x)=αβαxα-1,0≤x≤β, then Y=1/X~Par(α,1/β); that is, f(y)=(α/β)(β/y)α+1,1/β≤y≤∞.
(2)
The Power function distributionPow(α,β), with parameter α=1 and β, results in rectangular distribution defined on the interval [0,1/β].For simple random sampling MSEs are defined as(73)MSE(SRS)=∑i=110000(G^SRS-G)2N,MSE(SRS)=∑i=110000(B^SRS-B)2N,MSE(SRS)=∑i=110000(A^SRS-A)2N,
where G^SRS, B^SRS, and A^SRS denote the simple random sample estimates of Gini, Bonferroni, and Absolute Lorenz indices (Table 1).Similarly, for ranked set and systematic sampling the MSEs can be defined whereG^RSS, B^RSS, and A^RSS denote the ranked set sample estimates and G^SYS, B^SYS, and A^SYS denote the systematic random sample (SYS) estimates of Gini, Bonferroni and Absolute Lorenz indices (Tables 2 and 3).Gini index(G), Bonferroni index (B), and Absolute Lorenz index (A) for given distributions is given in the Tables 4–6.Efficiencies are defined as(74)E(1)=MSE(SRS)MSE(RSS).E(1)>1 implies that Ranked set sample estimates are better than Simple random sample estimates.Similarly,(75)E(2)=MSE(SRS)MSE(SYS).E(2)>1 implies that Systematic random sample (SYS) estimates are better than Simple random sample estimates. 						(76)E(3)=MSE(RSS)MSE(SYS)E(3)>1 implies that Systematic random sample (SYS) estimates are better than Ranked set sample estimates.The results have been presented in Tables7–15 and it can be seen that the simple random estimates have much less efficiency as compared to their ranked set or systematic sample estimates. Systematic sample estimates are more efficient than the ranked set sample estimates. The same trend is prevalent through all the distributions irrespective of the inequality index being used. We also observe that relative efficiency increases as m increases irrespective of the sampling technique used.


## 4. Conclusion

Ranked set sample estimates of inequality do improve the efficiency of all the inequality indices but compared to systematic sampling the ranked set sample estimates are less efficient. For example, in Table7 while looking at the relative efficiencies for Gini index using different sample estimates we see that the efficiency of RSS is almost two to three times more than SRS (ref. Col. E(1)) while efficiency of SYS is almost four to eight times more than RSS (ref. Col. E(3)). The efficiency of SYS versus SRS is noticeably high as is evident from col. E(2). The relative efficiency trend as given in the last three columns continues both for small as well as large sample sizes. One may conclude that SRS estimates though commonly used and simple to compute show less efficiency than their nonsimple random counterparts. One should use non-simple random estimate of inequality indices when the underlying data or practical situation so demands.



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*Source: 659580-2013-09-19.xml*