ISRN Probability and Statistics

Volume 2013 (2013), Article ID 659580, 14 pages

http://dx.doi.org/10.1155/2013/659580

Research Article

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

Pooja Bansal, Sangeeta Arora, and Kalpana K. Mahajan

Department of Statistics, Panjab University, Chandigarh 160014, India

Received 16 June 2013; Accepted 2 August 2013

Academic Editors: X. Dang and S. Lototsky

Copyright © 2013 Pooja Bansal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 Section 2, 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

where

is the Lorenz curve at ordinates [1] and

Since is distributed uniformly over , that is, ,

An unbiased estimator of is sample mean

where is the sample size.

Therefore, unbiased estimator of is

Using (6)

Algorithm for computing simple random estimate for Gini index is given as follows.

(i) Generate a sequence of random numbers from .(ii) Compute .(iii) Compute .(iv) Compute .

2.1.2. Bonferroni Curve

Bonferroni index is defined as

where

is the Bonferroni curve defined at ordinates [5]

Since is distributed uniformly over , that is, ,

An unbiased estimator of is sample mean

where is the sample size.

Therefore, unbiased estimator of is

Using (14),

Algorithm for computing simple random estimate for Bonferroni index is given as follows.

(i) Generate a sequence of random numbers from .(ii) Compute .(iii) Compute .(iv) Compute .

2.1.3. Absolute Lorenz Index

Absolute Lorenz index is defined as

where

is the Absolute Lorenz curve defined at ordinates [6]

Since is distributed uniformly over , that is, ,

An unbiased estimator of is sample mean

where is the sample size.

Therefore, unbiased estimator of is

using (22)

Algorithm for computing simple random estimate for Absolute Lorenz index is given as follows.

(i) Generate a sequence of random numbers from .(ii) Compute .(iii) Compute .(iv) Compute .

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 ()

Bonferroni index ()

Absolute Lorenz index ()

2.2. Ranked Set Sampling Procedure

General, ranked setsampling procedure involves the following steps [6].

(i) Select a simple random sample of units from the population and subject it to ordering on attribute of interestvia some ranking process, that is,

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 of 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 unmeasured units in the first random sample are not considered further.

This measurement is denoted by , the smallest orderstatistics or the smallest judgement order item. It may or may not actually have the smallest attribute measurement among the sampled units. Note that are not considered further in the selection of our ranked set sample. The sole purpose of units is to help in selecting an item for measurement that represents the smaller attribute values in the population.

(ii) Following selection of , a second independent random sample of size is selected from the population and judgement ranking without formal measurement on the attribute of interest is made and the second smallest item of units in second random sample is selected and is included in the ranked set sample for measurement of attribute of interest; that is,

The second measured observation is denoted by .

(iii) Similarly, from a third independent random sample, we select the unit judgement ranked to be the third smallest,, for measurement and inclusion in the rankedset sample, that is,

This process is continued until we have selected the unit judgement ranked to be largest of units in the th random sample, denoted by ; that is, 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, in our example, is called set size.

Thus, to complete a single ranked set cycle, we need to judge rank independent random samples of size involving a total of sample units in order to obtain measured observations. These observations represent a balanced ranked set sample with set size .

In practice, the sample size is kept small to ease the visual ranking, the RSS literature suggested or 6. Therefore, in order to obtain a ranked set sample with a desired total number of measured observation , we repeat the entire cycle process independent times, yielding the data

Algorithm for obtaining the RSS estimate of Gini index via method of Monte Carlo simulation is given as follows.

(a) Generate an RSS of size from .(b) Assign .(c) Compute .(d) Compute the ranked set sample estimate of Gini index as

Algorithm for obtaining the RSS estimate of Bonferroni index via method of Monte Carlo simulation is given as following.

(a) Generate a RSS of size from .(b) Assign .(c) Compute .(d) Compute the ranked set sample estimate of Bonferroni index as follows.

Algorithm for obtaining the RSS estimate of Absolute Lorenz index via method of Monte Carlo simulation is given as follows.

(a) Generate a RSS of size from .(b) Assign .(c) Compute .(d) Compute the ranked set sample estimate of Absolute Lorenz index as

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 ()

Bonferroni index ()

Absolute Lorenz index ()

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

Its unbiased estimator in case of stratified sampling is given as

where

where

Let region be divided into disjoint subregions ; that is, and where is null set.

Define

Define

where

Random variable is distributed according to on .

can be estimated by

Therefore, integral is estimated by

where is distributed according to on

where

In particular, if and , we get the systematic sampling estimate of Gini index as

Algorithm for computing systematic random sampling estimate is given as follows.

(i) Divide the range of the cumulative distribution into intervals each of width .(ii) Generate from .(iii) Compute .(iv) Compute .(v) Compute

2.3.2. Bonferroni Index

Similarly, Bonferroni index is defined as

Its stratified estimate is given as

where

where

Let region be divided into disjoint subregions , that is, and , where is null set.

Define

Define

where

Random variable is distributed according to on .

can be estimated by

Therefore, integral is estimated by

where is distributed according to on

where

In particular, if and , we get the systematic sampling estimate of Bonferroni index as

Algorithm for computing systematic random sampling estimate is given as follows.

(i) Divide the range of the cumulative distribution into intervals each of width .(ii) Generate from .(iii) Compute .(iv) Compute .(v) Compute

2.3.3. Absolute Lorenz Index

Absolute Lorenz index is defined as

Its stratified estimate is given as

where

where

Let region be divided into disjoint subregions ; that is, and where is null set.

Define

Define

where

Random variable is distributed according to on .

can be estimated by

Therefore, integral is estimated by

where is distributed according to on

where

In particular, if and , we get the systematic sampling estimate of Absolute Lorenz index as

Algorithm for computing systematic random sampling estimate is given as follows.

(i) Divide the range of the cumulative distribution into intervals each of width .(ii) Generate from .(iii) Compute .(iv) Compute .(v) Compute

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 ()

Bonferroni index ()

Absolute Lorenz index ()

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 , and Power ; Bonferroni index from exponential, Pareto , Power , and Absolute Lorenz index from exponential (3), Pareto , and rectangular distribution. The simulated ranked set samples with set size and number of repetitions, , 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 Tables 4, 5, and 6. The relation between Pareto, Power, and Rectangular distribution can be noted from the following remark.

Table 4: Gini index () for given distributions.

Distribution
cdf
Lorenz curve
Gini index ()

Exponential

Pareto

Power

Table 5: Bonferroni index () for given distributions.

Distribution
cdf
Bonferroni curve
Bonferroni index ()

Exponential

Pareto

Power

Where is the digamma function.

Table 6: Absolute Lorenz index () for given distributions.

Distribution
cdf
Absolute Lorenz curve
Absolute Lorenz index ()

Exponential

Pareto

Rectangular

Table 7: Relative efficiency of Gini index for exponential distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

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

2.179593
8.234807
3.778140

5
0.0002099437
0.0005397776

2.571059
12.460617
4.846492

6
0.0001524544
0.0004400118

2.886186
16.776824
5.812800

200
3
0.0002471209
0.0004459992

1.804781
5.079098
2.814246

4
0.000153431
0.0003294056

2.14693
8.126519
3.785181

5
0.0001031339
0.0002670593

2.589443
12.059777
4.657287

6

0.0002178877

2.866754
16.647886
5.807226

500
3
0.0001002982
0.0001764347

1.759102
4.777463
2.715854

4

0.0001320398

2.221716
8.274741
3.724482

5

0.0001076813

2.558395
12.233036
4.781526

6

2.907654
17.332479
5.960984

Table 8: Relative efficiency of Gini index for pareto () distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

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

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

2.452017
12.697695
5.178469

5
0.0001887224
0.0005433437

2.879063
19.466598
6.761435

6
0.0001309059
0.0004580686

3.499220
28.122036
8.036658

200
3
0.0002253251
0.0004546124

2.017584
7.540676
3.737477

4
0.0001392043
0.0003422813

2.458842
13.155315
5.350209

5

0.0002770614

2.995065
19.420718
6.484240

6

0.0002295553

3.439278
29.074772
8.453742

500
3

0.0001802806

1.902912
7.376671
3.876517

4

0.0001383186

2.467178
12.874499
5.218309

5

0.0001102943

2.981350
20.079053
6.734884

6

3.337793
27.314198
8.183310

Table 9: Relative efficiency of Gini index for power () distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

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

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

2.495118
16.550128
6.633003

5
0.0002358945
0.0007081798

3.002103
24.916370
8.299636

6
0.000173568
0.0005906876

3.403206
36.013606
10.582260

200
3
0.0002970297
0.0005835807

1.964722
8.632814
4.393912

4
0.0001840207
0.0004393907

2.387724
15.883226
6.652035

5
0.0001199216
0.0003552514

2.962364
24.988035
8.435168

6

0.0002953183

3.337786
35.949455
10.770449

500
3
0.0001217598
0.0002407522

1.977272
9.028816
4.566300

4

0.0001784493

2.457572
15.875114
6.459672

5

0.0001436364

2.966163
25.698317
8.663828

6

0.0001184066

3.482823
36.357032
10.438956

Table 10: Relative efficiency of Bonferroni index for rectangular () distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

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

2.906238
24.380856
8.389143

6
0.0003529966
0.001248915

3.538037
36.578305
10.338588

200
3
0.0006307183
0.001248089

1.978838
9.064110
4.580523

4
0.0003722346
0.0009302265

2.499033
15.602183
6.243289

5
0.0002493340
0.0007560325

3.032207
25.666468
8.464614

6
0.0001767726
0.000618749

3.500253
35.454007
10.128981

500
3
0.0002467436
0.0005050265

2.046767
9.133975
4.462637

4
0.0001511701
0.0003785248

2.503966
16.065722
6.416110

5

0.0002998347

3.002686
24.813461
8.263755

6

0.000252477

3.563258
36.113713
10.135026

Table 11: Relative efficiency of Bonferroni index for exponential distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

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

3.239447
22.904058
7.070361

20
3
0.0005728049
0.001073308

1.873776
6.362629
3.395619

4
0.0003477821
0.0008003468

2.301288
10.477729
4.552985

5
0.0002317084
0.0006640685

2.865966
16.117999
5.623932

6
0.0001682733
0.0005685107

3.378496
22.830737
6.757663

100
3
0.0001145552
0.0002185333

1.907668
6.321112
3.313528

4

0.0001665346

2.363148
10.187814
4.311121

5

0.0001299972

2.727103
13.133893
4.816060

6

0.0001080593

3.20796
16.650706
5.190422

200
3

0.0001113661

1.898254
6.062895
3.193932

4

2.292238
8.972654
3.914364

5

2.772329
11.990390
4.325024

6

3.120819
14.223219
4.557527

500
3

1.816505
5.313044
2.924873

4

2.250106
7.593678
3.374810

5

2.520520
8.659752
3.435701

6

2.813653
8.963752
3.185806

Table 12: Relative efficiency of Bonferroni index for pareto ( distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

2
3
0.0004945458
0.0009228851
0.0001830762
1.866127
5.040989
2.701311

4
0.0003013004
0.0007029501

2.333054
8.243527
3.533363

5
0.0002011407
0.0005511385

2.740064
11.504786
4.198728

6
0.0001507043
0.0004636178

3.076342
15.949032
5.184417

5
3
0.0001979686
0.0003752962

1.895736
5.144415
2.713677

4
0.0001211198
0.0002751505

2.271723
8.269590
3.640230

5

0.0002221785

2.683652
11.675306
4.350528

6

0.0001855068

3.123373
15.614433
4.999221

20
3

1.872889
5.040300
2.691190

4

2.330587
7.981155
3.424525

5

2.577291
11.092457
4.303921

6

3.032308
15.721910
5.184800

100
3

1.870386
5.062758
2.706797

4

2.269221
7.927018
3.493277

5

2.792905
11.242203
4.025270

6

3.105075
15.190778
4.892242

200
3

1.861257
5.097086
2.738518

4

2.303809
7.944157
3.448271

5

2.667423
11.446136
4.291084

6

3.173647
15.763539
4.967011

500
3

1.882087
5.128754
2.725036

4

2.250026
7.786863
3.460788

5

2.691062
11.383354
4.230060

6

3.108371
15.417209
4.959899

Table 13: Relative efficiency of absolute Lorenz index for pareto (, ) distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

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
3.713033

Table 14: Relative efficiency of absolute Lorenz index for rectangular () distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

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

1.401848
5.161965
3.682258

6
0.0002722183
0.0004232076

1.554662
7.621384
4.902275

20
3
0.0001900359
0.0002072712

1.090695
1.942699
1.781157

4
0.0001270530
0.0001569948

1.235664
3.484338
2.819811

5

0.0001217164

1.320345
5.043047
3.819492

6

0.0001039845

1.496521
7.259515
4.850928

100
3

1.084339
1.907901
1.759507

4

1.241990
3.295177
2.653143

5

1.373609
5.088021
3.704128

6

1.534990
7.168388
4.669990

200
3

1.108077
1.966270
1.774490

4

1.221911
3.328998
2.724419

5

1.367635
5.065401
3.703768

6

1.584246
7.512110
4.741755

500
3

1.106621
1.976053
1.785665

4

1.281139
3.418179
2.668078

5

1.384923
5.078039
3.666657

6

1.586569
7.412973
4.672330

Table 15: Relative efficiency of absolute Lorenz index for exponential () distribution.

MSE (RSS)
MSE (SRS)
MSE (SYS)

2
3
0.0001772438
0.0002139371

1.207022
2.141528
1.774225

4
0.0001160937
0.001597166

1.375756
3.319789
2.413065

5

0.0001291083

1.549330
4.923834
3.178040

6

0.0001062572

1.681354
6.549686
3.895483

5
3

1.214851
2.166290
1.783173

4

1.355857
3.390436
2.500585

5

1.554122
4.916173
3.163312

6

1.714612
6.860617
4.001266

20
3

1.180417
2.103296
1.781825

4

1.438963
3.546922
2.464915

5

1.536498
4.911821
3.196763

6

1.708839
6.722488
3.933953

100
3

1.197455
2.196855
1.834602

4

1.376887
3.345792
2.429967

5

1.573117
4.890472
3.108778

6

1.804759
6.973272
3.863825

200
3

1.187322
2.060826
1.735693

4

1.345617
3.308784
2.458935

5

1.578454
4.928511
3.122365

6

1.668682
6.755592
4.048462

500
3

1.170629
2.032041
1.735854

4

1.375981
3.323814
2.415595

5

1.539446
4.922766
3.197752

6

1.760022
6.661159
3.784702

Remark

(1)If random variable , that is, , then ; that is, .(2)The Power function distribution , with parameter and , results in rectangular distribution defined on the interval .

For simple random sampling MSEs are defined as

where , , and 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 where , , and denote the ranked set sample estimates and , , and denote the systematic random sample (SYS) estimates of Gini, Bonferroni and Absolute Lorenz indices (Tables 2 and 3).

Gini index , Bonferroni index , and Absolute Lorenz index for given distributions is given in the Tables 4–6.

Efficiencies are defined as

implies that Ranked set sample estimates are better than Simple random sample estimates.

Similarly,

implies that Systematic random sample (SYS) estimates are better than Simple random sample estimates.

implies that Systematic random sample (SYS) estimates are better than Ranked set sample estimates.

The results have been presented in Tables 7–15 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 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 Table 7 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. ) while efficiency of SYS is almost four to eight times more than RSS (ref. Col. ). The efficiency of SYS versus SRS is noticeably high as is evident from col. . 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.

References

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