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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue III, March 2026
Comparative Analysis of the Efficiency of Sampling Scheme
Estimators in Estimating Population Total
Faweya O, Akinyemi O, Ajayi T. A, Odukoya E. A
Department of Statistics, Ekiti State University, Ado-Ekiti, Ekiti State, Nigeria
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150300125
Received: 01 April 2026; Accepted: 06 April 2026; Published: 24 April 2026
ABSTRACT
Sampling is a fundamental tool in statistical research, providing a practical alternative to complete enumeration
where of time, cost, personal and accessibility are constraints Choosing the best estimator for population total
estimation is one of the main issues in survey sampling. Even though many different sampling strategies and
estimators are available, it is still difficult to assess how effective they are in diverse situations. This study
presents a comparative analysis of the efficiency of sampling scheme estimators (Hansen Hurwitz, Horvitz-
Thompson, Rao-Hartley-Cochran’s and Sen Yates-Grundy) in estimating population total using child birth data)
from Ekiti State . Population totals and variances of each estimator were obtained and the most efficient estimator
determined in terms of variance. The results revealed that the Rao–Hartley–Cochran estimator consistently
produced the lowest population total estimates with the least variance for 2 years, while in the other year, the
Sen–Yates–Grundy estimator demonstrated superior efficiency with minimal variance. (The study offers
empirical evidence regarding the relative efficiency of the probability proportional to size estimators with
replacement (Hansen–Hurwitz) and those without (Horvitz–Thompson, Rao–Hartley–Cochran, and Sen–Yates–
Grundy in estimating population totals). The study further showed that' efficiency of estimators is not constant
but rather fluctuates over time and across data distributions. The study also closes the gap between theoretical
sampling principles and real-world application in demographic and health statistics by using these sample
strategies on actual child birth registration data from Ekiti State.
Keyword: Relative efficiency, childbirth, Estimator, Sampling re-arrange for neatness
INTRODUCTION
Comparing the effectiveness of various sampling estimators requires empirical research. Although theoretical
characteristics are widely known, actual performance varies based on a number of variables, including sample
size limitations, data quality, and demographic variability and operational factors like cost, computational
efficiency, and ease of implementation are taken into account when choosing a sample estimator. A number of
sampling techniques have been created over time to improve population estimation accuracy Simple random
sampling (SRS), in which every unit in the population has an equal chance of being selected, is the most
straightforward and widely used technique. Even while SRS offers objective estimations, it frequently produces
considerable variability, especially when working with diverse populations (Singh & Mangat, 1996). Alternative
sample methods such cluster sampling, stratified sampling, and systematic sampling have been developed to
overcome this problem. Regression and ratio estimators have been created to increase the effectiveness of
population estimates in addition to these conventional methods. To lower variance and increase precision, these
estimators make use of auxiliary data that is connected to the research variable. One well-known example that
established the basis for probability-weighted estimate in survey sampling is the Horvitz-Thompson estimator
(1952) (Horvitz & Thompson, 1952). Accordingly, Dawodu, Adewara, and Oshungade (2013) confirmed that
one of the main justifications for the creation of the Rao-Hartley-Cochran sampling scheme was its
shortcomings, such as negative in variance estimates of the Horvitz-Thompson scheme. Claims that the
probability proportional to size with replacement is the least efficient scheme are also clear; yet, research like
Chaudry and Patra (2023) suggested that the probability proportional to size with replacement is the most
efficient estimator. Choosing the best estimator for population total estimation is one of the main issues in survey
sampling. Even though there are many different sampling strategies and estimators available, it is still difficult
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to assess how effective they are in diverse situations (Kish, 1965). Despite being simple to use, simple random
sampling (SRS) frequently produces significant variance when estimating population totals, especially in
populations that are heterogeneous (Singh & Mangat, 1996). By splitting the population into homogeneous
subgroups, stratified sampling, on the other hand, lowers variance; but it necessitates in-depth knowledge of
population characteristics, which is not always available (Cochran, 1977).
Even while systematic sampling is straightforward and frequently more effective than SRS, it may produce
biased results if the population contains hidden periodic patterns (Thompson, 2012). Compared to stratified or
systematic sampling, cluster sampling typically increases variance, but it might be helpful when population
elements are naturally grouped (Särndal et al., 2003). The selection of an estimator under probability
proportional to size has been controversial. Some of the options include probability proportional to size with
replacement estimator, Horvitz-Thompson without replacement estimator, and Rao-Hartley-Cochran's, Yates
Grundy, and Midzuno's special cases without replacement estimator. Although each estimator has theoretical
explanations from previous research, empirical validation across a variety of datasets is required to create useful
guidance for their application (Lohr, 2021).
METHODOLOGY
Probability Proportional to Size
Every unit in the population has an equal chance of being selected in a random sample produced using the basic
random sampling scheme. In some cases, giving the units in the population uneven probability of selection
yields more effective estimators.
Probability Proportion to Size with Replacement.
PPS sampling with replacement is the term used when the chosen unit with the corresponding size in the sample
is reexamined in the sampling frame in the probability proportional to size problem.
Probability Proportion to Size without Replacement.
PPS sampling without sampling occurs when the chosen unit with the corresponding size in the sample is
disregarded from the sampling frame.
Probability Proportional to Size with replacement (Hansen and Hurwitz Estimator)
The idea of probability proportional to size sampling was first given by Neyman (1934). Hansen and Hurwitz
(1943) developed the general theory of probability proportional to size with replacement. One unit was selected
at each of the n draws. They allocated the selection probability to the ith unit of the population given by
where
is the measure of size (auxiliary variable) for the ith population unit and
.
Unbiased estimate of Population total
If a sample of size n units is drawn with PPS of
and with replacement, then
Is an unbiased estimate of the population total Y. where
is the probability of selecting the ith unit in the
sample.
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Proof: Let
be the number of times that the i
th
unit appears in a specific sample of size n, where t
i
may have
any of the values 0,1,2,…,n. Consider the joint frequency distribution of the t
i
for all N units in the population.
The method of drawing the sample is equivalent to the standard probability problem in the balls are thrown
into N boxes, the probability that a ball goes into the
box being
at every throw. Consequently, the joint
distribution of the
is the multinomial expression
For the multinomial, the following properties of the distribution of
are well known:
We may therefore write (4) as
Where the sum extends over all units in the population. In repeated sampling the
are the random variables,
whereas the
and the
are a set of fixed numbers.
By taking the expectation of both sides of (3.4.7), we have
Hence, since
by (3.4.4) we have,
Therefore
is unbiased.
Variance of Hansen Hurwitz estimator
Taking the variance of both sides of the estimator in (3.4.2),
Concerning the variance, we have:
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Unbiased estimator of Variance of Hansen Hurwitz estimator
If a sample of units is drawn with PPS of
and with replacement, an unbiased sample estimate of
is,
for any
By the usual algebraic identity,
Hence (12) becomes
Hence, an unbiased variance estimator of
is given by Cochran (1953):
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Method II. We have by definition that
Taking the expectation,
Probability Proportional to Size Sampling Without Replacement (PPSWOR) – Horvitz Thompson
estimator
Horvitz and Thompson (1952) developed a general theory of sampling without replacement. Horvitz-Thompson
estimator of the population total Y is a linear estimator of the sample observations on the basis of n sample
observations
can be defined as
Unbiased estimator of Horvitz Thompson estimator
If a sample of size n units is drawn from population Y without replacement, then we show that Horvitz Thompson
estimator
is an unbiased estimate of the population total Y.
Variance of Horvitz Thompson estimator
From the definition of variance, we have
Since
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By definition of variance and covariance in equation 4 and 5, equation 8 becomes
24
Unbiased estimator of the Variance of Horvitz Thompson estimator
An unbiased estimator of the variance
of the Horvitz and Thompson (1952) estimator of the population
total Y is given by
Yates-Grundy Draw-by-Draw Procedure
This selection procedure is stated as select the first units with probability proportional to size select the second
unit with probability proportional to size of remaining units. This procedure is one of the simplest procedures as
it does not impose any restriction on initial probabilities of selection and final probabilities of inclusion.
Variance of Sen-Yates-Grundy estimator
Another form of the
, developed by Sen (1953), and Yates and Grundy (1953) independently, is given
by
Unbiased estimator of the Variance of Sen-Yates-Grundy estimator
An unbiased estimator of the variance of the Horvitz and Thompson (1952) estimator of the population total Y
in the Sen—Yates—Grundy (1953) form is given by
This implies that ++
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If
Rao-Hartley-Cochran Estimator
The estimator for estimating population total is given as
where
is the probability of the Tth unit being selected from the ith group.
Table 3.7.1: Rao Hartley Cochran Strategy
Structure of data in RHC-Sampling Strategy
1
st
Group
2
nd
Group
)
i
th
Group
(
n
th
Group
(
Value
Prob.
Value
Prob.
Value
Prob.
Value
Prob.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Where
. Denotes the sum of selection probability of the
random group.
Unbiased estimator of Rao Hartley Cochran estimator
The unbiased estimator of population total Y is given by
Variance of Rao Hartley Cochran estimator
The variance of the Rao Hartley Cochran estimator,
is given by
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Therefore (3.7.4) implies that
Unbiased estimator of the Variance of Rao Hartley Cochran estimator
An unbiased estimator of the variance
of Rao Hartley Cochran is given by
Therefore
Or
Superiority of Probability Proportional to Size without replacement over Probability Proportional to size
with replacement
The Rao Hartley Cochran (RHC) scheme is more efficient than PPSWR sampling if
Without loss of generality, we have
Combining these results, we have
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On substituting the optimum value of
in (3.8.1) we have
Note that
Therefore,
RESULTS
This research presents the findings of four sampling scheme estimators—Hansen Hurwitz, Horvitz-Thompson,
Rao-Hartley-Cochran, and Sen Yates Grundy—with regard to population total and variations in live births for
the three-year period 2022–2024. Thus, the outcomes are shown below.
Probability Proportional to Size Sampling Scheme
Range and Selection of Samples among the LGAs with cumulative total
Table 4.1.1 Range and Selection of Samples among the LGAs
S/N
LGA
No of
Reg.
Centre
Cum
Total
Ranges
Total
Birth
(2022
)
Total
Birth
(2023
)
Total
Birth
(2024)
Total
Double
Birth
(2022)
Total
Double
Birth
(2023)
Total
Double
Birth
(2024)
1
Ado*
11
11
1 – 11
10419
9553
10800
331
286
410
2
Efon
4
15
12 – 15
1470
1689
1532
23
30
24
3
Ekiti East*
11
26
16 – 26
2086
3618
1918
77
116
72
4
Ekiti South
West
4
30
27 – 30
1112
1360
1106
37
70
46
5
Ekiti West
5
35
31 – 35
1461
1644
1422
41
64
30
6
Emure
4
39
36 – 39
902
1261
849
29
46
34
7
Gbonyin*
6
45
40 – 45
1710
1917
1611
36
59
36
8
Ido Osi
4
49
46 – 49
2190
2173
1730
50
60
56
9
Ijero
6
55
50 – 55
2244
2342
2602
80
133
96
10
Ikere*
5
60
56 – 60
1866
1994
1953
62
128
14
11
Ikole
7
67
61 – 67
2106
2202
1783
49
60
16
12
Ilejemeje
3
70
68 – 70
786
789
814
32
35
15
13
Irepodun/
Ifelodun
6
76
71 – 76
2348
2430
1897
55
64
70
14
Ise Orun*
4
80
77 – 80
1601
1553
1145
34
36
22
15
Moba
6
86
81 – 86
1620
1692
1500
59
108
63
16
Oye
4
90
87 – 90
1753
1677
1610
44
50
41
Note: 5 LGAs were selected based on random sampling with replacement
Source: National Population Commission, Ado Ekiti
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Estimation of Total Population
An unbiased estimator of the population total under probability proportional to size with replacement, Y is given
by
Table 4.1.2 Probabilities and Yearly data of child’s birth for the sampled LGAs
LGAs
Ado
Ekiti East
Gbonyin
Ikere
Ise Orun
= 0.1222
= 0.1222
= 0.0667
= 0.0555
= 0.0444
331
77
36
62
34
286
116
59
128
36
410
72
36
14
22
Total Population Estimate for 2022
Total Population Estimate for 2023
Total Population Estimate for 2024
Estimation of Variance
Variance Estimate for 2022
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Standard Error (SE) is
Variance Estimate for 2023
Standard Error (SE) is
Variance Estimate for 2024
Standard Error (SE) is
Horvitz-Thompson Sampling Scheme estimator
The first number, which is random number 10 falls within the range of 1 – 11 which is for Ado, the second
number 28 lies in range 27 – 30 which stand for Ekiti South West, the third number, 46 falls within 46 – 49 which
is for Ido/Osi, 64 number falls in the range of 61 – 67 is for Ikole and lastly, 82 falls within 81 – 86 which is for
Moba. Hence, the five (5) selected Local Government Areas under Horvitz-Thompson sampling scheme are
Ado, Ekiti South West, Ido/Osi, Ikole and Moba.
Probability of Inclusion
The numbers of Registration Centre under each of the selected Local Government Area are 11, 4, 4, 7, and 6,
based on the presentation in table 4.1.2, as well as k which is 18, the probability for each of the selected Local
Government Area is given below;
Table 4.2.1: Probability of the selected samples among all the Local Government Areas.
Registration
Centre
Ado
Ekiti South
West
Ido/Osi
Ikole
Moba
Range
1 – 11
27 – 30
46 – 49
61 – 67
81 – 86
Samples of 5
18
36
54
72
90
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Estimation of Total Population by Horvitz-Thompson Sampling Scheme
An unbiased estimator of the population total for PPSWOR sampling as given by Horvitz-Thompson is
Table 4.2.2: Probability of the selected samples among all the LGAs with annual enrolment data
Sample LGA
Nos. of
Reg.
Centres
Serial no
Total Double
Birth (2022)
Total
Double
Birth (2023)
Total Double
Birth (2024)
Ado
11
1
331
286
410
Ekiti South
West
4
2
37
70
46
Ido/Osi
4
3
50
60
56
Ikole
7
4
49
60
16
Moba
6
5
59
108
63
Based on the Horvitz-Thompson sampling scheme criteria, Table 4.2.2 reveals the probability of selecting each
of the Local Government which are sampled, Ado, Ekiti South West, Ido/Osi, Ikole and Moba based on the
number 10 selected from the random. In the Table 4.2.2 probabilities stood at
,
,
,
and
for Ado, Ekiti South West, Ido/Osi, Ikole and Moba respectively, in regards to k and the number of units
(Registration Centres) in each of the LGA.
Estimation of Variance by Horvitz-Thompson Sampling Scheme
Rao-Hartley-Cochran’s (RHC) Sampling Scheme
Table 4.3.1: Total Registered Live Births by Type of Birth by Registration Centre Across All LGA in Ekiti
State for 2022, 2023 & 2024
S/N
LGA
No of Registration
Centres
Total Double
Birth (2022)
Total Double
Birth (2023)
Total Double Birth
(2024)
1
Ado
11
331
286
410
2
Efon
4
23
30
24
3
Ekiti East
11
77
116
72
4
Ekiti South West
4
37
70
46
5
Ekiti West
5
41
64
30
6
Emure
4
29
46
34
7
Gbonyin
6
36
59
36
8
Ido Osi
4
50
60
56
9
Ijero
6
80
133
96
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10
Ikere
5
62
128
14
11
Ikole
7
49
60
16
12
Ilejemeje
3
32
35
15
13
Irepodun/ Ifelodun
6
55
64
70
14
Ise Orun
4
34
36
22
15
Moba
6
59
108
63
16
Oye
4
44
50
41
Total
1039
1345
1045
Following the numbers of Local Government Areas in the population collected above, we are to select a sample
of size 4 by using RHC scheme. Thus, the population (LGAs) will be divided into four (4) random groups. To
do this, we selected 16 distinct random numbers between 1 and 16 from 36
th
to 43
rd
rows
This random numbers came in the sequence; 10, 11, 01, 02, 08, 16, 05, 12, 09, 13, 04, 14, 06, 15, 03, 07.
The LGAs bearing the serial numbers corresponding to the first four selected random numbers constitute the
first random group, whereas, the next four random numbers form the second random group and so on.
Let
Therefore, the following are the 4 random groups of units along with the initial selection of probability.
S/N
RANDOM
NO.
LGA
1
10
Ikere
5
62
128
14
0.0556
2
11
Ikole
7
49
60
16
0.0778
3
01
Ado
11
331
286
410
0.1222
4
02
Efon
4
23
30
24
0.0444
Total
27
465
504
464
0.3000
Table 4.3.2: first random group by RHC selection
S/N
Random No.
LGA
1
08
Ido/Osi
4
50
60
56
0.0444
2
16
Oye
4
44
50
41
0.0444
3
05
Ekiti West
5
41
64
30
0.0556
4
12
Ilejemeje
3
32
35
15
0.0333
Total
16
167
209
142
0.1778
Table 4.3.3: second random group by RHC selection
S/N
Random No.
LGA
1
09
Ijero
6
80
133
96
0.0667
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2
13
Irepodun/
Ifelodun
6
55
64
70
0.0667
3
04
Ekiti South
West
4
37
70
46
0.0444
4
14
Ise/Orun
4
34
36
22
0.0444
Total
20
206
303
234
0.2222
Table 4.3.4: third random group by RHC selection
S/N
Random No.
LGA
1
06
Emure
4
29
46
34
0.0444
2
15
Moba
6
59
108
63
0.0667
3
03
Ekiti
East
11
77
116
72
0.1222
4
07
Gbonyin
6
36
59
36
0.0667
Total
27
201
329
205
0.3000
Table 4.3.5: fourth random group by RHC selection
The next thing is to select one unit independently in each group using a method of selection of sample (we make
use of Lahiri’s method).
The first random group consists of
units and maximum value of the variable
. Choosing
, we select random number
by starting with 4
th
and 5
th
columns and another random number
by starting from 23
rd
and 24
th
columns of the Random Number Table. Then the first effective pair
of random number is (04, 05) as shown in the table of effective pairs as shown in the table below. Thus, from the
first random group, Ikere LGA will be included in the sample.
Trial No.
Group S/N
(
)
LGA
Decision
R = Rejected
S = Selected
1
04
Efon
05
4
R
2
03
Ado
14
11
R
3
02
Ikole
12
7
R
4
01
Ikere
04
5
S
Table 4.3.6: selected LGA for group 1 by RHC selection
The second random group consists of
units and maximum value of the variable
. Choosing
, we select random number
by starting with 13
th
and 14
th
columns and another random number
by starting from 22
nd
to 24
th
columns of the Random Number Table. Then the first effective pair
of random number is (02, 04) as shown in the table of effective pairs as shown in the table below. Thus, from the
first random group, Oye LGA will be included in the sample.
Trial No.
Group S/N
LGA
Decision
R = Rejected
S = Selected
1
01
Ido/Osi
09
4
R
2
04
Ilejemeje
07
3
R
3
02
Oye
02
4
S
4
03
Ekiti West
06
5
R
Table 4.3.7: selected LGA for group 2 by RHC selection
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The third random group consists of
units and maximum value of the variable
. Choosing
, we select random number
by starting with 19
th
and 20
th
columns and another random number
by starting from 15
th
and 16
th
columns of the Random Number Table. Then the first effective pair
of random number is (03, 01) as shown in the table of effective pairs as shown in the table below. Thus, from the
first random group, Ekiti South West LGA will be included in the sample.
Trial No.
Group S/N
LGA
Decision
R = Rejected
S = Selected
1
03
Ekiti South
West
01
4
S
2
02
Irepodun/
Ifelodun
08
6
R
3
01
Ijero
10
6
R
4
04
Ise/ Orun
06
4
R
Table 4.3.8: selected LGA for group 3 by RHC selection
The fourth random group consists of
units and maximum value of the variable
. Choosing
, we select random number
by starting with 41
st
and 42
nd
columns and another random number
by starting from 31
st
to 32
nd
columns of the Random Number Table. Then the first effective pair of
random number is (03, 11) as shown in the table of effective pairs as shown in the table below. Thus, from the
first random group, Ekiti East LGA will be included in the sample
Trial No.
Group S/N
LGA
Decision
R = Rejected
S = Selected
1
03
Ekiti East
07
11
S
2
02
Moba
04
6
S
3
04
Gbonyin
14
6
R
4
01
Emure
10
4
R
Table 4.3.9: selected LGA for group 4 by RHC selection
Result presented in Table 4.3.6 – Table 4.3.9 reveals that there are four random groups for the Local Government
Areas covered in the study. For the first random group, there are Ikere, Ikole, Ado and Efon. In the second
random group we have Ido/Osi, Oye, Ekiti West and Ilejemeje.
The third random group consist of Ijero, Irepodun/Ifelodun, Ekiti South West, Ise/Orun while the last random
group is made up of Emure, Moba, Ekiti East and Gbonyin. In line with selection of one LGA from each of the
groups, based on random sampling and Lahiri’s selection method, the LGAs selected were Ikere, Oye, Ekiti
South West and Ekiti East for first, second, third and fourth group respectively.
Probability of Selection under Rao-Hartley-Cochran’s estimator
Table 4.3.10: Sampled LGAs and Selection Probabilities for double birth in 2022
S/N
LGA
1
Ikere
5
62
0.0556
0.3000
0.1852
334.80
3844
0.0031
0.0103
373636.80
2
Oye
4
44
0.0444
0.1778
0.2500
176.02
1936
0.0020
0.0111
174261.78
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3
Ekiti
South
West
4
37
0.0444
0.2222
0.2000
184.98
1369
0.0020
0.0089
153997.10
4
Ekiti East
11
77
0.1222
0.3000
0.4074
189.00
5929
0.0149
0.0498
119070.00
Total
884.80
820965.68
Note that
Population Total for RHC is given as
The Total population estimate of double births for the year 2022 is given by
An estimate of variance of the estimator
is given by
Table 4.3.11: Sampled LGAs and Selection Probabilities for double birth in 2023
S/N
LGA
1
Ikere
5
128
0.0556
0.3000
0.1852
691.20
16384
0.0031
0.01028807
1592524.8
2
Oye
4
50
0.0444
0.1778
0.2500
200.03
2500
0.0020
0.01110972
225028.125
3
Ekiti South
West
4
70
0.0444
0.2222
0.2000
349.97
4900
0.0020
0.00888978
551194.875
4
Ekiti East
11
116
0.1222
0.3000
0.4074
284.73
13456
0.0149
0.04979424
270232.0661
Total
1525.92
2638979.87
Note that
The Total population estimate of double births for the year 2023 is given by
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An estimate of variance of the estimator
is given by
Table 4.3.12: Sampled LGAs and Selection Probabilities for double birth in 2024
S/
N
LGA
1
Ikere
5
14
0.055
6
0.300
0
0.185
2
75.60
196
0.003
1
0.0102880
7
19051.2
2
Oye
4
41
0.044
4
0.177
8
0.250
0
164.02
168
1
0.002
0
0.0111097
2
151308.9113
3
Ekiti
South
West
4
46
0.044
4
0.222
2
0.200
0
229.98
211
6
0.002
0
0.0088897
8
238026.195
4
Ekiti
East
11
72
0.122
2
0.300
0
0.407
4
176.73
518
4
0.014
9
0.0497942
4
104108.4298
Total
646.32
512494.74
Note that
The Total population estimate of double births for the year 2024 is given by
An estimate of variance of the estimator
is given by
Sen – Yates – Grundy Sampling Scheme
Estimation of Variance by Sen – Yates – Grundy Estimator
Sen – Yates – Grundy form of the variance of the Horvitz and Thompson estimator of the population total Y is
given by
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Comparison of Estimated Population and Variance for the four PPS estimators
Table 4.5.1: Comparison of Estimated Population Total for HH, HT and RHC
Estimated Population Total
YEAR
HH
HT
RHC
2022
1152
1236
885
2023
1458
1531
1526
2024
1046
1360
646
Note: HH= Hansen Hurwitz; HT= Horvitz-Thompson; RHC= Rao-Hartley-Cochran’s
Bar Chart Visualizations for PPSWR and PPSWOR Estimates (2022–2024)
Figure 4.5.1: Estimated Population Total for PPS, HT and RHC
Table 4.5.1 and Figure 4.5.1 present the estimated population total of double child birth enrollments in all LGAs
in Ekiti State under the Rao-Hartley-Cochran's sampling scheme, the Horvitz-Thompson sampling scheme, and
the probability proportional to size sampling scheme (Hansen Hurwitz) for the years 2022, 2023, and 2024. The
results show that the Horvitz-Thompson sampling scheme without replacement has the highest estimated
population total of double child births enrolled in Ekiti State over the period covered, the Hansen Hurwitz
sampling scheme with replacement has the lowest estimated population total for 2023, and the Rao-Hartley-
Cochran sampling scheme without replacement has the lowest estimated population total for 2022 and 2024.
Table 4.5.2: Comparison of Estimated Variance for HH, HT, RHC and SYG
Estimated Variance
YEAR
HH
HT
RHC
SYG
2022
161308.54
391184.02
12696.15
40356.96
1152
1458
1046
885
1526
646
0
200
400
600
800
1000
1200
1400
1600
1800
2022 2023 2024
Estimated Population Total
Year
Comparison Of Population Estimate Using PPSWR and
PPSWOR (2022-2024)
Y-hatPPSWR Y-hatPPSWOR
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2023
125701.32
791623.40
103518.78
16102.79
2024
336621.23
431017.39
31586.34
79283.25
Comparison of Variance Estimates Using PPSWR and PPSWOR (2022–2024)
Figure 4.5.2: Estimated Variance for HH, HT, RHC and SYG
Estimated variance of double child birth enrollment in all LGAs in Ekiti State under probability proportional to
size sampling schemes with replacement (Hansen Hurwitz) and probability proportional to size sampling
schemes without replacement (Horvitz-Thompson, Rao-Hartley-Cochran, and Sen – Yates – Grundy) for 2022,
2023, and 2024 is shown in Table 4.5.2 and Figure 4.5.2. It turns out that Sen-Yates-Grundy has the lowest
projected variance for 2023, while Rao-Hartley-Cochran's has the lowest estimated variance for 2022 and 2024.
Additionally, it was discovered that throughout the entire time under review, Horvitz-Thompson had the biggest
variance in double childbirth enrollment among all Ekiti State LGAs.
CONCLUSION
In order to estimate the population total of double child birth registrations across the Local Government Areas
of Ekiti State, this study conducted a comparative investigation of the efficiency of sampling scheme estimators.
Hansen-Hurwitz's probability proportional to size with replacement and Horvitz-Thompson, Rao-Hartley-
Cochran, and Sen-Yates-Grundy's probability proportional to size without replacement estimators were the main
subjects of the analysis.
The results showed that in 2022 and 2024, the Rao–Hartley–Cochran estimator consistently yielded the lowest
estimated population totals and achieved the least variance in those years. However, by producing the least
variance in 2023, the Sen–Yates–Grundy estimate demonstrated greater efficiency. These findings imply that
efficiency differed by year and the underlying data distribution, rather than any one estimator being consistently
better over all time periods.
161309
125701
336621
12696
103519
31586
0 50000 100000 150000 200000 250000 300000 350000 400000
2022
2023
2024
Variance of Population Estimate
Year
Comparison of Variance Estimates Using PPSWR and PPSWOR (2022-
2024)
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The study concludes that, in terms of variance reduction, probability proportional to size sampling without
replacement estimators—specifically, Rao–Hartley–Cochran and Sen–Yates–Grundy—generally performs
better than the conventional Hansen–Hurwitz estimator. Because it has a substantial impact on the precision and
dependability of population total estimates, this conclusion emphasizes the significance of carefully choosing an
estimator based on data feature
REFERENCES
1. Anderson, M., and Scott, D. (2024). Investigating the Rao-Hartley-Cochran (RHC) method’s
effectiveness in estimating labor force participation rates in informal economies. Journal of Labor
Economics, 41(2), 75-88.
2. Brown, P., Clark, E., and Thompson, J. (2024). The Horvitz-Thompson (HT) estimator’s application in
estimating malnutrition prevalence among children in low-income regions. Public Health Statistics
Review, 34(2), 56-70.
3. Cochran, W. G. (1977). Sampling techniques (3rd ed.). Wiley. Chowdhury, S., and Ahmed, N. (2021).
Assessment of the Midzuno-Sen sampling scheme in estimating deforestation rates in tropical regions.
Environmental Monitoring and Assessment, 172(4), 210-221.
4. Horvitz, D. G., and Thompson, D. J. (1952). A generalization of sampling without replacement from a
finite universe. Journal of the American Statistical Association, 47(260), 663–685.
5. Kish, L. (1965). Survey sampling. Wiley.Singh, D., and Mangat, N. S. (1996). Theory and analysis of
sample survey designs.
6. Wiley Yates, F., and Grundy, P. (2018). The Yates-Grundy draw-by-draw method’s application in
environmental migration studies. Population and Environment Studies Journal, 30(3), 45
