# Ratio Estimator in Adaptive Cluster Sampling without Replacement of Networks

Authors: Nipaporn Chutiman; Monchaya Chiangpradit
Journal: Journal of Probability and Statistics
 (2014)

---

## Abstract

In this paper, we study the estimators of the population total in adaptive cluster sampling by using the information of the auxiliary variable. The numerical examples showed that the ratio estimator in adaptive cluster sampling without replacement of networks is more efficient than the ratio estimators in adaptive cluster sampling without replacement of units.


---

## Body

## 1. Introduction

Adaptive cluster sampling, proposed by Thompson [1], is an efficient method for sampling rare and hidden clustered populations. In adaptive cluster sampling, an initial sample is selected by simple random sampling without replacement of units (Thompson [1]). The unbiased estimators for adaptive cluster sampling with an initial sample taken by simple random sampling without replacement of units are Hansen-Hurwitz and Horvitz-Thompson estimator. Although the initial sample is selected by simple random sampling without replacement of units, some networks in the sample may be selected more than once. An adaptive cluster sampling design exists with selection without replacement of networks (Salehi and Seber [2]). The unbiased estimator for adaptive cluster sampling without replacement of networks is Des Raj estimator.The estimators mentioned above are provided by the variable of interest(
y
) only. Sometimes other variables are related to the variable of interest. We can obtain additional information for estimating the population total. The use of an auxiliary variable is a common method to improve the precision of estimates of a population total. Dryver and Chao [3] proposed ratio estimators based on Hansen-Hurwitz and Horvitz-Thompson estimator in adaptive cluster sampling without replacement of units. In this study, we will study the estimator of population total in adaptive cluster sampling without replacement of networks using an auxiliary variable. Some comparisons are made using a simulation.


## 2. Adaptive Cluster Sampling without Replacement of Units

As in the finite population sampling situation, the population consists ofN units (
u
1
,
u
2
,
…
,
u
N
) indexed by their labels (
1,2
,
…
,
N
). Unit i is associated as a variable of interest y
i. The vector of the population total y-values may be written as y
=
{
y
1
,
y
2
,
…
,
y
N
}. The estimator of the population total τ
y
=
∑
i
=
1
N
y
i is studied.For adaptive cluster sampling, an initial sample of units is selected by simple random sampling without replacement. The unbiased estimator of the population total of the variable of interest is formed by using Hansen-Hurwitz estimator and Horvitz-Thompson estimator.


### 2.1. The Hansen-Hurwitz Estimator

The Hansen-Hurwitz estimator is based on draw-by-draw probabilities that a unit’s network is intersected by the initial sample.Letn denote the initial sample size and ν denote the final sample size. Let ψ
i denote the network that includes unit i and let m
i be the number of units in that network.The Hansen-Hurwitz estimator of the population total of the variable of interest can be written as (Thompson and Seber [4])
(1)
(
τ
^
y
)
HH
=
N
n
∑
i
=
1
n
(
w
y
)
i
,
where (
w
y
)
i is the average of the y-values in the network that includes unit i of the initial sample; that is,
(2)
(
w
y
)
i
=
1
m
i
∑
j
∈
ψ
i
y
j
.
The variance of (
τ
^
y
)
HH is
(3)
V
(
τ
^
y
)
HH
=
N
(
N
-
n
)
n
(
N
-
1
)
∑
i
=
1
N
(
(
w
y
)
i
-
τ
y
N
)
2
.
The unbiased estimator of V
(
τ
^
y
)
HH is
(4)
V
^
(
τ
^
y
)
HH
=
N
(
N
-
n
)
n
(
n
-
1
)
∑
i
=
1
n
(
(
w
y
)
i
-
(
τ
^
y
)
HH
N
)
2
.


### 2.2. The Horvitz-Thompson Estimator

The Horvitz-Thompson estimator is based on probabilities of the initial sample intersecting networks. The unbiased estimator of the population total of the variable of interest can be written as (Thompson and Seber [4])
(5)
(
τ
^
y
)
HT
=
∑
k
=
1
K
y
k
.
I
k
α
k
=
∑
k
=
1
κ
y
k
.
α
k
,
where y
k
. is the sum of y-values in network k; that is, y
k
.
=
∑
i
∈
ψ
k
y
i. K is the total number of distinct networks in the population and κ is the total number of distinct networks in the sample. I
k is an indicator variable that takes a value of 1 with probability α
k if the initial sample intersects network k and 0 otherwise. The inclusion probability of network k is α
k
=
1
-
[
(
N
-
m
k
n
)
/
(
N
n
)
], and the probability that networks j  and k are both intersected is
(6)
α
j
k
=
1
-
{
[
(
N
-
m
j
n
)
+
(
N
-
m
k
n
)
-
(
N
-
m
j
-
m
k
n
)
]
(
N
n
)
}
.
The variance of (
τ
^
y
)
HT is
(7)
V
(
τ
^
y
)
HT
=
∑
j
=
1
K
∑
k
=
1
K
y
j
.
y
k
.
(
α
j
k
-
α
j
α
k
)
α
j
α
k
,
and the unbiased estimator of V
(
τ
^
y
)
HT is
(8)
V
^
(
τ
^
y
)
HT
=
∑
j
=
1
κ
∑
k
=
1
κ
y
j
.
y
k
.
α
j
k
(
α
j
k
α
j
α
k
-
1
)
.


### 2.3. Ratio Estimators in Adaptive Cluster Sampling

Dryver and Chao [3] proposed the ratio estimator in adaptive cluster sampling based on Hansen-Horvitz’s ratio estimator as
(9)
(
τ
^
y
)
HH
_
R
=
(
τ
^
y
)
HH
(
τ
^
x
)
HH
τ
x
=
R
^
HH
τ
x
,
where R
^
HH
=
(
τ
^
y
)
HH
/
(
τ
^
x
)
HH, (
τ
^
x
)
HH
=
(
N
/
n
)
∑
i
=
1
n
(
w
x
)
i, and (
w
x
)
i is the average of the auxiliary variable x in the network that includes unit i of the initial sample; that is, (
w
x
)
i
=
(
1
/
m
i
)
∑
j
∈
ψ
i
x
j.The approximated variance of(
τ
^
y
)
HH
_
R is
(10)
V
(
τ
^
y
)
HH
_
R
=
N
(
N
-
n
)
n
(
N
-
1
)
∑
i
=
1
N
(
(
w
y
)
i
-
R
(
w
x
)
i
)
2
.
The unbiased estimator of V
(
τ
^
y
)
HH
_
R is
(11)
V
^
(
τ
^
y
)
HH
_
R
=
N
(
N
-
n
)
n
(
n
-
1
)
∑
i
=
1
n
(
(
w
y
)
i
-
R
^
HH
(
w
x
)
i
)
2
.The ratio estimator in adaptive cluster sampling based on Horvitz-Thompson’s ratio estimator is(12)
(
τ
^
y
)
HT
_
R
=
(
τ
^
y
)
HT
(
τ
^
x
)
HT
τ
x
=
R
^
HT
τ
x
,
where
(13)
R
^
HT
=
(
τ
^
y
)
HT
(
τ
^
x
)
HT
,
(
τ
^
x
)
HT
=
∑
k
=
1
κ
(
x
k
.
α
k
)
.
The approximated variance of (
τ
^
y
)
HT
_
R is
(14)
V
(
τ
^
y
)
HT
_
R
=
∑
j
=
1
K
∑
k
=
1
K
u
j
.
u
k
.
(
α
j
k
-
α
j
α
k
)
α
j
α
k
,
where
(15)
u
j
.
=
y
j
.
-
R
x
j
.
.
The unbiased estimator of V
(
τ
^
y
)
HT
_
R is
(16)
V
^
(
τ
^
y
)
HT
_
R
=
∑
j
=
1
κ
∑
k
=
1
κ
u
j
.
′
u
k
.
′
α
j
k
(
α
j
k
α
j
α
k
-
1
)
,
where
(17)
u
j
.
′
=
y
j
.
-
R
^
HT
x
j
.
.


## 2.1. The Hansen-Hurwitz Estimator

The Hansen-Hurwitz estimator is based on draw-by-draw probabilities that a unit’s network is intersected by the initial sample.Letn denote the initial sample size and ν denote the final sample size. Let ψ
i denote the network that includes unit i and let m
i be the number of units in that network.The Hansen-Hurwitz estimator of the population total of the variable of interest can be written as (Thompson and Seber [4])
(1)
(
τ
^
y
)
HH
=
N
n
∑
i
=
1
n
(
w
y
)
i
,
where (
w
y
)
i is the average of the y-values in the network that includes unit i of the initial sample; that is,
(2)
(
w
y
)
i
=
1
m
i
∑
j
∈
ψ
i
y
j
.
The variance of (
τ
^
y
)
HH is
(3)
V
(
τ
^
y
)
HH
=
N
(
N
-
n
)
n
(
N
-
1
)
∑
i
=
1
N
(
(
w
y
)
i
-
τ
y
N
)
2
.
The unbiased estimator of V
(
τ
^
y
)
HH is
(4)
V
^
(
τ
^
y
)
HH
=
N
(
N
-
n
)
n
(
n
-
1
)
∑
i
=
1
n
(
(
w
y
)
i
-
(
τ
^
y
)
HH
N
)
2
.


## 2.2. The Horvitz-Thompson Estimator

The Horvitz-Thompson estimator is based on probabilities of the initial sample intersecting networks. The unbiased estimator of the population total of the variable of interest can be written as (Thompson and Seber [4])
(5)
(
τ
^
y
)
HT
=
∑
k
=
1
K
y
k
.
I
k
α
k
=
∑
k
=
1
κ
y
k
.
α
k
,
where y
k
. is the sum of y-values in network k; that is, y
k
.
=
∑
i
∈
ψ
k
y
i. K is the total number of distinct networks in the population and κ is the total number of distinct networks in the sample. I
k is an indicator variable that takes a value of 1 with probability α
k if the initial sample intersects network k and 0 otherwise. The inclusion probability of network k is α
k
=
1
-
[
(
N
-
m
k
n
)
/
(
N
n
)
], and the probability that networks j  and k are both intersected is
(6)
α
j
k
=
1
-
{
[
(
N
-
m
j
n
)
+
(
N
-
m
k
n
)
-
(
N
-
m
j
-
m
k
n
)
]
(
N
n
)
}
.
The variance of (
τ
^
y
)
HT is
(7)
V
(
τ
^
y
)
HT
=
∑
j
=
1
K
∑
k
=
1
K
y
j
.
y
k
.
(
α
j
k
-
α
j
α
k
)
α
j
α
k
,
and the unbiased estimator of V
(
τ
^
y
)
HT is
(8)
V
^
(
τ
^
y
)
HT
=
∑
j
=
1
κ
∑
k
=
1
κ
y
j
.
y
k
.
α
j
k
(
α
j
k
α
j
α
k
-
1
)
.


## 2.3. Ratio Estimators in Adaptive Cluster Sampling

Dryver and Chao [3] proposed the ratio estimator in adaptive cluster sampling based on Hansen-Horvitz’s ratio estimator as
(9)
(
τ
^
y
)
HH
_
R
=
(
τ
^
y
)
HH
(
τ
^
x
)
HH
τ
x
=
R
^
HH
τ
x
,
where R
^
HH
=
(
τ
^
y
)
HH
/
(
τ
^
x
)
HH, (
τ
^
x
)
HH
=
(
N
/
n
)
∑
i
=
1
n
(
w
x
)
i, and (
w
x
)
i is the average of the auxiliary variable x in the network that includes unit i of the initial sample; that is, (
w
x
)
i
=
(
1
/
m
i
)
∑
j
∈
ψ
i
x
j.The approximated variance of(
τ
^
y
)
HH
_
R is
(10)
V
(
τ
^
y
)
HH
_
R
=
N
(
N
-
n
)
n
(
N
-
1
)
∑
i
=
1
N
(
(
w
y
)
i
-
R
(
w
x
)
i
)
2
.
The unbiased estimator of V
(
τ
^
y
)
HH
_
R is
(11)
V
^
(
τ
^
y
)
HH
_
R
=
N
(
N
-
n
)
n
(
n
-
1
)
∑
i
=
1
n
(
(
w
y
)
i
-
R
^
HH
(
w
x
)
i
)
2
.The ratio estimator in adaptive cluster sampling based on Horvitz-Thompson’s ratio estimator is(12)
(
τ
^
y
)
HT
_
R
=
(
τ
^
y
)
HT
(
τ
^
x
)
HT
τ
x
=
R
^
HT
τ
x
,
where
(13)
R
^
HT
=
(
τ
^
y
)
HT
(
τ
^
x
)
HT
,
(
τ
^
x
)
HT
=
∑
k
=
1
κ
(
x
k
.
α
k
)
.
The approximated variance of (
τ
^
y
)
HT
_
R is
(14)
V
(
τ
^
y
)
HT
_
R
=
∑
j
=
1
K
∑
k
=
1
K
u
j
.
u
k
.
(
α
j
k
-
α
j
α
k
)
α
j
α
k
,
where
(15)
u
j
.
=
y
j
.
-
R
x
j
.
.
The unbiased estimator of V
(
τ
^
y
)
HT
_
R is
(16)
V
^
(
τ
^
y
)
HT
_
R
=
∑
j
=
1
κ
∑
k
=
1
κ
u
j
.
′
u
k
.
′
α
j
k
(
α
j
k
α
j
α
k
-
1
)
,
where
(17)
u
j
.
′
=
y
j
.
-
R
^
HT
x
j
.
.


## 3. Adaptive Cluster Sampling without Replacement of Networks

Although the initial sample is selected by simple random sampling without replacement of units, some networks in the sample may be selected more than once. An adaptive cluster sampling design exists with selection without replacement of networks (Salehi and Seber [2]).Salehi and Seber [2] proposed a new sampling design, that is, adaptive cluster sampling with networks selected without replacement. Their procedure is as follows.The first initial sample unit is selected by simple random sampling from the population. Then the first network is created and this network is removed from the population. Next, the second initial unit is selected by simple random sampling without replacement of the remaining units; then a second network is created. And finally, the process is continued until all ofn networks are selected.Letn denote the initial sample size. Let ψ
i denote the network that includes unit i and let m
i be the number of units in that network.Further, letp
i be the first-draw probabilities for network which includes unit i. Thus p
i
=
m
i
/
N, where m
i is the number of units in the network which includes unit i. So p
i
/
(
1
-
∑
j
=
1
i
-
1
p
j
) is the conditional ith draw probability for the ith network which includes unit i in the sample given the first i
-
1 networks selection.By using a modified Des Raj estimator (Salehi and Seber [2]), an unbiased estimator for the population total of the variable of interest is
(18)
(
τ
^
y
)
Raj
=
1
n
∑
i
=
1
n
(
z
y
)
i
,
where, for i
=
1, (
z
y
)
i
=
y
1
.
/
p
1 and for i
=
2
,
3
,
…
,
n let
(19)
(
z
y
)
i
=
∑
j
=
1
i
-
1
y
j
.
+
(
1
-
∑
j
=
1
i
=
1
p
j
)
y
i
.
p
i
.
From Raj [5] we find that (
z
y
)
i are mutually uncorrelated so that the variance of (
τ
^
y
)
Raj (Salehi and Seber [2]) is
(20)
V
(
τ
^
y
)
Raj
=
1
n
2
∑
i
=
1
n
V
(
z
y
)
i
,
and the unbiased estimator of V
(
τ
^
y
)
Raj is
(21)
V
^
(
τ
^
y
)
Raj
=
1
n
(
n
-
1
)
∑
i
=
1
n
(
z
i
y
-
(
τ
^
y
)
Raj
)
2
.The variance of the modified Des Raj estimator,V
(
τ
^
y
)
Raj, is less than that of the Hansen-Hurwitz estimator, V
(
τ
^
y
)
HH (Salehi and Seber [2]).


## 4. Proposed Ratio Estimator in Adaptive Cluster Sampling without Replacement of Networks

Some other variables are related to the variable of interest. We can obtain additional information for estimating the population total. The use of an auxiliary variable is a common method to improve the precision of estimates of a population total. In this section, proposed ratio estimator in adaptive cluster sampling without replacement of networks will be described.The proposed ratio estimator based on Des Raj’s estimator is(22)
(
τ
^
y
)
Raj
_
R
=
(
τ
^
y
)
Raj
(
τ
^
x
)
Raj
τ
x
=
R
^
Raj
τ
x
,
where R
^
Raj
=
(
τ
^
y
)
Raj
/
(
τ
^
x
)
Raj and (
τ
^
x
)
Raj is the modified Des Raj estimator for τ
x.To obtain the variance of(
τ
^
y
)
Raj
_
R, we write
(23)
R
^
Raj
-
R
=
(
τ
^
y
)
Raj
(
τ
^
x
)
Raj
-
R
=
(
τ
^
y
)
Raj
-
R
(
τ
^
x
)
Raj
(
τ
^
x
)
Raj
,
where R
=
τ
y
/
τ
x is the population ratio.Ifn is reasonably large, (
τ
^
x
)
Raj is closed to τ
x and E
(
R
^
Raj
) is approximately equal to R; therefore,
(24)
R
^
Raj
-
R
≈
(
τ
^
y
)
Raj
-
R
(
τ
^
x
)
Raj
τ
x
,
V
(
R
^
Raj
)
=
E
[
R
^
Raj
-
E
(
R
^
Raj
)
]
2
≈
E
[
R
^
Raj
-
R
]
2
i
i
i
i
i
i
i
i
i
i
i
i
=
E
[
(
τ
^
y
)
Raj
-
R
(
τ
^
x
)
Raj
τ
x
]
2
i
i
i
i
i
i
i
i
i
i
i
i
=
1
τ
x
2
E
[
(
τ
^
y
)
Raj
-
R
(
τ
^
x
)
Raj
]
2
.
Let
(25)
d
i
=
(
z
y
)
i
-
R
(
z
x
)
i
,
1
n
∑
i
=
1
n
d
i
=
1
n
∑
i
=
1
n
(
z
y
)
i
-
R
1
n
∑
i
=
1
n
(
z
x
)
i
=
(
τ
^
y
)
Raj
-
R
(
τ
^
x
)
Raj
.
So,
(26)
V
(
R
^
Raj
)
=
1
τ
x
2
E
(
1
n
∑
i
=
1
n
d
i
)
2
,
V
[
1
n
∑
i
=
1
n
d
i
]
=
E
(
1
n
∑
i
=
1
n
d
i
)
2
-
[
E
(
1
n
∑
i
=
1
n
d
i
)
]
2
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
=
E
(
1
n
∑
i
=
1
n
d
i
)
2
,
V
(
R
^
Raj
)
=
1
τ
x
2
×
V
[
1
n
∑
i
=
1
n
d
i
]
.
Therefore,(27)
V
(
(
τ
^
y
)
Raj
_
R
)
=
V
(
R
^
Raj
τ
x
)
=
τ
x
2
V
(
R
^
Raj
)
.
The approximated variance of (
τ
^
y
)
Raj
_
R is
(28)
V
(
τ
^
y
)
Raj
_
R
=
1
n
2
[
1
n
2
∑
i
=
1
n
V
(
d
i
)
]
,
and the estimator of V
(
τ
^
y
)
Raj
_
R is
(29)
V
^
(
τ
^
y
)
Raj
_
R
=
1
n
2
[
1
n
(
n
-
1
)
∑
i
=
1
n
(
(
z
y
)
i
-
R
^
Raj
(
z
x
)
i
)
2
]
.


## 5. Examples

### 5.1. The Blue-Winged Teal Example

As an example, the blue-winged teal data (Smith et al. [6]) were used in this study. The population region of 5,000 km2 in an area of Central Florida was partitioned into fifty units of 100 km2. These data were set to be y-values. The total of y-values, τ
y, is 14,121 (see Figure 1).Figure 1
Blue-winged teal data (y-values).Thex-values were simulation results from the model x
i
=
4
y
i
-
ε
i, where ε
i
~
N
(
0
,
y
i
) (see Figure 2).Figure 2
Simulated data (x-values) from the model x
i
=
4
y
i
-
ε
i.For each iteration, an initial sample of units is selected by simple random sampling without replacement. They-values are obtained for keeping the sample network. In each of the sample networks, the x-values are obtained. The condition for added units in the sample is defined by C
=
{
y
:
y
>
0
}.For each estimator, 20,000 iterations were performed to obtain an accuracy estimate. Initial SRS sizes were varied withn
=
5, 8, 10, 15, and 20. The estimated variance of the estimate total is
(30)
M
S
^
E
(
τ
^
y
)
=
1
20,000
∑
i
=
1
20,000
(
(
τ
^
y
)
i
-
τ
y
)
2
,
where (
τ
^
y
)
i is the value for the relevant estimator for sample i.For the blue-winged teal example, the estimatedMSE of estimators under the adaptive cluster sampling without replacement of unit were calculated and listed in Table 1. The estimatedMSE of estimators under adaptive cluster sampling without replacement of networks were shown in Table 2.Table 1
The estimatedMSE of the Hansen-Horvitz estimators and Horvitz-Thompson estimators for the population total of blue-winged teal.
n
E
(
ν
)
M
S
^
E
(
τ
^
y
)
HH
M
S
^
E
(
τ
^
y
)
HT
M
S
^
E
(
τ
^
y
)
HH
_
R
M
S
^
E
(
τ
^
y
)
HT
_
R
5
17.09
2,359,716,325
1,855,257,461
444,212,079
444,211,995
8
23.44
1,369,069,022
814,459,237
155,321,999
155,321,710
10
26.14
1,076,080,659
527,029,260
79,499,876
79,499,398
15
31.27
575,418,109
163,370,814
11,895,009
11,894,258
20
34.71
348,333,275
42,718,623
1,512,773
1,511,986Table 2
The estimatedMSE of the modified Des Raj estimators for the population total of blue-winged teal.
n
E
(
ν
)
M
S
^
E
(
τ
^
y
)
Raj
M
S
^
E
(
τ
^
y
)
Raj
_
R
5
19.90
1,897,377,719
22,982,048
8
26.66
931,732,897
15,329,005
10
29.84
654,226,056
11,162,171
15
35.74
305,945,821
3,298,687
20
39.32
171,618,612
624,795


### 5.2. Simulation Example

As a second illustration, we have used the simulationx-values and y-values from Chutiman and Kumphon [7] were studied. The populations were shown in Figures 3 and 4.Figure 3
They-values.Figure 4
Thex-values.For each iteration, an initial sample of units is selected by simple random sampling without replacement. They-values are obtained for keeping the sample network. In each sample network, the x-values are obtained. The condition for added units in the sample is defined by C
=
{
y
:
y
>
0
}.For each estimator, 20,000 iterations were performed to obtain an accuracy estimate. Initial SRS sizes were variedn
=
5, 10, 15, 20, 30, 40, and 50.For the simulation example, the estimatedMSE of estimators under the adaptive cluster sampling without replacement of unit were calculated and listed in Table 3. Table 4 shows the estimatedMSE of estimators under adaptive cluster sampling without replacement of networks.Table 3
The estimatedMSE of the Hansen-Horvitz estimator and Horvitz-Thompson estimator for the population total of the variable of interest.
n
E
(
ν
)
M
S
^
E
(
τ
^
y
)
HH
M
S
^
E
(
τ
^
y
)
HT
M
S
^
E
(
τ
^
y
)
HH
_
R
M
S
^
E
(
τ
^
y
)
HT
_
R
5
28.999
407,050.393
398,429.963
113,906.753
113,871.977
10
50.596
197,774.349
187,902.766
55,691.349
55,554.041
15
68.027
130,815.296
120,108.018
28,580.328
28,341.920
20
82.526
95,153.494
84,773.793
15,778.559
15,465.085
30
104.183
61,714.960
52,527.501
7,242.755
6,890.563
40
120.132
45,383.260
36,945.984
5,263.785
4,962.297
50
133.058
33,434.958
25,939.191
4,451.101
4,167.722Table 4
The estimatedMSE of the modified Des Raj estimators for the population total of the variable of interest.
n
E
(
ν
)
M
S
^
E
(
τ
^
y
)
Raj
M
S
^
E
(
τ
^
y
)
Raj
_
R
5
31.590
395,296.216
8,146.702
10
56.182
181,264.543
7,641.379
15
75.244
116,781.850
7,033.291
20
91.807
83,718.824
6,417.127
30
116.093
51,340.406
5,359.412
40
133.820
35,017.241
4,606.211
50
148.388
25,772.744
4,041.559From the examples, we see that the estimatedMSE of adaptive cluster sampling without replacement of networks is smaller than the estimatedMSE of adaptive cluster sampling without replacement of units; that is,
(31)
M
S
^
E
(
τ
^
y
)
Raj
≤
M
S
^
E
(
τ
^
y
)
HT
≤
M
S
^
E
(
τ
^
y
)
HH
.For the population total in adaptive cluster sampling by using the information of the auxiliary variable, we see that the estimatedMSE of the ratio estimator in adaptive cluster sampling without replacement of networks is smaller than the estimatedMSE of the ratio estimator in adaptive cluster sampling without replacement of units; that is,
(32)
M
S
^
E
(
τ
^
y
)
Raj
_
R
≤
M
S
^
E
(
τ
^
y
)
HT
_
R
≤
M
S
^
E
(
τ
^
y
)
HH
_
R
.


## 5.1. The Blue-Winged Teal Example

As an example, the blue-winged teal data (Smith et al. [6]) were used in this study. The population region of 5,000 km2 in an area of Central Florida was partitioned into fifty units of 100 km2. These data were set to be y-values. The total of y-values, τ
y, is 14,121 (see Figure 1).Figure 1
Blue-winged teal data (y-values).Thex-values were simulation results from the model x
i
=
4
y
i
-
ε
i, where ε
i
~
N
(
0
,
y
i
) (see Figure 2).Figure 2
Simulated data (x-values) from the model x
i
=
4
y
i
-
ε
i.For each iteration, an initial sample of units is selected by simple random sampling without replacement. They-values are obtained for keeping the sample network. In each of the sample networks, the x-values are obtained. The condition for added units in the sample is defined by C
=
{
y
:
y
>
0
}.For each estimator, 20,000 iterations were performed to obtain an accuracy estimate. Initial SRS sizes were varied withn
=
5, 8, 10, 15, and 20. The estimated variance of the estimate total is
(30)
M
S
^
E
(
τ
^
y
)
=
1
20,000
∑
i
=
1
20,000
(
(
τ
^
y
)
i
-
τ
y
)
2
,
where (
τ
^
y
)
i is the value for the relevant estimator for sample i.For the blue-winged teal example, the estimatedMSE of estimators under the adaptive cluster sampling without replacement of unit were calculated and listed in Table 1. The estimatedMSE of estimators under adaptive cluster sampling without replacement of networks were shown in Table 2.Table 1
The estimatedMSE of the Hansen-Horvitz estimators and Horvitz-Thompson estimators for the population total of blue-winged teal.
n
E
(
ν
)
M
S
^
E
(
τ
^
y
)
HH
M
S
^
E
(
τ
^
y
)
HT
M
S
^
E
(
τ
^
y
)
HH
_
R
M
S
^
E
(
τ
^
y
)
HT
_
R
5
17.09
2,359,716,325
1,855,257,461
444,212,079
444,211,995
8
23.44
1,369,069,022
814,459,237
155,321,999
155,321,710
10
26.14
1,076,080,659
527,029,260
79,499,876
79,499,398
15
31.27
575,418,109
163,370,814
11,895,009
11,894,258
20
34.71
348,333,275
42,718,623
1,512,773
1,511,986Table 2
The estimatedMSE of the modified Des Raj estimators for the population total of blue-winged teal.
n
E
(
ν
)
M
S
^
E
(
τ
^
y
)
Raj
M
S
^
E
(
τ
^
y
)
Raj
_
R
5
19.90
1,897,377,719
22,982,048
8
26.66
931,732,897
15,329,005
10
29.84
654,226,056
11,162,171
15
35.74
305,945,821
3,298,687
20
39.32
171,618,612
624,795


## 5.2. Simulation Example

As a second illustration, we have used the simulationx-values and y-values from Chutiman and Kumphon [7] were studied. The populations were shown in Figures 3 and 4.Figure 3
They-values.Figure 4
Thex-values.For each iteration, an initial sample of units is selected by simple random sampling without replacement. They-values are obtained for keeping the sample network. In each sample network, the x-values are obtained. The condition for added units in the sample is defined by C
=
{
y
:
y
>
0
}.For each estimator, 20,000 iterations were performed to obtain an accuracy estimate. Initial SRS sizes were variedn
=
5, 10, 15, 20, 30, 40, and 50.For the simulation example, the estimatedMSE of estimators under the adaptive cluster sampling without replacement of unit were calculated and listed in Table 3. Table 4 shows the estimatedMSE of estimators under adaptive cluster sampling without replacement of networks.Table 3
The estimatedMSE of the Hansen-Horvitz estimator and Horvitz-Thompson estimator for the population total of the variable of interest.
n
E
(
ν
)
M
S
^
E
(
τ
^
y
)
HH
M
S
^
E
(
τ
^
y
)
HT
M
S
^
E
(
τ
^
y
)
HH
_
R
M
S
^
E
(
τ
^
y
)
HT
_
R
5
28.999
407,050.393
398,429.963
113,906.753
113,871.977
10
50.596
197,774.349
187,902.766
55,691.349
55,554.041
15
68.027
130,815.296
120,108.018
28,580.328
28,341.920
20
82.526
95,153.494
84,773.793
15,778.559
15,465.085
30
104.183
61,714.960
52,527.501
7,242.755
6,890.563
40
120.132
45,383.260
36,945.984
5,263.785
4,962.297
50
133.058
33,434.958
25,939.191
4,451.101
4,167.722Table 4
The estimatedMSE of the modified Des Raj estimators for the population total of the variable of interest.
n
E
(
ν
)
M
S
^
E
(
τ
^
y
)
Raj
M
S
^
E
(
τ
^
y
)
Raj
_
R
5
31.590
395,296.216
8,146.702
10
56.182
181,264.543
7,641.379
15
75.244
116,781.850
7,033.291
20
91.807
83,718.824
6,417.127
30
116.093
51,340.406
5,359.412
40
133.820
35,017.241
4,606.211
50
148.388
25,772.744
4,041.559From the examples, we see that the estimatedMSE of adaptive cluster sampling without replacement of networks is smaller than the estimatedMSE of adaptive cluster sampling without replacement of units; that is,
(31)
M
S
^
E
(
τ
^
y
)
Raj
≤
M
S
^
E
(
τ
^
y
)
HT
≤
M
S
^
E
(
τ
^
y
)
HH
.For the population total in adaptive cluster sampling by using the information of the auxiliary variable, we see that the estimatedMSE of the ratio estimator in adaptive cluster sampling without replacement of networks is smaller than the estimatedMSE of the ratio estimator in adaptive cluster sampling without replacement of units; that is,
(32)
M
S
^
E
(
τ
^
y
)
Raj
_
R
≤
M
S
^
E
(
τ
^
y
)
HT
_
R
≤
M
S
^
E
(
τ
^
y
)
HH
_
R
.


## 6. Conclusions

We discuss using the auxiliary information, that is, the ratio estimators in adaptive cluster sampling without replacement of units and without replacement of networks. For the adaptive cluster sampling without replacement of units, Horvitz-Thompson’s ratio estimator is better than Hansen-Hurwitz’s ratio estimator (Dryver and Chao [3]).Although the initial units in adaptive cluster sampling are selected by simple random sampling without replacement, networks may be selected more than once. So the adaptive cluster sampling without replacement of units is equivalent to the adaptive cluster sampling without replacement of networks. Therefore, the modified Des Raj ratio estimator is better than both Hansen-Hurwitz’s ratio estimator and Horvitz-Thompson’s ratio estimator.



---

*Source: 726398-2014-05-27.xml*