In the present paper, an attempt has been made to conduct a limited simulation study to examine the relative performance of calibration approach based estimators. The real data has been taken from Appendix-C of Sarndal et al., (2003). The study variate y and the auxiliary variate x are the population of the Sweden in year 1985 and year 1975 which is divided in 284 municipalities (MU284). The results have been found that the calibration estimators have outperformed the usual estimator of finite population mean in two-stage stratified random sampling.
Trang 1Original Research Article https://doi.org/10.20546/ijcmas.2018.701.219
Calibration Approach Based Estimators of Finite Population Mean in
Two - Stage Stratified Random Sampling
Department of Agricultural Statistics, Narendra Deva University of Agriculture and
Technology, Kumarganj, Faizabad-224229 (UP), India
*Corresponding author
A B S T R A C T
Introduction
The auxiliary information has been effectively
used in sample surveys at selection,
stratification and estimation stage for bringing
about the improvement in the estimate of
population parameters
The different fundamental approaches are
used in finite population survey sampling
These are design based approaches, model
based approach and model assisted approach
Under design based approach, the most
common unbiased estimator of finite population total Y of study variable y is the
well-known Horvitz-Thompson (HT) estimator is given by
s i i i
With variance
i N
j
j i ij
Y V
1 1
2 2
1
International Journal of Current Microbiology and Applied Sciences
ISSN: 2319-7706 Volume 7 Number 01 (2018)
Journal homepage: http://www.ijcmas.com
Deville and Sarndal (1992) developed calibration estimator by using the auxiliary
following Deville and Sarndal (1992) calibration approach, calibration estimators of finite population mean in two-stage stratified random sampling have been developed The variances and unbiased estimator of their variances have been derived In the present paper, an attempt has been made to conduct a limited simulation study to examine the relative performance of calibration approach based estimators The real data has been
taken from Appendix-C of Sarndal et al., (2003) The study variate y and the auxiliary
variate x are the population of the Sweden in year 1985 and year 1975 which is divided in
284 municipalities (MU284) The results have been found that the calibration estimators have outperformed the usual estimator of finite population mean in two-stage stratified random sampling.
K e y w o r d s
Finite population,
Auxiliary information,
Two-stage stratified
random sampling,
Calibration estimators,
Population mean
Accepted:
14 December 2017
Available Online:
10 January 2018
Article Info
Trang 2Where D ij ij ij, d i 1/i, i being
the inclusion of probability of ith unit in the
sample s which has been drawn from the
finite population of size N by a probability
sampling design P ) and y is the observed i
value of y corresponding to the ith unit
selected in sample s
Deville and Sarndal (1992) developed
calibration estimator of finite population total
is given by
s
i
i i
i
s
i
i i i
i
x q
d
y x
q
d
2 ˆ
Which is equivalent to GREG estimator of
Y (See, Cassel et al., 1976)
An approximate variance of Yˆ of Y for a C
large sample is given by (Deville and Sarndal,
1992)
i
N
j
j j i i ij
Y
V
1 1
2 2
1
Where E i y i x i, and
N
i
i i i
N
i
i i i i
x q d
y x q d
1
2 1
An attempt has been made in the present paper
to develop calibration estimator of finite
population mean in two-stage stratified
random sampling Following the calibration
approach of Deville and Sarndal (1992) &
calibration based estimator in two-stage
sampling Aditya et al., (2016) when auxiliary
information on a single auxiliary variable x
related to study variable y is known at first
stage unit (fsu) level, that means the total of x
i.e
N I
i i
x X
1 , at th
i fsu level is known The usual estimator of population mean of the study variate y in two-stage stratified random
sampling has been developed in section-2 The calibration estimators of the population mean in two-stage stratified random sampling
by calibrating the design weight have been developed in the section-3 The variances and variance estimators of the calibration estimators have been developed in the
section-4 and a limited simulation study has been conducted in the section-5
The usual estimator of population mean in two-stage stratified random sampling
Consider the finite population
) , ,
, , (U U1 U2 U3 U N consists of N
first stage units (fsu) and is stratified into L
strata such that U consists of t N fsu and t
L
t t
1
.We also consider that each fsu in
tth stratum (t 1,2,3, ,L) has M t
number of second stage units (ssu) Now the
following terms are define as
tij
Y value of the characteristic y under study
on jth ssu(j 1,2,3, ,M t)corresponding
to the ith fsu (i1,2,3, ,N t) in the tth stratum
i ti t t
j tij t
ti
Y
1 1
.
1
of ith fsu in tth stratum
tij t
t t
Y
1 1
1
, the population mean
of y in tth stratum
Trang 3
t
t t
L
t
t t
t
1
1
, the population mean of Y
in the population
1
2
1
1
j
ti tij t
tiy
i
t ti t
tby
S
1
2
2
) 1
(
1
Now, consider that a sample of size n fsu’s t
out of N fsu’s is selected from t t th stratum and
sub-samples of size m out of t M ssu’s from t
the selected n fsu’s are drawn by SRSWOR t
(simple random sampling without
replacement) This process is carried out
independently in each stratum We further
define
ti
y =
t
m
j
tij
t
y
m 1
1
, sample mean from ith selected fsu (i1,2,3, ,n t) in tth stratum
ts
y =
t
n
i
ti
t
y
n 1
1
, the overall sample mean in tth stratum
1
2
1
1
j
ti tij t
m
s
1
2
1
1
i
ts ti t
n
s
Obviously, the y is an unbiased estimator of ts
t
The variance of y is obtained as ts
1 1 2 1 1 1 2
twy t t t tby t t
m n
S n
y
i tiy t
S
1
2
An unbiased estimator of V(y ts)is obtained as
2
1 1 ) (
ˆ
twy t t t tby t t
m
s n
y
i tiy t
n
s
1
2
An unbiased estimator of in stratified two stage random sampling is given by
ts L
t t
1 (7)
Where
t
1
L
t t
W
The variance of y is obtained as s
ts
L
t t
y
1
2
2 2
1
twy t t t tby t t
L
t
m n
S n
An unbiased estimator of V y s is obtained as
L
t t
y
1 2
2 2
1
twy t t t tby t
L
t
m
s n
Trang 4Proposed calibration estimators of
population mean in two-stage stratified
random sampling
We have described details of development of
an estimator of population
tij
t t y
1 1 1 0
L
t
t
1
simple random sampling without replacement
(SRSWOR) independently in each stratum
The estimator of is given by
L
t
ts
t
y
1
(10)
Such that
L
t
t
W
1
1.The weight W is self t
design weight and it is given by
t
,
t
t
1
The variance of y is obtained as s
2 2
1
twy t t t tby t t
L
t
t
m n
S n
W
y
An unbiased estimator of V y s is obtained as
2 2
1
ˆ
twy t t t tby t
L
t
t
m
s n
W
y
The weight W can be calibrated if the t
information of an auxiliary variable xrelated
to the study variate y is available in order to
improve the efficiency of the estimatory The s
information of an auxiliary variable x related
to y may be available at fsu’s level in
two-stage stratified random sampling
In this case, the population mean of the auxiliary variable can easily be obtained, i.e
t N
i ti
(13)
Where X is the value of the auxiliary ti
variable corresponding to ith fsu in the tth stratum
Let W be calibrated weight.The calibration t' estimator of Y is therefore, given by
ts L
t t
1 ' (14)
Where W is calibrated weight obtained by t'
minimizing a distance measure
L
t
t t
W
1
2 ' , where q is positive t
quantity unrelated to W , subject to calibration t
constraint
ts
L
t
t x W
1 ' (15)
Where xts is an estimator oft, developed similarly as y For the ready reference, ts x is ts
given by
i ti t
n
x
1
1
j tij t
m
x
1
1
, (16)
The following function
X x W W
q
W W
L
t t t
' 1
2 ' '
2
Is minimized with respect to W , where t' is Langrangian multiplier This yields W as t'
Trang 5
ts t t
ts t t
x q W
x W X
W
The developed calibration estimator is given
by
L
t ts t L
t
t ts t
L
t
t ts ts t L
t
ts
t
q x W
q y x W y
W
y
1 1
2 1 1
(19)
Which is a combined regression estimator in
stratified random sampling y is a class of sc
estimators depending upon the value of q t
For q t 1, we get an estimator as
L
t ts t L
t ts t
L
t
ts ts t ts
L
t
t
x W
y x W y
W
y
1
1 2 1
1
Which is a combined regression estimator in
two-stage stratified random sampling
For
ts
t
x
q 1 , we get another estimator as
L
t ts t ts
t
L
t ts t L
t
ts t
x W
y W y
W
y
1 1
1
2
x
W
y
W
L
t
ts
t
L
t
ts
t
1
1
(21)
Which is a combined ratio estimator in
two-stage stratified random sampling
Variance and variance estimator of proposed calibration estimators
The approximate variance of ysc1 has been derived following the procedure given by
Sarndal et al., (2003, chapter 4&8), and is
given by
tpsu tssu
L
t
N M
W y
1
2 2
2
1)
N t t
i N
j i Iij tpsu
D D V
1 1 ,
1
t t
t t t Iij
N N
n N n
,
t
t Ii
N
n
ti t
t ti
t
t t
L
t
t t t
X W
X Y W B
1 2
1
Ui kl i
V / ' / '/ ,
1
2 /
t t
t t t i
kl
M M
m M m
and /i m t M t
The estimator of variance of y SC1 following
Sarndal et al., (2003), is obtained as
t
t j t
t t t L
t t t t
n N V n
d n
n N N M N W y
2 2 2 2
1 2 2 2
1
1
ˆ
2 2
tij
si tij t
t
t t t
m m
M m M
ts ti
i y y B x x
2 1 1
ˆ
ts L
t t L
t
ts ts
W
B
Trang 6Table.1 The estimate of Y based on y s, y sc1and y sc2 along with their estimate of variance
Estimator Estimate %RB Estimate of
variance
PSE
s
1
sc
2
sc
The approximate variance of ysc 2 following
Sarndal et al., (2003), and Singh et al.,
(1998), is obtained as
1
2 2 2
1
Et L
t t
s
n
f W x
X
y
(24)
t ti t
N
S
1
2 2
1
1
,
ti t
t
ti
, f t n t N t
and
ti L
t
L
t t ti
W
The approximate consistent and unbiased
estimator of V ysc2 is obtained
as
1
2 2
2
1 ˆ
et L
t t s
n
f W x
X y
(25)
Where
1 i
2 ti t
2
1 n
1 s
,
ts
ti
,
t
L
t ts t ts
W
B
ˆ
L
t t
1 is
an estimate of X in two-stage stratified
random sampling
Simulation
A limited simulation study has been carried
out with real data The population MU284
given in Appendix-C of Sarndal et al., (2003)
has been used There are 50 fsu’s of varying size The variable under study y is
population of 1985 and an auxiliary variable
x is the population of 1975 The 50 fsu’s
are stratified into 4 strata considering the value of x in ascending order The stratum I consists of 13 fsu’s, stratum II consists of 14 fsu’s, stratum III consists of 12 fsu’s, stratum
IV consists of 11 fsu’s respectively
The samples of size 4 fsu’s were drawn by SRSWOR independently from strata 1 to 4, respectively This process has been repeated
300 times independently That means, we obtained 300 samples of size 4 fsu’s from each stratum Sub samples of size 3 ssu’s are drawn by SRSWOR from each sample of
fsu’s in each stratum The values of y and x
in sub samples were used to compute the population mean In this process, we get 300 estimates of Y t
from 300 sub samples in each stratum
The values of y and x in sub samples were
used to compute the population mean of Y
In this process, we get 300 estimates of Y t
from 300 sub-samples in each stratum The averages of these 300 estimates from each
stratum are used to get the estimate of Y
Mathematically, let ˆti
be the estimate of Y t
from ith stratum We compute
Trang 7
1
ˆ
300
1
ˆ
t
ti
(26)
The average of 300 sub-samples in the tth
stratum So, we get simulated estimate of
Y as follows
t
L
t
t
W
1
(27)
The percent relative bias (%RB) of the
estimate has been computed as follows
100
ˆ
RB
(28) Similarly, the approximate variances of the
usual estimator ys
, ysc 1
and ysc 2
are computed The percent standard error (PSE)
of the estimate has been computed as follows:
100
ˆ
ˆ
SE
PSE
(29)
The simulation studies results are presented in
the Table 1
It has been found from the results of the Table
1 that the regression type calibration estimator
1
sc
y and ratio type calibration estimator ysc2
have performed better than the usual
estimatorys in two-stage stratified random
sampling However, estimator ysc 1
has been found best in comparison to estimator ysc 2 as
it has minimum PSE of 9.74 as against 11.24
for ysc 2 It may be noted that the calibration
estimator ysc 1
is equivalent to combined weighted regression estimator and ysc 2 is the
usual combined ratio estimator It has also
been found that the calibration estimator ysc 1
is relatively less biased than estimators ysc 2 and ys
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How to cite this article:
Sandeep Kumar 2018 Calibration Approach Based Estimators of Finite Population Mean in
Two - Stage Stratified Random Sampling Int.J.Curr.Microbiol.App.Sci 7(01): 1808-1815
doi: https://doi.org/10.20546/ijcmas.2018.701.219