Exponential Ratio Type Estimators In Stratified Random Sampling
ABSTRACT Kadilar and Cingi (2003) have introduced a family of estimators using
auxiliary information in stratified random sampling. In this paper, we propose
the ratio estimator for the estimation of population mean in the stratified
random sampling by using the estimators in Bahl and Tuteja (1991) and Kadilar
and Cingi (2003). Obtaining the mean square error (MSE) equations of the
proposed estimators, we find theoretical conditions that the proposed
estimators are more efficient than the other estimators. These theoretical
findings are supported by a numerical example.
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Dataset: Aamir et al. (2014)
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Dataset: Aamir et al. (2014)
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Page 1
Exponential Ratio–Type Estimators In Stratified Random
Sampling
†Rajesh Singh, Mukesh Kumar, R. D. Singh, M. K. Chaudhary
Department of Statistics, B.H.U., Varanasi (U.P.)India
†Corresponding author
Abstract
Kadilar and Cingi (2003) have introduced a family of estimators using auxiliary information in
stratified random sampling. In this paper, we propose the ratio estimator for the estimation of
population mean in the stratified random sampling by using the estimators in Bahl and Tuteja
(1991) and Kadilar and Cingi (2003). Obtaining the mean square error (MSE) equations of the
proposed estimators, we find theoretical conditions that the proposed estimators are more
efficient than the other estimators. These theoretical findings are supported by a numerical
example.
Key words: Stratified random sampling, exponential ratiotype estimator, bias, mean squared
error.
1. Introduction
Let a finite population having N distinct and identifiable units be divided into L strata.
Let nh be the size of the sample drawn from hth stratum of size Nh by using simple random
sampling without replacement. Let
= n and = N.
∑
=
h
L
1
h
n
∑
=
h
L
1
h
N
Page 2
Let y and x be the response and auxiliary variables respectively, assuming values yhi
and xhi for the ith unit in the hth stratum.
Let the stratum means be
h
Y =
h
N
1
and
∑
=
i
h
N
1
hi
y
h
X =
h
N
1
respectively.
∑
=
i
h
N
1
hi
x
as –
A commonly used estimator for Y is the traditional combined ratio estimator defined
st
st
st
CR
X.
x
y
y
=
(1.1)
where,
st
y =
,yw
1
=
h
L
∑
h
h
st
x =
,xw
1
=
h
L
∑
h
h
h
y =
h
n
1 and
∑
=
i
h
n
1
hi
y
h
x =
h
n
1
,x
h
n
∑
=
1i
hi
wh = Nh/N and .Xw
1
=
X
L
∑
h
hh
=
The MSE of
CR
y, to a first degree of approximation, is given by
] RS2SR S [
h
w) y ( MSE
yxh
2
xh
1h
22
yh
2
h CR
−+γ≅ ∑
=
L
(1.2)
where
),
11
(
−=γ
Nn
hh
h
st
XX
st
Y
Y
R
==
2
yh
S
2
xh
S
is the population ratio, is the population
variance of a variate of interest in stratum h and is the population variance of auxiliary
Page 3
variate in stratum h and is the population covariance between auxiliary variate and variate
of interest in stratum h.
yxh
S
Auxiliary variables are commonly used in survey sampling to improve the precision of
estimates. Whenever there is auxiliary information available, the researchers want to utilize it
in the method of estimation to obtain the most efficient estimator. In some cases, in addition to
mean of auxiliary, various parameters related to auxiliary variable, such as standard deviation,
coefficient of variation, skewness, kurtosis, etc. may also be known (Koyncu and Kadilar
(2009). In recent years, a number of research papers on ratio type and regression type
estimators have appeared, based on different types of transformation. Some of the
contributions in this area are due to Sisodiya and Dwivedi (1981), Upadhyaya and Singh
(1999), Singh and Tailor (2003), Kadilar and Cingi (2003, 2004, 2006), Singh et.al.(2004),
Khoshnevisan et.al. (2007), Singh et.al. (2007) and Singh et.al. (2008). In this article, we
study some of these transformations and propose an improved estimator.
2. Kadilar and Cingi Estimator
Kadilar and Cingi (2003) have suggested following modified estimator
ab, st
st
KC
X
y
y
=
ab, st
x
(2.1)
(),bw
h
L
1h
+
∑
=
xa
h
where
ab, st
x
=
ab, st
X
(
=
) bXaw
h
L
h
h
+
=1
) x (
2
∑
.
and a, b suitably chosen scalars, these are either functions of the auxiliary variable x such as
coefficient of variation Cx, coefficient of kurtosis β
etc or some other constants.
Page 4
The MSE of the estimator
KC
y
is given by
MSE (
KC
y
) =
h
1h
2
h
W
γ
∑
=
L
[
S
]
yxh ab
2
xh
2
ab
2
yh
SR2SR
−+
(2.2)
where
)X .(X
st ab, st
b , a
R
a .
h
Xw.Y
hh st∑
=
.
Bahl and Tuteja (1991) suggested an exponential ratio type estimator
⎥⎦
⎤
⎢⎣
⎡
+
−
=
xX
xX
expyyBT
(2.3)
BT
y
The estimator is more efficient than the usual ratio estimator under certain conditions. In
recent years, many authors such as Singh et. al. (2007), Singh and Vishwakarma (2007) and
Gupta and Shabbir (2007) have used Bahl and Tuteja (1991) estimator to propose improved
estimators.
Following Bahl and Tuteja (1991) and Kadilar and Cingi (2003), we have proposed some
exponential ratio type estimators in stratified random sampling.
3. Proposed estimators
The Bahl and Tuteja (1991) estimator in stratified sampling takes the following form
⎥⎦
⎤
⎢⎣
⎡
+
−
=
st st
st st
st
xX
xX
exp yt
(3.1)
The bias and MSE of t, to a first degree of approximation, are given by
)S
2
1
t (Bias
yxh
2
xh−
S
8
R3
(w
1
=
X
1
)
h
L
∑
h
2
h
st
L
γ=
(3.2)
]S
4
R
RS S [w
1
=
) t ( MSE
2
xh
2
yxh
2
yh
h
h
2
h
+−γ= ∑
(3.3)
Page 5
3.1 Sisodia Dwivedi estimator
When the population coefficient of variation Cx is known, Sisodia and Dwivedi (1981)
suggested a modified ratio estimator for Y as
x
x
SD
Cx
CX
yy
+
+
=
(3.4)
In stratified random sampling, using this transformation the estimator t will take the form
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎢
+++
=
∑
=
h
∑
=
h
L
1
L
1
xhhh xhhh
SD
)Cx ( w)CX(w
expyt
⎥
⎤
⎢
⎡
+−+
∑
=
h
∑
=
h
L
1
L
1
xhhh xhhh
)Cx (w)CX(w
⎥⎦
⎤
⎢⎣
⎡
+
−
SDSD
SD SD
st
xX
xX
expy
= (3.5)
where
=
SD
X
)CX(w
xh
L
1h
hh
+
∑
=
and
=
SD
x
).C x (
h
w
1
=
xh
L
∑
h
h
+
The bias and MSE of tSD, are respectively given by –
)
yxh
S
2
1
S
8
R
(
X
2
xh
SD
1h
st
−
=
w
1
SDh
L
∑
2
h
θγ
Bias (tSD) = (3.6)
MSE (tSD) =
]S
4
R
SRS [
h
w
2
xh
2
SD
yxhSD
2
yh
L
2
h
+−θ
1h=∑
(3.7)
where
∑
=
h
+
1
xhhh
SD
)CX(w
∑
=
h
==
L
L
1
hh
st
SD
Yw
X
Y
R
SD
st
SD
X
X
=
and θ
.
Page 6
3.2 SinghKakran estimator
Motivated by Sisodiya and Dwivedi (1981), Singh and Kakran (1993) suggested another ratio
type estimator for estimating Y as
) x (
2
x
yy
2
SK
β+
=
) x (X
β+
(3.8)
Using (3.8), the estimator t at (3.1) will take the following form in stratified random sampling
⎥
⎦
⎥
⎢
⎣
⎢
β++β+
∑
=
h
∑
=
h
L
1
L
1
h2hhh2hh
))x ( x (w ))x (X(w
⎥
⎥
⎤
⎢
⎢
⎡
β+−β+
=
∑
=
h
∑
=
h
L
1
L
1
h2hhh2hh
SK
)) x (x (w)) x (X(w
expyt
⎥⎦
⎤
⎢⎣
⎡
+
−
=
SKSK
SKSK
st
xX
xX
expy
(3.9)
where,
=
SK
x
))x (
h
x (
h
w
2
L
h
β+
∑
h=
1
and
=
SK
X
)). x (
h
X(w
1
=
2
L
∑
h
hh
β+
Bias and MSE of tSK, are respectively given by
)S
2
1
yxh
xh−
S
8
R
(w
1
=
X
1
2
SK
SKh
L
∑
h
2
h
st
θγ
Bias (tSK) = (3.10)
MSE (tSK) =
]S
4
R
SRS [
h
w
1
=
2
xh
2
SK
yxhSK
2
yh
L
∑
h
2
h
+−γ
(3.11)
.
X
X
SK
st
SK=θ
∑
=1h
β+
=
L
h2hh
st
SK
))x (X(w
Y
where
and
R
Page 7
3.3 UpadhyayaSingh estimator
Upadhyaya and Singh (1999) considered both coefficients of variation and kurtosis in their
ratio type estimator as
x2
x2
) x (
1 US
Cx
C ) x (X
yy
+β
+β
=
(3.12)
We adopt this modification in the estimator t proposed at (3.1)
⎟⎟
⎠
⎟
⎜⎜
⎝
⎜
+β
=
∑
=
h
L
1
xhh2hh
st1 US
C) x ( x (w
expyt
⎟
⎟⎞
⎜
⎜⎛
+β
∑
=
h
xhh2
L
1
hh
)C) x (X(w
(3.13)
tUS1 at (3.13) can be rewritten as
⎥
⎦
⎤
⎢
⎣
⎡
+
−
=
1 US1 US
1 US1 US
st1 US
xX
xX
expyt
(3.14)
)
xh
L
C ) x (X(w
1
=
X where
h
h2hh1 US
+β= ∑
).Cx(w
1
x
xh
L
h
h2hh1 US
+β= ∑
=
Bias and MSE of tUS1, to first degree of approximation, are respectively given by
()
∑
=
h
⎜⎛
⎝
θγ=
L
1 US
8
1 USh
2
h
1 US
R
w
1
1
tBias
⎟
⎠
⎞
−
yxh
2
xh
st
S
2
1
S
X
(3.15)
()
⎥
⎦
⎥
⎤
⎢
⎣
⎢
⎡
+−γ= ∑
h
=
2
xh
2
US
4
1
yxh 1US
2
yh
h
L
1
2
h
1 US
S
R
SRSwtMSE
(3.16)
where
()
∑
=
h
+β
=
L
1
stxhh2hh
1us
XC) x (Xw
R
∑
.
βh2
.
hhst
) x (XwY
.
X
) x (
h
.Xw
1US
2hh
1US
β
=
∑
θ
and
Page 8
Upadhyaya and Singh (1999) proposed another estimator by changing the place of coefficient
of kurtosis and coefficient of variation as
) x (
2
Cx
) x (
2
CX
yy
x
x
2 US
β+
β+
=
(3.17)
Incorporating this modification in the proposed estimator t, we have
⎥
⎦
⎤
⎢
⎣
⎡
−
+
=
2 US2 US
x
x
X
X
expyt
2 US2 US
st2 US
(3.18)
where
∑
=
h
β+=
L
1
h2xhhh2 us
)) x (CX(wx and
∑
=
h
β+=
L
1
h2xhhh2us
))x (CX(wX
Bias and MSE of tUS2, are respectively given by –
)S
2
1
S
8
R
(w
1
X
1
) t (Bias
yxh
2
xh
2 US
2 USh
L
∑
=
h
2
h
st
2 US
−θγ=
(3.19)
]S
4
R
SR S [
h
w
1
=
) t (MSE
2
xh
2
US2
yxh2 US
2
yh
L
h
2
h
2 US
+−γ= ∑
(3.20)
()
∑
=1h
∑
.
β+
=
L
sth2xhhh
xhhh st
2US
X. ) x (CXw
C .XwY
R
where
.
X
CXw
2US
xhhh
2US
∑
=θ
and
3.4. G.N. Singh Estimator
),x (
2
β
we propose following two estimators. Following Singh (2001), using values ofand
⎥⎦
⎤
⎣
=
1 GNS1GNS
1
1 GNS
t
⎢
⎡
+
−
GNS1GNS
st
xX
xX
expy
(3.21)
Page 9
where
=
1 GNS
X
)X(w
1
=
xh
L
∑
h
hh
σ+
and
=
1 GNS
x
) x (
h
w
1
xh
L
∑
=
h
h
σ+
The Bias and MSE of to a first degree of approximation, are respectively given by –
1 GNS
t
)S
2
1
S
8
R
(w
1
=
X
1
) t (Bias
yxh
2
xh
1 GNS
1GNS hθγ
L
∑
h
2
h
st
1 GNS
−
(3.22)
]S
4
R
SR S [2
h
yh
w
1
=
) t (MSE
2
xh
2
GNS1
yxh1GNS
L
h
2
h
1GNS
+−γ= ∑
(3.23)
where
()
,
Xw
1
=
Y
R
L
∑
h
xhhh
st
1 GNS
σ+
=
.
X
X
1GNS
st
1GNS=θ
Similarly, we propose another estimator
⎥⎦ ⎢⎣
+
2 GNS2 GNS
st2 GNS
xX
⎤⎡
−
=
2 GNS2 GNS
xX
expyt
(3.24)
where,
),) x (
h
X(w
1
=
X
xh
L
h
2hh2GNS
σ+β= ∑
)x(w
1
x
xh
L
h
h2hh2GNS
σ+β= ∑
=
The Bias and MSE of to a first degree of approximation are respectively given by –
2GNS
t
)S
2
1
S
8
3R
(
X
1
)t ( Bias
yxh
2
xh
GNS2
2GNS
L
∑
=
2
hγ
st
2GNS
−θ=
w
1
h
h
(3.25)
]S
4
R
SRS [
h
w
1
=
)t ( MSE
2
xh
2
GNS2
yxh2GNS
2
yh
L
h
2
h
2 GNS
+−γ= ∑
(3.26)
()
,
) x (
h
Xw
1
Y
R
L
∑
h=
xh2hh
st
1GNS
σ+β
=
2GNS
h2
L
∑
=
1i
hh
2GNS
X
) x (.Xw
β
=θ
where,
Page 10
4. Improved Estimator
Motivated by Singh et. al. (2008), we propose a new family of estimators given by
α
⎤
⎥
⎦
⎡
⎢
⎣
−
+
ab, stab, st
x
x
X
X
⎥⎢
=
ab, st ab, st
st MK
expyt
(4.1)
where a and b are suitably chosen scalars and αis a constant.
The bias and MSE of MK
t
()
up to first order of approximation, are respectively given by
()
∑
=
h
⎟
⎠
⎞
⎜
⎝
⎛
−
+αα
=
L
1
yxh
2
xh
ab
ab
2
h
st
MK
S
2
1
S
8
R2
X
1
t Bias
θγh
w
(4.2)
()
⎥
⎦
⎥
⎤
⎢
⎣
⎢
⎡
α
+α−γ= ∑
h
=
2
xh
ab
yxhab
2
yh
h
L
1
2
h
MK
S
4
R
SRSwt MSE
22
) t (MSE
(4.3)
The is minimized for the optimal value of
MK
α given by
∑
=1i
h
ab
wR
∑
=
i
γ
=α
L
1
yxhh
2
h
Sw
γ
L
2
xh
h
2
S
2
M
Putting this value of αin equation (4.3), we get the minimum MSE of the estimator
K
t
ρ−γ
2
c
2
yh
h
2
h
)1 (S
c
as
∑
=
i
=
L
1
. minMK
w)t (MSE
(4.4)
where,ρ is combined correlation coefficient in stratified sampling across all strata. It is
2
L
xhyhhh
h
2
c
=ρ
wSw
111i
==
calculated as
.
S
SSw
2
xh
1i
2
∑
=
⎟⎟
⎠
⎞
⎝
ργ
M
⎜⎜⎛
L
∑
L
∑
h
2
h
2
yh
h
2
h
γγ
We note here that min MSE of
same for any (all) values of a and b.
K
t
is independent of a and b. therefore, we conclude that it
Page 11
5. Efficiency comparisons
First we compare the efficiency of the estimator t at (3.1) with estimator tSD. We have
MSE(tSD) < MSE(t)
∑
=
h
∑
=
h
⎥
⎦
⎥
⎤
⎢
⎣
⎢
⎡
+γ<
⎥
⎦
⎥
⎤
⎢
⎣
⎢
⎡
+−γ
L
2
xh
2
yxh
2
yhh
2
h
L
2
xh
2
SD
4
yxhSD
2
yhh
2
h
S
4
R
RSSwS
R
SRSw
11
(5.1)
∑
=
h
⎥
⎥⎦
⎤
⎢
⎢⎣
⎡
+−γ
L
2
xh
2
SD
4
2
h
S
R
SRw
<
1
yxh SDh
∑
=
h
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
+−γ
L
1
2
xh
2
yxhh
2
h
S
4
R
RSw
γ
2
h
w
1
Let and
∑
=
h
γ=
L
1
yxhh
2
h
SwA
∑
=
h
=
L
B
2
xh
hS
Then equation (5.1) can be rewritten as
B .
R2
4
B .
R
A
2
SD
4
SD
+
<
R
−
RA +−
A (RSD – R) + B/4(RSD  R) (RSD + R) < 0 (5.2)
From (5.2), we get two conditions
∑
=
h
⎢
⎣
⎡−
θ
L
2
h
SRw
⎥
⎦
⎤
+
1
2
xh
2
SD
4
yxhSDh
S
R
< ∑
=
⎥
⎦
⎥
⎤
⎢
⎣
⎢
⎡
+−θ
L
1h
2
xh
2
yxhh
2
h
S
4
R
RSw
∑
=
h
θ=
1
yxhh
2
h
SwA
∑
=
h
θ=
L
1
2
xh h
2
h
SwB
(i) When (RSDR) (RSD+R) > 0
B < 4A /(RSD+R) (5.3)
(ii) When (RSD – R) (RSD + R) < 0
B > 4A / (RSD+R) (5.4)
where and
L
When either of these conditions is satisfied, estimator tSD will be more efficient than the
estimator t.
Page 12
The same conditions also holds true for the estimators tSK, tUS1, tUS2, tGNS1 and tGNS2 if we
replace RSD by RSK, RUS1, RUS2, RGNS1 and RGNS2 respectively in conditions (i) and (ii).
Next we compare the efficiencies of topt with the other proposed estimators.
∑
=
i
∑
=
i
⎥
⎦
⎥
⎤
⎢
⎣
⎢
⎡
−+γ<ρ−γ
<
L
1
yxab
2
x
2
ab
4
2
h
h
2
h
L
1
2
c
2
yh
h
2
h
ab. minMK
SRS
R
Sw) 1 (Sw
) t (MSE) t (MSE
(5.5)
On putting the value of and rearranging the terms we get
c
ρ
0Sw
2
R
Sw
2
L
∑
=
1i
2
xh
h
2
h
ab
L
∑
=
1i
yxhh
2
h
>
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
γ−γ
(5.6)
This is always true. Hence the estimator MK
t
under optimum condition will be more efficient
than other proposed estimators in all conditions.
6. Data description and results
For empirical study we use the data set earlier used by Kadilar and Cingi (2003). Y is
apple production amount in 854 villages of turkey in 1999, and x is the numbers of apple trees
in 854 villages of turkey in 1999.
The data are stratified by the region of turkey from each stratum, and villages are selected
randomly using the Neyman allocation as
h
L
1h
h
hh
h
SN
n
∑
=
=
SN
Page 13
Table 6.1: Data Statistics
N1=106
N4=171
n1=9
n4=67
24375X1=
74365X4=
536iY1=
5588Y4=
. 25
1
β
. 97
4
β
Cx1=2.02
Cx4=3.84
Cy1=4.18
Cy4=5.13
Sx1=49189
Sx4=285603
Sy1=6425
Sy4=28643
82 . 0
=ρ
1
99. 0
ρ
102. 0
1=γ
009. 0
4=γ
015 . 0w2
04. 0w2
N=854
Cx=3.85
Sy=17106
2930Y =
RSK=0.07784
RGNS1= 0.06632
N2=106
N5=204
n2=17
n5=7
27421X2=
26441X5=
2212Y2=
967Y5=
57. 34
2
β
47. 27
5
β
Cx2=2.10
Cx5=1.72
Cy2=5.22
Cy5=2.47
Sx2=57461
Sx5=45403
Sy2=11552
Sy5=2390
86. 0
=
2
ρ
71 . 0
ρ
049. 0
2=γ
138. 0
5=γ
015 . 0w2
057. 0w2
n=140
Cy=5.84
92 . 0
=ρ
R =0.07793
RUS1=.07789
RGNS2=0.07765
N3=94
N6=173
n3=38
n6=2
72409X3=
9844X6=
9384Y3=
404Y6=
26
3
β
28
6
β
Cx3=2.22
Cx6=1.91
Cy3=3.19
Cy6=2.34
Sx3=160757
Sx6=18794
Sy3=29907
Sy6=946
90 . 0
ρ
89. 0
ρ
016 . 0
3=γ
006. 0
6=γ
. 0w2
. 0w2
312
β
Sx=144794
37600X =
RSD=0.07792
RUS2=0.07786
71
60
14
10.
x=
x =
x =
x =
.
x =
x =
3=
6=
4=
5=
1=
4=
2=
5=
012
041
07.
3=
6=
x=
Page 14
Table 6.2 : Estimators with their MSE values
Estimators MSE values
t
tSD
tSK
tUS1
tUS2
tGNS1
tGNS2
tMK(opt)
359619.594
359649.688
359890.313
359739.875
359830.125
360007.985
360479.192
218374.8898
From Table 6.2, we conclude that the estimator tMK has the minimum MSE and hence it is
most efficient among the discussed estimators.
7. Conclusion
In the present paper we have examined the properties of exponential ratio type estimators
in stratified random sampling. We have derived the MSE of the proposed estimators and also
that of some modified estimators and compared their efficiencies theoretically and
empirically.
Page 15
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