Bajopas Volume 10 Number 1 June, 2017
http://dx.doi.org/10.4314/bajopas.v10i1.2
Bayero Journal of Pure and Applied Sciences, 10(1): 6 – 17 Received: November, 2016 Accepted: June, 2017 ISSN 2006 – 6996
A MODIFIED RATIO-PRODUCT PRODUCT ESTIMATOR OF POPULATION MEAN IN THE PRESENCE OF MEDIAN AND COEFFICIENT OF VARIATION OF THE AUXILIARY VARIABLE IN STRATIFIED RANDOM SAMPLING Abubakar Yahaya and Umar Kabir Abdullahi Department of Statistics, Ahmadu Bello University, Zaria, Ni Nigeria.
[email protected] [email protected], +2348069334466.
[email protected] [email protected], +2348109554170.
ABSTRACT For the past decades, the estimation of population mean is one of the challenging aspects in sampling survey techniques and much effort has been employed to improve the precision of estimates. In this research work, we proposed a modified ratio ratio-product product estimator of population mean of the variable of interest using median and coefficient of variation of the auxiliary variable in stratified random sampling scheme. The expression of bias and MSE of the proposed estimator have been obtained under large sample approximation, asymptotically optimum estimator (AOE) is identified with its approximate MSE formula. Estimator based on “estimated optimum values” was also investigated. Theoretical and empirical comparison of proposed estimator with some other ratio and product estimator justified the performance of the proposed estimat estimator. or. There is a minimum of 15 percent reduction in the MSE from each of the existing ratio and product estimators considered. Thus most preferred over the existing estimators for the use in practical application. Keywords: bias, mean square error, auxiliar auxiliary y variable, optimum estimator, stratified random sampling, study variable. INTRODUCTION Auxiliary variable(s) has been widely discussed in sampling theory and population study study. Auxiliary variables are in used in survey sampling to obtain improved sampling designs and to achieve more precision in the estimates of some population parameters such as the mean and the vari variance of the variable of interest.. This information may be used at both the design stage, execution stage and estimation stage of designing a survey. The estimation of population mean is a burning issue in sampling theory and many efforts have been made mad to improve the precision of the estimate. In survey sample literature, a great variety of techniques for using auxiliary information by means of ratio, regression and product methods have been used in the presence of one or more auxiliary variables. It I is also established that if the regression line of the variable under study and the auxiliary variable is through the origin and are positively correlated the best estimator to be used is ratio estimator likewise if the regression line of the variable underr study and the auxiliary variable is through the origin and are negatively correlated the best estimator to be used is product estimator, on the other hand when the regression line does not pass through the origin but makes an intercept along the y-axis and there is weak correlation either positive or negative between the auxiliary variable and the variable of interest the best to used is the linear regression estimator (Okafor, 2002). In modern surveys, the scientific technique for selecting a sample iss that of selecting a probability
sample that is usually based on a stratification of the population. It is well known that stratification is one of the design tools that gives increased in precision. In stratified design the population under investigation is divided into different strata so as to obtain the homogeneity with each stratum and sample observation are drawn within each stratum by well known simple random sampling. The disadvantage of using simple random sampling (SRS) when the population iss not homogeneous have been comprehensively documented in the well known literature (see for instance, (Cochran 1977), more so, research by several authors reveal that ratio product estimator performs better than ratio and product type estimators in simple imple random sampling (SRS) under stratification and other certain conditions. These therefore motivate us to propose an estimator in stratified random sampling design and study its properties. Background of the Study Consider a finite population
P = ( P1 , P2 ,..., PN ) be
N divided into L homogeneous strata of size Nh ( h = 1, 2,3,...L ) . A sample of size nh is drawn
of
from each stratum using simple random sampling without replacement (SRSWOR). Let y be the study variate taking values stratum) and let values
6
xhi
.
x
yhi
(
i th observation
from
hth
be the auxiliary ary var variate taking the
Bajopas Volume 10 Number 1 June, 2017 L
L
h =1
h =1
yst = ∑Wh yh , xst = ∑ Wh xh
Moreover, let
study variate) and Where,
be the unbiased estimators of the population mean
Y
(the
X (the auxiliary variate, respectively).
Nh th , is the weight of h stratum, N 1 nh yh = ∑ yhi , is the mean of the study variate y nh h =1
Wh =
1 nh
xh =
in
hth
nh
∑x
hi
h =1
, is the mean of the auxiliary variate
x
in
stratum,
hth
stratum,
Remarks: 1 To ensure the applicability of the estimator, we assume the population values of the study variate are known in the entire stratum. This is a reasonable assumption as survey samplers usually obtain such information inexpensively through pilot survey or past experience. 2
A ratio-product estimate of population mean Y can be made in two ways. One is to make a separate ratio estimate of the total of each stratum and add their totals. An alternative estimate is to derive a single combined ratio.
Y of the study variable Y , assuming the knowledge of the ( h = 1, 2,3,..., L ) of the auxiliary variable X , we define a separate
To have a survey estimate of the population mean
Xh
population mean
of the
h
th
stratum
ratio estimator as L ) yRst1 = ∑ Wh Rh X h
,
(2.1)
h =1
) y Rh = h
Where
xh
, xh ≠ 0 is
the estimate of ratio Rh
=
Yh Xh
, Xh ≠ 0 ,
hth −
of the
stratum in the
population. The estimator is only efficient if the variables are strongly positively correlated. The separate product estimator
) Ph , (2.2) yPst1 = ∑ Wh Xh h =1 ) th Where Ph = y h xh is the estimate of product Ph = Yh X h , X h ≠ 0 of the h − stratum in the population. The L
estimator is only efficient if the variables are strongly negatively correlated. To the first order of approximation (i.e, to terms of order
0 ( nh−1 ) ), the variance of
equation (2.1)
and
(2.2) above are respectively given by
(
)
(
)
L
(1 − f h )
h =1
nh
MSE y Rst1 = ∑ Wh2 L
(1 − f h )
h =1
nh
MSE y Pst1 = ∑ Wh2
Y h Chy2 + C hx2 {1 − 2 K h }
(2.3)
Y h Chy2 + C hx2 {1 + 2 K h }
(2.4)
2
2
where
fh =
Chy Shxy S hy nh hhx , K h = ρh , ρh = , Chy = , Chx = Chx Shx ShY Nh Xh Yh
∑( x Nh
Shxy =
i =1
hi
− Xh
)( y
( N h − 1)
hi
−Yh
)
∑( x Nh
,
Shx2 =
i =1
hi
− Xh
( Nh − 1) 7
)
.
∑( y Nh
2
,
Shy2 =
i =1
hi
−Yh
( Nh − 1)
)
2
.
Bajopas Volume 10 Number 1 June, 2017 The direct generalization of Sisodia and Dwivedi (1981) transformation of the auxiliary variable in stratified random sampling design by Kadilar and Cingi (2003), is defined as L
∑( X
yst .SDR = y
+ C xh )
h
h =1 st L
∑(x
+ C xh )
h
h =1 L
∑(x
yst .SDP = y st
h =1 L
+ C xh )
h
∑( X
(2.6)
+ C xh )
h
h =1
(2.5)
L
L
h =1
h =1
X SD = ∑ ( X h + C xh ) , xSD = ∑ ( xh + C xh )
They defined
. Then equation (2.5) and (2.6) will be
) X SD = R SD X SD (2.7) x SD ) yst Where RSD = yst . It should be noted that the difference between combined ratio and Sisodia and Dwivedi xSD ) (1981) estimator is only RSD . Thus, bias and MSE of those estimators can be given in the same way like y st . SD = y st
equations
(2.8)
Bias y st .SDR
(
)
(
)
(
)
(
)
and
(2.9),
equation
(2.10)
and
(2.11)
respectively
1 K 2 2 = ∑Wh γ h ( RSD S xh − S yxh ) X SD h =1
as (2.8)
2 2 MSE y st .SDR = ∑Wh2 γ h ( S yh − 2 RSD S yxh + RSD S xh2 ) K
(2.9)
h =1
1 K 2 ∑Wh γ h ( RSD S xh + 2S yxh ) X SD h =1
Bias y st .SDP =
(2.10)
2 2 MSE y st .SDR = ∑Wh2 γ h ( S yh + 2 RSD S yxh − RSD S xh2 ) K
(2.11)
h =1
K
Y st = = X SD
R SD R = R SD P
∑W h =1
∑ (X
h
K
h
h =1
X
(2.12)
h
+ C xh
)
The Subramani and Kumarapandiyan (2012) stratified ratio and product estimator are define in equation (2.13) and (2.14) respectively
∑ (X K
y R st 2 = y st
h
h =1 K
∑ (x
h
h =1
C xh + M
C xh + M
∑ (x K
y Pst 2 = y st
h =1 K
∑ (X h =1
h
)
d
d
(2.13)
)
C xh + M d
)
C xh + M d
)
h
(2.14)
The corresponding bias and MSE of the estimator in equation (2.13) and (2.14) are given in equation (2.15) and (2.16) respectively
(
)=
(
) ∑W γ (S
B ia s y R st 2
MSE y Rst 2 =
1 X
R st 2
K 2 2 ∑ W h γ h ( R R st 2 S xh − S yxh ) h =1
K
h =1
2 h
h
2 yh
(2.15)
2 2 − 2 RRst 2 S yxh + RRst 2 S xh )
8
(2.16)
Bajopas Volume 10 Number 1 June, 2017
(
)=
(
) ∑W
B ia s y P s t 2
MSE y Pst 2 =
K 2 ∑ W h γh h =1
1 X
P st 2
K
h =1
2 h
(R
S x2h + S y x h )
P st 2
(2.17)
2 2 2 γ h ( S yh − RPst 2 S xh + 2 R Pst 2 S yxh )
(2.18)
K
∑W
R Rst 2 = R Pst 2 =
h =1
K
∑W h =1
h
X h C xh
h
(2.19)
X h C xh + M d
Proposed Estimator Motivated by the direct generalization of Sisodia and Dwivedi (1981), Housila and Neha-Agnihotri (2008), Singh and Vishwakarma (2011), Subramani and Kumarapandiyan (2012), Using the various definitions above we proposed the separate and combine ratio-product estimator respectively as (S )
TRP
T R(P ) C
L L W h X h C xh + M d h W h x h C xh + M dh ∑ ∑ h =1 h =1 = ∑ Wh y h δ h L + (1 − δ h ) L h =1 W h x h C xh + M dh W h X h C xh + M dh ∑ ∑ h =1 h =1 L L X st C xst + M d st x st C x st + M d st ∑ ∑ = y st δ st h =L1 + (1 − δ st ) hL= 1 x st C xst + M dst X st C x st + M d st ∑ ∑ h =1 h =1 L
Where y = h
nh
1 nh
∑ h=i
(
)
(
)
(
)
(
)
y hi and x h =
1 nh
(3.1)
(3.2)
(Y , X )
nh
∑x h =i
hi
are unbiased estimators of the population means
h
h
respectively, ‘ Cxh ’ and ‘ M dh ’ are the known characteristics positive scalars of the auxiliary variable respectively and
δst
is a real constant to be determined such that the mean square error of
The family of estimators
(S )
TRP
(S )
TRP
X
is minimum.
reduces to the following set of known estimators,
( Cxh , M dh , δh ) = ( 0,1, δh ) , TRP( S ) → y st (usual unbiased stratified estimator) (S) For ( Cxh , M dh , δh ) = ( 0,1, δh ) , TRP → Q1 ∑ W X
i For ii
L
= δ
st
y
st
∑
h
h
h =1 L
W
h
h =1
x
+
h
(1
− δ
st
)y
st
L
∑
W
h
h =1 L
∑
h =1
W
h
x X
h
h
which is due to Singh and Espejo (2003) iii For
(S) RP
T
( Cxh , M dh , δh ) = ( 0,1, δh ) ,
→ Q2
= δh y
st
(X
L
∑
W
∑
W
h =1 L h =1
h
h
(x
h
h
+ C + C
xh
xh
) )
+ (1 − δ h
envisaged by Singh and Tailor 2005 , where the median of the auxiliary
X
Cxh
) y st
L
∑
W
∑
W
h =1 L
h =1
h
h
(x (X
h
h
+ C + C
xh
xh
) )
is the known population coefficient of variation and
respectively, many other ratio-product estimators can be generated from
putting any suitable parameters rather than values of
( Cxh , M dh , δh ) .
bias and Mean Square Error (MSE) To obtain the bias and mean square error M Let e 0 h = Then
(y
h
−Yh Yh
)
y h = Y h (1 + e0 h )
and e1 h = and
(x
h
− X X
of the proposed family of estimator h
)
h
x h = X h (1 + e1h )
9
( ) TRP S
in
M dh
is
( ) TRP
by
S
( 3.71) , we write
Bajopas Volume 10 Number 1 June, 2017
E ( e0 h ) = E ( e1h ) = 0
Such that
E ( e02h ) =
1 − fh 2 1 − fh 2 1 − fh C yh , E ( e12h ) = Cxh , E ( e0 h e1h ) = K h Cxh2 nh nh nh Cy S xyh S S Nh n , ρh = , C yh = yh , C xh = xh . fh = h , K h = ρ N Cx S xh S Yh Nh Yh Xh
where W h =
∑ (x L
S xyh =
h =1
− X
hi
h
(N h
)( y
hi
−Y
h
− 1)
),
∑ (x L
S x2h =
h =1
hi
− X
(N h
h
)
∑ (y L
2
, S2 = yh
− 1)
( 3.71) in terms of eh ' s , we have −1 (S ) TRP = Y h (1 + e0 h ) δ h (1 + θh e1h ) + (1 − δ h )(1 + θh e1h )
h =1
hi
(N h
−Y
h
)
2
.
− 1)
And expanding
Where
θh =
(X
X h C xh
We assume that series
C xh + M dh
)
θh e1h < 1 ,
so that the expression
h
using
binomial
(1+ θh e1h )
theorem.
(3.3)
−1
can be expanded to a convergent infinite
Hence
from
(3.3)
we
TRP = Y st (1 + e0h ) δh (1− θh e1h + θ e − θ e + θ e ...) + (1 − δh )(1+ θh e1h ) (S)
2 2 h 1h
3 3 h 1h
have.
4 4 h 1h
= Y st δh (1 + e0h ) (1− θh e1h + θ2h e12h − θ3h e13h + θ4h e14h ...) + (1 − δh )(1 + e0h )(1+ θh e1h )
δh (1 + e0 h − θh e1h + θ2h e12h − θh e1h e0 h + θ2h e1h e0 h + θ2h e12h e0 h − θ3h e13h + θ4h e14h − θ3 e13h e0 h + ...) + = Y st (1 − δh )(1 + e0 h + θh e1h + θh e1h e0 h ) 1 + e0 h − θe1h − θ h e1h e0 h + θ 2h e12h + θ 2h e12h e0 h − θ3h e13h + θ 4h e14h − θ3h e13h e0 h + = Y st 1 + e0 h + θ h e1h + θ h e1h e0 h + δ h ... − 1 − e0 h − θ h e1h − θ h e0 h e1h
= Y st 1 + e0h + θh e1h + θh e1he0h + δh ( −2θhe1h e0h − 2θh e0he1h + θ2h e12h + θ2h e12h e0h − θ3h e13h + θ4h e14h − θ3he13h e0h + ...)
= Yst 1+ e0h − (1− 2δh ) θhe1h + (1− 2δh ) θhe1he0h +δhθ2he12h +δh ( θ2he12he0h −θ3he13h +θ4he14h −θ3he13he0h +...) We assume that the contribution of terms involving powers in
S
and
e1h higher than the second is negligible,
, where v > 1 . Thus, from the above expression we write to a first order of approximation, nv ≅ Y st 1 + e0 h − (1 − 2δh ) θh e1h + (1 − 2δh ) θh e1h e0 h + δh θ2h e12h + δh θ2h e12h e0 h , or
being of order
( ) TRP
e0h
1
(T ( ) − Y ) = Y S RP
st
st
e0h − (1 − 2δh ) θh e1h + (1 − 2δh ) θh e1h e0h + δθ2h e12h + δh θ2h e12h e0h ,
Taking the expectation of both side of (3.4), we obtained the bias of approximation as
(
B i a s T R(PS )
)= ∑ L
h =1
(1 −
fh
)
nh
Equation (3.5) will vanishes if δ = h Thus for δ = h
Kh (2Κ h − θ h )
( ) TRP S
θhY
h
Κ
h
+ δh
Kh ( 2Κ h − θ h ) is almost unbiased.
10
(θ h
− 2K
h
) C x2h
(T ( ) ) S RP
(3.4)
to the first degree of
(3.5)
Bajopas Volume 10 Number 1 June, 2017 Squaring both side of the equation (3.4), and neglecting the terms of
eh ' s
having power greater than two we
have
(T ( ) − Y ) S RP
2
h
= Y h e02h + (1 − 2δh ) θh {(1 − 2δh ) θh e12h + 2e1h e0h } 2
(3.6)
Taking the expectation of both sides of (3.6), we get the mean square error MSE of
( ) TRP to the first order of S
approximation as
) ∑(
(
1 − fh )
L
M S E T R(PS ) =
nh
h =1
δh
To obtain the value of
2 2 Y h C yh + θ h (1 − 2 δ h ) C xh {(1 − 2 δ h ) θ h + 2 K h } h
( ) , we take the partial derivative of the MSE of
( ) MSE of TRP S
that minimizes the
(T ( ) ) with respect to δ and equate it to zero. S RP
δh =
(3.7)
h
1 2
Kh (Optimum Value) 1 + = δ0h θh
(3.8)
Putting (3.8) in (3.7), we get the Asymptotically Optimum Estimator (AOE) as
T R(PS O) =
L
∑
h =1
yh 2
K h X h C xh + M d h K h x h C xh + M d h 1 + + 1 − θ θ h X h C xh + M d h + x C M h h xh dh
MSE of (T R(PS ) ) or
Substitution of (3.8) in (3.7) yield the minimum
( AOE ) (T ( ) ) as
(3.9)
MSE of
the
asymptotically optimum
S RP
estimator
(
) ∑(
M S E m in T R(P ) = S
L
1 − fh
h =1
nh
)
S y2h (1 − ρh2
which is equal to the approximate
y lrS T = y st +
∑ β$ ( X L
∑( L
h =1
h
h =1
s Where β$ h = xyh
s x yh =
(nh
− xh
)(
y hi − y h
)
(3.10)
of stratified regression estimator as
− 1)
)
, s2 = xh
(
practice only when
K h and θh are
h =1
hi
( nh
− xh
)
βh
of
yh
on
xh
2
− 1)
.
) in (3.9) depends on K
h
and θh , so the
known. Here it should be mention that
So only the value of
K h can
∑ (x L
(S ) AOE of TRPO
The value of
S
)
It is to be noted the
( C xh , M dh , δ h ) .
(
M S E T R(P O)
, is the sample estimate of the population regression coefficient
2 s xh
x hi − x h
h
MSE
)=
AOE of
θh is
(T ( ) ) can be used in S RP
a function of known quantities
( S ) in practice. K h should be known for making the use of AOE of (TRPO )
be made known quite accurately either from pilot study or past data or experience
gathered in due course of time. This problem has been discussed among others by Murthy (1967), Reddy (1978), Srivankataramana and Tracy (1980). Thus, the value of
Kh
can be guessed quite accurately and such an
estimator can be used in practice. Allowable Departure Let k 0 h be an estimate or guessed value of K h with
k 0 h = K h (1 + η h ) , then
δh =
Kh Kh 1 1 1 + 1 + = 1 + 2 θh 2 θh θh
Putting (4.1) in (3.7) we obtain the
1 − fh (S ) (S ) MSE (TRP ) = (TRPO )+ nh
(k 0h
η K − K h ) = δ0h + h h 2θh
MSE of (T R(PS ) )
2 2 2 ρh η h S yh
11
as
(4.1)
Bajopas Volume 10 Number 1 June, 2017 1 − fh 2 2 2 ⇒ M S E ( T R(PS ) ) − M S E ( T R(PS O) ) = ρh η h S yh nh M S E ( T R(PS ) ) − M S E ( T R(PS O) ) ρh2 η h2 ⇒ = S (1 − ρh2 ) M S E ( T R(P O) ) It follows from (4.2) that the proportional increase in
MSE of
(T ( ) ) over that of S RP
AOE of
(T ( ) ) is less S RPO
γ h if,
than
ρ h2 η
(1
(4.2)
− ρ
2 h 2 h
< γ
)
h
(1 − ρ ) γ 2 h
i.e. η < h
ρ
2 h
,
(4.3)
h
Which clearly shows that to ensure only a small relative increase in neighborhood of “zero” if
MSE of
(T ( ) ) , S RP
ηh
must be in the
ρ is high but can depart substantially from ”zero” if ρh is moderate.
Efficiency Comparison It is well known under SRSWOR that
(y ) = ∑ W
1 − fh 2 S yh = nh
L
Var
st
2 h
h =1
From (3.7) and (5.1) we have
(
( )
) ∑W L
V a r y st − M S E T R(PS ) =
h =1
2 h
Y
2
(1 − nh
L
∑W h =1
fh
)
2 h
Y
2 h
1 − fh 2 S yh nh
(5.1)
2 θ h C xh ( 2 δ h − 1 ) (1 − 2 δ h ) θ h + 2 K h
which is non- negative if L 2∑ K h 1 1 h =1 m in , 1 + L 2 2 θh ∑ h =1
< δ st < m a x
It is to be noted that for δ h
(T R st 2
) = y st
where for
(T P s t 2
)
L
∑W h =1 L
h
∑W h =1
h
= 1,
X h C xh + M x h C xh + M
= y
st
∑
W
∑
W
h =1 L
h =1
h
h
x hC
dh
X
h
+ M
xh
C
xh
dh
(5.2)
the estimator T R(PS ) reduces to the stratified ratio-type estimator (5.3)
turns out to be the stratified product-type estimator
dh
+ M
dh
δh = 0 , the estimator T RP( S ) L
L 2∑ K h 1 1 h =1 , 1 + L 2 2 θh ∑ h =1
(5.4)
To the first degree of approximation the mean squared errors of T R st 2 , T P st 2 are respectively given by L
MSE (TRst 2 ) = ∑ Wh2 Y h 2
h =1
(1 − f h )
2 2 C yh + θ h C xh {θ h − 2 K h }
nh
(5.5)
fh ) 2 2 C yh + θ h C xh {θ h + 2 K h } n h =1 h From (3.7), (5.5) and (5.6), we have M S E (T P st 2 ) =
L
∑W
2 h
Y
2 h
(1 −
4 (1 − f h ) 2 2 2 θ hW h Y h C xh (1 − δ h )( δ h θ h − K h ) n h =1 h L 4 1− f 2 ( h ) 2 2 =∑ θ hW h Y h C xh (1 − δ h )( δ h θ h + K h ) nh h =1
(5.6)
(
) ∑
(5.7)
(
)
(5.8)
(S ) M SE (T Rst 2 ) − M SE TRP =
M SE ( T Pst 2 ) − M SE T R(PS )
L
It follows from (5.7) and (5.8) that the ratio-product estimator T R(PS ) is more efficient than i The ratio type estimator
TRst 2
if
12
Bajopas Volume 10 Number 1 June, 2017 m in
,1 < δ
L
∑
K
h =1 L
∑
θ
h =1
h
h
< m ax
st
L
,1
∑
K
∑
θ
1 +
∑
K
∑
θh
h =1 L
h
h
h =1
(5.9)
ii The product- type estimator T P 2 if m in
1 +
L
∑
K
∑
θh
h =1 L h =1
h
Further, if we set
< δ st < m a x
,0
( C xh , M d h ) = (1, 0 )
L
h =1 L h =1
h
,0
(5.10)
in (5.3) and (5.4) the ratio-type estimator
TRst 2
TPst 2
and product-type
estimators respectively reduces to
X st (usual stratified ratio estimator) x st
T R st 2 → T R st 2 = y st
(5.11)
X st (usual stratified product estimator) (5.12) x st ) = (1, 0 ) in (5.2) and (5.3) we get the mean squared errors of usual stratified ratio and
T R st 2 → T R st 2 = y st Putting
( C xh , M d h
product estimators respectively as L 2 (1 − f h M S E ( T R s t 1 ) = ∑ W h2 Y h nh h =1
M S E (T R s t 1 ) =
L
(1 −
fh
)
C
2 yh
+ C x2h {1 − 2 K
h
}
(5.13)
)
C y2h + C x2h {1 + 2 K h } nh From (3.7), (5.13) and (5.14) we have L 2 (1 − f h ) M S E (T R st 1 ) − M S E T R(PS ) = ∑ W h2 Y h (1 + θ h − 2 δ h θ h )(1 − θ h − 2 K h + 2 θ h δ h ) C x2h nh h =1
∑W h =1
2 h
2 h
Y
(
(5.14)
)
(
) ∑W
M S E (T P st 1 ) − M S E T R(PS ) =
L
h =1
2 h
2
Yh
(1 −
fh )
nh
(1 − θ h
(5.15)
2 + 2 δ h θ h )(1 + θ h + 2 K h − 2 θ h δ h ) C xh
(5.16)
From (5.15) and (5.16), we note that the ratio–product estimator T R(PS ) is better than i The stratified ratio type estimator 1 + ∑ W h θ h h =1 m in L 2∑ W h θ h h =1 L
,
L
∑ (W h =1
h
TRst1 if
L 2K h + θh 2) − 1 1 + ∑ W h θ h h =1 < δ st < m ax L L 2∑ W h θ h 2∑ Whθh h =1 h =1
,
L
∑ (W h =1
h
2 K h + θh 2 ) − 1 L 2∑ Whθh h =1
(5.17)
ii The product- type estimator T P s t 1 if L L L L ∑ W h θ h − 1 ∑ (W h 2 K h + θ h 2 ) + 1 (5.18) ∑ W h θ h − 1 ∑ (W h 2 K h + θ h 2 ) + 1 h =1 h =1 h =1 h =1 , < δ st < max , min L L L L 2∑ W θ 2∑ Wh θh 2∑ Wh θh 2∑ Wh θh h h h =1 h =1 h =1 h =1
Estimator based on the Optimum value The optimum value of
δ0h
δh
at (3.8) is
Kh 1 = 1 + 2 θh
(6.1)
Where θh is a known quantity and
K h = ρh
C yh C xh
Replacing
βh
=
ρh S y h X
and
S xh Y
Rh
h
h
=
S xyh R S x2h
=
βh Rh
by their consistent estimators
) s xyh y β$ h = 2 and R h = h respectively. xh s xh From (6.1) we get a consistent estimator of
) 1 K δ0h = 1 + h 2 θh
δ0 h
as (6.2)
13
Bajopas Volume 10 Number 1 June, 2017 where
) ) βh Kh = Rh
If the experimenter is unable to guess the value of
K h , then it is worth advisable to replace K
Thus, the estimator based on the estimated ‘optimum’ value. L ) ) W h ( X h C xh + M dh ) ) (S ) y st Kh ∑ Kh = 1 h 1 + + 1 − TRPO = 2 θh L θh ( + ) W x C M ∑ h h x h d h h =1
MSE
To obtain the Let e = 2h
( k$
h
of
− Kh
)(S ) TRPO
with E ( e 2 h
W h (xhC
∑
W h ( X hC
h =1 L
h =1
xh
xh
) Kh
in (3.9).
) + M dh )
+ M
(6.3)
dh
we write
) ,then k$
Kh
L
∑
by
h
= K h (1 + e 2 h )
) = K h + o ( n −1 ) , expanding (6.3) in terms of eh ' s we have
) ( S ) Yst K K TRPO = (1 + e0 h ) 1 + h (1 + e2 h ) (1 + θ h e1h ) −1 + 1 − h (1 + e2 h ) (1 + θ h e1h ) θh θh 2
(6.4)
From (6.4) we have
Kh K (1 + e2 h ) (1 − θ h e1h + θ 2h e1h − θ 3h e1h + θ 4h e1h + ... ) + 1 − h (1 + e2 h ) (1 − θ h e1 h ) θh θh Kh Kh 2 2 Y st 1 + θ (1 + e2 h ) − θ h e1 h − K h ( e1h + e1h e2 h ) + K h θ h ( e1h + e2 h e1 h ) + ... + 1 − θ (1 + e2 h ) + θ h e1h = + 1 e ( oh ) h h 2 − K h ( e1h + e1h e2 h ) =
Y st (1 + eoh 2
=
Y st (1 + eoh ) 2 − 2 K h ( e1h + e1h e2 h ) + K h θ h ( e12h + e2 h e12h ) + ... 2
) 1 +
K θ = Y st (1 + eoh ) 1 − K h ( e1h + e1h e2 h ) + h h ( e12h + e2 h e12h ) + ... 2
Kθ = Y st 1 + eoh − Kh ( e1h + e0h e1h + e1h e2h + e0h e1h e2h ) + h h ( e12h + e2h e12h + e0h e12h + e0h e2h e12h ) + ... 2 Neglecting the terms of
eh ' s
having power greater than two we have
) (S ) Kθ TRPO = Yst 1 + e0 h − K h e1h + K h ( e0 h e1h + e1h e2 h ) + h h e12h or 2 L )(S ) Kθ (TRPO − Yst ) = ∑ Wh Yh e0 h − K h e1h + K h ( e0 h e1h + e1h e2 h ) + h h e12h 2 h =1 Now squaring both sides of (6.5) and neglecting the terms of
eh ' s
(6.5)
having power greater than the second we
have L )(S ) (TRPO − Yst ) 2 = ∑ Wh2Yh2 e02h − K h2 e02h − 2 K h e0 h e1h
(6.6)
h =1
Taking the expectation of both sides of (6.6) we get the mean square error
)(S ) MSE of TRPO
to the first order of
approximation as L )(S ) )(S ) MSE (TRPO ) = E (TRPO − Yst ) 2 = ∑ Wh2Yh2 E e02h − K h2 e02h − 2 K h (e0 h e1h ) L
(1 − f h )
h =1
nh
= ∑ Wh2 =
L
∑
h =1
W h2
(1 − nh
fh
)
h =1
2
2 Y h C yh + K h2 C xh2 − 2 K h ρh C yh C xh
Y
2 h
2 C yh C yh 2 C y2h + ρh C x h − 2 ρh ρh C y h C x h C xh C xh
14
(6.7)
Bajopas Volume 10 Number 1 June, 2017 =
L
∑W h =1
=
L
∑W h =1
=
2 h
2 h
L
∑W h =1
2 h
(1 −
fh
)
fh
)
fh
)
nh
(1 −
nh
(1 −
Y
2 h
(C
− ρh2 C y2h
2 yh
2 Y h C yh (1 − ρh2
nh
2
S y2h (1 − ρ h2
)
)
(6.8)
MSE of TRP( S )
Which is equal to the minimum the MSE
)
or the
)
(S ) given by (3.10). Thus, we established that MSE of TRPO
)(S ) of the estimator TRPO in (6.3), based on ‘estimated optimum value’ to the first degree of
approximation is same as that of be used as an alternative to the
)(S ) )(S ) TRPO given by (3.9). So it is interesting to note the estimator TRPO )(S ) TRPO given by (3.9) if the value of the parameter is not known.
in (6.3) can
Data Presentation for Stratified Random Sampling we consider a natural population data earlier considered by Singh and Chaudary 1986 given in page 162. Y : Total number of Trees, X : Area under orchards in hectares. Table 7.1: Statistics of the Dataset Total Stratum →
N = 25
1 6
2 8
3 11
3
3
4
n = 10
Nh nh
X = 8.379 Y = 4 0 1 .8 4 0
X h Yh
6.813
10.12
7.967
417.33
503.375
340.00
S = 59.737
2 xh
15.97
132.66
38.438
2 yh
74775.467
259113.70
65885.60
2 x
S
S y2 = 12377.1
S
S xy = 2524.8
S xyh
1007.055
5709.1629
1404.71
ρxy = 0.9285
ρxyh
0.92152
0.9738
0.8827
R = 49.031
γh
0.16667
0.2083
0.15909
0.0576
0.1024
0.1936
4.34932
4.34932
4.34932
ρ + = 0 .9 4 1
W
Μ d = 4.349
Μd
2 h
The merit of the proposed estimator T R(PS ) is illustrated using a real-life dataset. We compare the efficiency of the proposed estimator T R(PS ) with some other ratio and product estimators, i.e.
yst , TRst1 , TPst1 , TRst 2 and TPst 2
in
Table 7.2 and Table 7.3
(S)
Table 7.2: Ranges of δst under which the proposed estimator TRP is better than yst , TRst1 , TPst1 , TRst 2 and TPst 2 . Estimators 1
yst
0.5 < δ st < 1.933 1.0 < δ st < 1.433
TRst 2 TPst 2
0.0 < δ st < 2.433
TRst 1
1.138 < δ st < 1.295
TPst1
−0.295 < δ st < 2.728
δ st
1.2164
Percentage Relative Efficiency (PRE) We compute the percent relative efficiency of yst , TRst1 , TPst1 , TRst 2 , TPst 2 and the proposed estimator respect to yst . The Percentage relative efficiency (PRE) of different estimators V ( y st ) P R E (T . , y s t ) = Χ 100 V (T . )
15
T
respect to
yst is
(S) TRP
with
defined as,
Bajopas Volume 10 Number 1 June, 2017 Table 7.3: Percentage relative efficiency of various estimators with respect to yst Estimator Variance
PRE (T., yst )
yst
8274.879
100
TRst1
1014.8035
815.42
TPst1
34998.554
23.64
TRst2
1437.01685
575.84
TPst2
21,538.7342
38.42
(S) TRP
842.5991
982.07
DISCUSSION We have proposed an estimator of the separate ratioproduct estimator and obtained the asymptotically optimum estimator (AOE) with its approximate MSE formula for the proposed estimator using the coefficient of variation and median of the auxiliary variable X in stratified random sampling. Theoretically, we have demonstrated that the proposed estimator is always more efficient than other
yst , TRst1 , TPst1 , TRst 2 and TPst 2 ,
δst
chosen
δst
to obtain better estimators than yst , TRst1 ,
in efficiency by using proposed estimator the
shows that even if the scalar
δst
optimum values
estimator TRP is more efficient than other estimators
TRst2 and TPst2 .
) ∑W
(
2 h
h =1
(1 − nh
) ∑W
2 h
(
)= ∑W
2 h
(
) ∑W
( ) M SE (T Rst 1 ) − M SE T RP = S
M S E ( T P s t 1 ) − M S E T R(PS )
( )
h =1
h =1 L
h =1
fh )
(1 −
(1 −
fh
nh fh
nh
(1 − f h )
h =1
nh
T
2
nh
L
( ) = ∑ Wh2 MSE (TPst 2 ) − MSE TRP S
2 h
The suggested estimator
2 Y h C yh >0
(1 −
L
L
M S E (T R st 2 ) − M S E T R(PS ) =
fh )
opt .
This
δst deviates from with its y T
T
(
(δ ) .
over
(S) RP will yield better estimate than st , Rst1 , Pst1 ,
(S)
( )
(S) TRP
estimators yst , TRst1 , TPst1 , TRst 2 and TPst 2 .
along with its optimum values for which the proposed
(S ) M SE y st − M SE T RP =
mean
Table 7.3 provides that there is a considerable gain
In addition, we support these theoretical results numerically using the data sets as shown in Table 7.1.
L
as
TPst1 , TRst2 and TPst2
and its optimum values.
Table 7.2 provides us with the wide ranges of
far
squared error criterion is considered. It is also observed from Table 4.6 that there is a scope for
estimators y st , TRst 1 , TPst 1 , TRst 2 and TPst 2 under
the effective ranges of
as
)
)
2 Y h C yh (1 − K h ) > 0 2
2
Y h C y2h (1 + K h
)
Y h C y2h ( θ h − K h
)
2
2
2
2
> 0 > 0
2 Y h C yh ( θh + K h ) > 0 2
2
It was observed from the analysis that in stratified random sampling that the proposed estimator have the minimum MSE and bias compared to some ratio and product estimators in existence and mean per unit estimator, and the proposed estimator attain its minimum MSE at its optimum value. Evidence from the study revealed that the proposed estimator is more efficient than the already existing ratio, product and ratio- product type estimators based on some certain conditions and efficiency conditions. Therefore, there is always need to ensure that the auxiliary variable is highly correlated with the study variable. Also where there is correlation between the auxiliary variable and the study variable
and such population is non-homogeneous, stratified random sampling will be more appropriate. when there is no correlation between the auxiliary variable and the study variable, the application of singlephase sample will not yield more efficient or the mean per unit will be more efficient. Hence, we conclude that the proposed class of estimator
( ) TRP is S
more
efficient than the other estimators in terms of its optimality. Thus, it is preferred to use the proposed estimator
( ) TRP over yst , TRst1 , TPst1 , TRst 2 and TPst 2 S
estimators in stratified random sampling.
16
Bajopas Volume 10 Number 1 June, 2017 REFERENCES Cochran, W.G. (1940). The estimation of the yield of the cereal experiments by sampling for ratio of grain to total produce. The journal of Agricultural Science,30: 262-275. Das, A.K. and Tripathi, T.P. (1981). A class of sampling strategies for population mean using information on mean and variance of an auxiliary character. Proceedings of Italian
Singh, H.P and Tailor, R. (2003). Estimation of finite population mean using known correlation coefficient of auxiliary characters. Statistica, Anno LXV, 4:407-418. Singh, H.P and Tailor, R. (2005). Estimation of finite population mean with known coefficient of variation of auxiliary characteristic. Statistica, anno LXV, (3):301-313. Singh, D. and Chaudary, F.S. (1986). Theory and analysis of sample survey Designs. New Age International Publisher.pp. 177-178. Singh, H.P. and Kakaran, M.S. (1993). A modified ratio estimator using known coefficient of kurtosis of an auxiliary character. Journal of
Statistical Institute, Golden Jubilee International Conference of Statistics. Applications and new directions,174-181. Housila, P.S. and Neha-Agnihotri (2008). A general procedure of estimating population mean using auxiliary information in sample surveys. Statistics in Transition-new series, 9(1):71-87. Kadilar, C. and Cingi, H. (2003). Ratio estimators in stratified random sampling. Biometrical Journal, 2:218-225. Okafor, F. C. (2002). Sample Survey Theory and Applications (1st edition). Nsukka, Nigeria. Perri, P.F. (2005). Combination of auxiliary variables in ratio-cum-product type estimators. In:
Indian Society of Agricultural Statistics, New Delhi, India. Singh, H.P. and Ruiz Espejo, M. (2003). On linear regression and ratio-product estimation of a finite population mean. The Statistician, 52(1): 59-67. Singh, H.P. and Vishwakarma, G.K. (2011). Seperate ratio- product estimator for estimating population mean using auxiliary information. Journal of Statistical Theory and Application. 10(4): 653-664. Subramani, J. and Kumarapandiyan, G. (2012). Estimation of population mean using coefficient of variation and median of the auxiliary variable. International Journal of Probability Statistics, 1(4): 111-118, DOI 10.5923/j. ijps.20120104.04.
Proceedings of Italian Statistical Society, Intermediate meeting on Statistics and Environment, Messina, Italy, September 2005,pp. 193-196. Reddy, V.N. and Singh, H.P. (1978). A study on the use of prior knowledge on certain population parameters in estimation, Sankhya C, 40: 2937.
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