A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator
for Finite Population Mean in Sample Surveys
Housila P. Singh
School of Studies in Statistics, Vikram University, Ujjain- 456010, M.P., India
hpsujn@gmail.com
Anita Yadav
School of Studies in Statistics, Vikram University, Ujjain- 456010, M.P., India
yadavd.anita@gmail.com
Abstract
This paper suggests a two-parameter ratio-product-ratio type exponential estimator for a finite population
mean in simple random sampling without replacement (SRSWOR) following the methodology in the
studies of Singh and Espejo (2003) and Chami et al (2012). The bias and mean squared error of the
suggested estimator are obtained to the first degree of approximation. The conditions are obtained in which
suggested estimator is more efficient than the sample mean, classical ratio and product estimators, ratiotype and product type exponential estimators. An empirical study is given in support of the present study.
Keywords: Finite populations mean, Study Variable, Auxiliary Variable, Bias, Mean
Squared Error.
1. Introduction
In sample surveys it is well established fact that the improvement in the precision of an
estimator of the population mean Y of the study variable y is possible by using
information on an auxiliary variable x (highly correlated with the study variable y ) at the
estimation stage. It is known that if the correlation between study variate y and the
auxiliary variate x is positive (high) the usual ratio estimator is employed. On the other
hand if this correction is negative (high), the product method of estimation can be
employed. Consider the finite population U = (U1 ,U 2 ,...,U N ) of size N. Let ( y , x ) be the
study, auxiliary variates respectively likely values ( yi , xi ) , i = 1,2,..., N respectively. Let
1 N
1 N 

 Y =  yi , X =  xi  be the population means of the variates ( y , x ) respectively. It
N i =1
N i =1 

is desired to estimate the population means of the variates ( y , x ) respectively. It is desired
to estimate the population mean Y of the study variable y using known population mean
X of the auxiliary variable x. For estimating the population mean Y , a simple random
sample of size n is drawn without replacement from the population U.
The usual unbiased estimator for population mean Y is defined by
1 n
(1)
y =  yi
n i =1
The classical ratio and product estimators for population mean Y are respectively given
by
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
Housila P. Singh, Anita Yadav
X
y R = y 
x



(1.2)
and
 x
y P= y  
X
(1.3)
It can be shown that the ratio estimator y R is more efficient than the unbiased estimator
y if
1
(1.4)
C
2
and the product estimator y P is more efficient than the usual unbiased estimator y if
1
(1.5)
C− ,
2
S yx
Sy
Cy
S
where C =
,Cy =
, Cx = x ,  =
,
X
S ySx
Y
Cx
S y2 = ( N − 1 )−1  ( yi − Y ) , S x2 = ( N − 1 )−1  (xi − X ) and
N
2
N
2
i =1
i =1
S yx = ( N − 1 )−1  ( yi − Y )(xi − X ) .
N
i =1
Bahl and Tuteja (1991) suggested the ratio-type and product-type exponential estimators
for the population Y respectively as
X −x
(1.6)

y Re = y exp 
X +x
and
x−X
y Pe = y exp
x+X

 .

(1.7)
The ratio-type exponential estimator y Re is better than unbiased estimator y if
1
(1.8)
C
4
and the product-type exponential estimator y Pe is better than the unbiased estimator y if
1
(1.9)
C−
4
In this paper taking motivation from Singh and Ruiz Espejo (2003) and Chami et al
(2012), we have suggested a class of ratio-product-ratio-type exponential estimators for
estimating the population mean Y and its properties are studied under large sample
approximation. We have compared the proposed class of ratio-product-ratio-type
exponential estimators with the three traditional estimators ( y , y R , y P ) , ratio-type
exponential estimator y Re and product-type exponential estimator y Pe and conditions are
216
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A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys
obtained in which the proposed class of ratio-product-ratio-type exponential estimators is
preferred. We carry out an empirical study showing that the proposed class of ratioproduct-ratio-type exponential estimators out performs the estimators y , y R , y Re , y P and
y Pe . In this context reader is referred to Shirley et al (2014), Singh and Pal (2015, 2017)
Singh et al (2016) and Sharma and Singh (2015).
2. The Suggested Two Parameter Ratio-Product-Ratio Type Exponential Estimator
For estimating the population mean Y of the main variable y , we suggest the following
two-parameter ratio-product-ratio type exponential estimator:

 (1 − 2 )(x − X )
 (1 − 2 )(X − x )
(2.1)
Te( , ) = y  exp 
 + (1 −  ) exp 
(x + X ) 
(x + X ) 



where , are real constants. The aim of this paper is to derive values for these constants
, such that the bias and / or the mean squared error (MSE) of Te(, ) is minimal.
We mention that Te( , ) = Te(1− ,1− , ) , that is the estimator Te(, ) is invariant under point
1 1
1 1
reflection through the point ( ,) =  ,  . In the point of symmetry ( ,) =  ,  , the
2 2
2 2
estimator reduces to the sample mean y , that is, we have T  1 1  = y . In fact, on the
e , 
2 2
whole line  =
T
1
e  , 
 2
= y.
1
our proposed estimator reduces to the sample mean estimator y , that is
2
For ( ,  ) = (1, 1) the proposed estimator Te(, ) reduces to the Bahl and Tuteja (1991)
ratio-type exponential estimator y Re defined at (1.6) while for ( ,  ) = (1, 0),(0, 1) it
reduces to the Bahl and Tuteja (1991) product-type exponential estimator y Pe defined at
(1.7). Owing to the simplicity of the proposed estimator Te(, ) and that all three known
estimators y , y Re , y Pe can be obtained from it by choosing appropriate parameters why
we study the estimator in (2.1) and compare it to the three known estimators ( y , y Re ,
y Pe ), usual ratio estimator y R and product estimator y P .
2.1 Bias and Mean Squared Error (MSE) of the Proposed Estimator
Applying the standard techniques we evaluate the first degree of approximation (upto
terms of order n −1 ) to the bias and mean squared error (MSE) of the suggested estimator
Te( , ) .
Let
y = Y (1 + e0 ) , and x = X (1 + e1 )
( )
so that E(e0 ) = E(e1 ) = 0 , E e02 =
(1 − f ) C 2 ,
n
y
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
( )
E e12 =
(1 − f ) C 2 , E (e e ) = (1 − f ) C C ,
x
0 1
y x
n
n
217
Housila P. Singh, Anita Yadav
where, f =
n
is the sampling fraction. Further, it is assumed that the sample is so large
N
as to make e0 and e1 small, justifying the first degree approximation considerd wherein
we ignore the terms involving e0 and/or e1 in a degree greater than two, see Sahai and
Ray(1980, p.272).
Bias of the Estimator Te(, )
Expressing (2.1) in terms of e’s we have

 (1 − 2 )e1 
 − (1 − 2 )e1 
Te( , ) = Y (1 + e0 ) exp 
 + (1 −  ) exp 

 (2 + e1 ) 
 (2 + e1 ) 

−1
−1 



 (1 − 2 )e1  e1  

 − (1 − 2 )e1  e1  

(
)
= Y (1 + e0 ) exp 
1
+
+
1
−

exp
1
+

 

 

2
2 
2
2 









(2.2)
Expanding the right hand side of (2.2) and neglecting terms of e’s having power greater
than two we have
(1 − 2 )(1 − 2 ) e + (3 − 4 − 2 ) (1 − 2 )e 2 − (1 − 2 )(1 − 2 ) e e 

Te( , )  Y 1 + e0 −
1
1
0 1
2
8
2

 (2.3)
or
(Te( , ) − Y ) = Y e0 − (1 − 2 )(1 − 2 ) (e1 + e0 e1 ) + (3 − 4 − 2 )(1 − 2 ) e12 
(2.4)
2
8


Taking expectation of both sides of (2.4) we get the bias of Te(, ) to the first degree of
approximation as
(1 − f ) YC 2 (1 − 2 )3 − 4 − 2 − 4C + 8C 
(2.5)
B(Te( , ) ) =
x
8n
The bias of Te(, ) would be zero if
=
1
or
2
 = 1.5 − 2 − 2C + 4C .
(2.6)
The suggested ratio-product –ratio-type exponential estimator Te( , ) , substituted with
the values of  from (2.6), becomes an (approximately) unbiased estimator for the
ˆ
3
population mean Y .In the three dimensional parameter space ( ,,C )  R , these
unbiased estimators lie on a plane (in the case  =
1
). We mention further that as the
2
sample size n approaches the population size N, the bias Te(, ) becomes negligible, since
(1 − f ) tends to zero.
the factor
n
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Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys
Mean Squared Error of Te ( , )
Squaring both sides of (2.4) and neglecting terms of e' s having power greater than two
we have
2
2
 2
(
1 − 2 ) (1 − 2 ) 2 
2
2
(Te( , ) − Y ) = Y e0 − (1 − 2 )(1 − 2 )e0 e1 +
e1 
4


(2.7)
Taking expectation of both sides of (2.7) we get the MSE of the estimator Te(, ) to the
first degree of approximation as

MSE (Te( , ) ) = E Te( , ) − Y
=
2
(1 − f ) Y 2 C 2 + (1 − 2 )(1 − 2 ) C 2 (1 − 2 )(1 − 2 ) − 4C
n

y
4
x

(2.8)
   
Taking the gradient  = 
,  of (2.6), we get
   
2(1 − f ) 2 2  (1 − 2 )(1 − 2)

MSE (Te ) =
Y Cx 
− C (1 − 2 ,1 − 2) (2.9)
n
2


Equating (2.9) to zero to obtain the critical points, we get the following solutions:
1
1
(2.10)
= , =
2
2
or
2C = (1 − 2 )(1 − 2 )
(2.11)
It can be easily shown that the critical point in (2.10) is a saddle point unless C = 0 , in
which case we get a local minimum. However, the critical points determined by (2.11)
are always local minima; for a given C , (2.11) is the equation of a hyperbola symmetric
1 1
through ( ,  ) =  ,  . Inserting (2.10) into the estimator Te( , ) gives the unbiased
2 2
estimator y (sample mean) of the population mean Y .Thus we get the MSE of the
sample mean y as


(1 − f ) Y 2 C 2 = (1 − f ) S 2 .
MSE Te  1 ,1   = MSE ( y ) =
y
y
n
n
 2 2 
(2.12)
1
in (2.1), the class of estimators Te( , ) reduces to the estimator for the
2
population mean Y as
For  =
 (1 − 2 ) (x − X )
 − (1 − 2 ) (x − X )
y
(2.13)
 + exp 

exp 
2
(x+ X )
 (x+ X )



whose bias and MSE to the first degree of approximation are respectively given by


(1 − f ) (1 − 2 )2 C 2
B T  1   = Y
(2.14)
x
 e  ,  
8
n
 2  
and
T
1 
e  , 
2 
=
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
219
Housila P. Singh, Anita Yadav

 (1 − f )
MSE  T  1   =
S y2 = MSE ( y )
 e  ,  
n
 2  
(2.15)
1
is
2
positively biased though it will be negligible for sufficiently large sample size n.
However it has MSE equal to the sample mean y . So the estimator Te (1 2, ) at (2.13) is
It is observed from (2.14) and (2.15) that the proposed estimator Te( , ) at  =
not advisable to use in practice.
By substituting (2.10) into the estimator, an asymptotically optimum estimator (AOE)
T e((0 ),  ) is obtained. For the first degree approximation of the MSE, we find (independent of
 and  )
(
)
MSE Te((0) , ) =
(1 − f ) S 2 (1 −  2 ) .
n
(2.16)
y
In fact Srivastava (1971, 1980) has shown that
(1 − f ) S 2 (1 −  2 ) is the minimal possible
y
n
MSE up to first degree of approximation for a large class of estimators to which the
x
estimator (2.1) also belongs, for example, for estimators of the form y g = y .g  
X
where g is a C 2 function with g (1) = 1 . Further Srivastava and Jhajj (1981) incorporating
sample and population variances of the auxiliary variable x might yield an estimator that
(1 − f ) S 2 1 −  2 especially when the relationship between the
has a lower MSE than
y
n
study variate y and the auxiliary variate x is markedly non-linear. Thus irrespective of
value of C, we are always able to select an AOE y e(0(),  ) from the two-parameter family in
(
)
(2.1).
3. Comparison of Mean Squared Errors and Choice of Parameters
3.1 Comparing the MSE of the Sample Mean y to the Suggested Estimator Te( ,  )
From (2.8) and (2.12) we have
(1 − f ) Y 2 C 2 (1 − 2 )(1 − 2 )(1 − 2 )(1 − 2 ) − 4C
MSE (Te( , ) ) − MSE ( y ) =
x
4n
which is less than zero if
(1 − 2 )(1 − 2 ) (1 − 2 )(1 − 2 ) − 4C  0
Therefore, either
(1 − 2 )(1 − 2 )
1
1
(i)   ,   and C 
(3.1)
2
2
4
(1 − 2 )(1 − 2 )
1
1
(ii)   ,   and C 
2
2
4
(1 − 2 )(1 − 2 )
1
1
(iii)   ,   and C 
2
2
4
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A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys
1
2
1
2
(iv)   ,   and C 
(1 − 2 )(1 − 2 )
4
Combining these with the condition
−
1
1
C  ,
4
4
we get the following explicit ranges :
1
4
1
2
(
1
2 + 4C − 1)
 
, (from (i))
4
2(2 − 1)
1
1
(ii) If 0  C  and   , then
4
2
(2 + 4C − 1)    1 , (from (iv))
2(2 − 1)
2
1
1
(iii) If −  C  0 and   , then
2
2
(2 + 4C − 1)    1 , (from (ii))
2(2 − 1)
2
1
1
(iv) If −  C  0 and   , then
2
2
(
1
2 + 4C − 1)
 
, (from (iii))
2
2(2 − 1)
(i) If 0  C  and   , then
We note that the case C=0 implies  = 0, and thus the sample mean y is the estimator
with minimal MSE.
3.2
Comparing the MSE of the Ratio-Type Exponential Estimator y Re to the
Suggested Estimator Te ( , )
1
4
For C  , the ratio-type exponential estimator y Re due to Bahl and Tuteja (1991) is used
instead of the sample mean y or product-type exponential estimator y Pe .Thus, we are
concerned with a range of plausible values for  and  , where the suggested estimator
Te( ,  ) performs better than the ratio-type exponential estimator y Re .
Putting ( ,  ) = (1, 1) in (2.8) we get the MSE of the ratio-type exponential estimator y Re
as
MSE( y Re ) =
(1 − f ) Y 2 C 2 + C x2 (1 − 4C )
n


y
4


(3.2)
From (2.8) and (3.2) we have
(
) (1 4−nf ) Y 2C x2 1 − (1 − 2 )(1 − 2 ) 1 + (1 − 2 )(1 − 2 ) − 4C 
MSE( y Re ) − MSE Te( , ) =
(3.3)
which is non negative if
(2 −  −  )2C − 1 − (2 −  −  )  0
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
(3.4)
221
Housila P. Singh, Anita Yadav
Therefore,
(i) (2C − 1)  (2 −  −  )  0
or
(ii) (2C − 1)  (2 −  −  )  0
Hence, from solution (i), where C 
1
2
1
(ii) If   , then
2
(i) If   , then
1
, we have the following:
2
( + 2C − 1)     .
(2 − 1)
(2 − 1)
( + 2C − 1) .

 
(2 − 1)
(2 − 1)
1
1
 C  , we obtain the following:
4
2
(

 + 2C − 1)
 
.
(2 − 1)
(2 − 1)
( + 2C − 1)     .
(2 − 1)
(2 − 1)
Further, from solution (ii), where
1
2
1
(ii) If   , then
2
(i) If   , then
3.3
Comparing the MSE of the Product-Type Exponential Estimator y Pe to the
Suggested Estimator Te ( , )
1
It is known that C  − , the product-type exponential estimator y Pe is preferred to the
4
sample mean estimator y and the ratio-type exponential estimator y Re .Putting
( ,  ) = (0, 1) or (1, 0) in (2.9) we get the MSE of the product-type exponential estimator
y Pe to the first degree of approximation as
MSE( y Pe ) =
(1 − f ) Y 2 C 2 + C x2 (1 + 4C )


n
y
4


(3.5)
We, therefore, seek a range of  and  values, where the suggested estimator Te( , ) has
lesser MSE than the product-type exponential estimator y Pe due to Bahl and Tuteja
(1991).
From (2.8) and (3.5) we have
(
) (1 −n f ) Y 2C x2 1 + 2 −  −   2C − (2 −  −  ) (3.6)
MSE( y Pe ) − MSE Te( , ) =
which is positive if
1 + 2 −  −   2C − (2 −  −  )  0
(3.7)
We obtain the following two cases:
(i) 2C  (2 −  −  )  −1 (if both factors in (3.7) are positive) or
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A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys
(ii) 2C  (2 −  −  )  −1 (if both factors in (3.7) are negative)
1
Observing that C  − , so we get from (i),
4
1
−  2C  (2 −  −  )  −1
2
(3.8)
1
1
 C  − , and the range for  and  where these
2
4
inequalities hold are explicitly given by the following two cases:
We mention that this implies −
(i)
( − 1)    ( + 2C ) .
(2 − 1)
(2 − 1)
( + 2C )    ( − 1) .
(2 − 1)
(2 − 1)
1
2
If   , then
1
2
(ii) If   , then
For any given C, we again note that the two regions determined here are symmetric
1 1
through ( ,  ) =  ,  . We also note that the parameters ( ,  ) which yields an AOE [see
2 2
(2.11)] which for a fixed C lie on a hyperbola, are contained in these regions.
1
1
In case (ii), where 2C  −1  i.e. C  −  (and therefore automatically C  − ), the range
of  and  are given by
2
4
( − 1)    ( + 2C ) .
(2 − 1)
(2 − 1)
( + 2C )    ( − 1) .
(2 − 1)
(2 − 1)
1
2
(i) If   , then
1
2
(ii) If   , then
3.4

Comparing the MSE of the Classical Ratio Estimator y R to the Suggested
Estimator Te ( , )
To the first degree of approximation, the MSE of the usual ratio estimator y R is given by
MSE ( y R ) =
(1 − f ) Y 2 C 2 + C 2 (1 − 2C )
n
y
(3.9)
x
From (2.8) and (3.9) we have
(
) (1 −n f ) Y 2C x2 1 − (1 − 2 )(21 − 2 ) 1 + (1 − 2 )(21 − 2 ) − 2C 
MSE( y R ) − MSE Te( , ) =



(3.10)
which is non-negative if
 (1 − 2 )(1 − 2 )
1 −

2
(1 − 2 )(1 − 2 )  0

1 − 2C +

2
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
(3.11)
223
Housila P. Singh, Anita Yadav
Therefore,
(i)
(ii)
(2C − 1)  (1 − 2 )(1 − 2 )  1
2
(
1 − 2 )(1 − 2 )
(2C − 1) 
1
or
2
Hence from solution (i), where C  1, we have the following:
(2 + 4C − 3)    (2 + 1) .
2(2 − 1)
2(2 − 1)
(2 + 1)    (2 + 4C − 3) .
2(2 − 1)
2(2 − 1)
1
, then
2
1
(ii) If   , then
2
(i) If  
1
 C  1, we obtain the following:
2
(2 + 1)    (2 + 4C − 3) .
2(2 − 1)
2(2 − 1)
(2 + 4C − 3)    (2 + 1) .
2(2 − 1)
2(2 − 1)
Also from solution (ii), where
1
, then
2
1
(ii) If   , then
2
(i) If  
3.5
Comparing the MSE of the classical Product Estimator y P to the Suggested
Estimator Te ( , )
1
It is well known that, for C  − , the product estimator y p is more efficient than the
2
sample mean y and ratio estimator y R . Thus, we are concerned with a range of plausible
values for  and  where the proposed estimator Te( , ) acts better than the classical
product estimator y P .
The MSE of the usual product estimator y p to the first degree of approximation as
MSE ( y P ) =
(1 − f ) Y 2 C 2 + C 2 (1 + 2C )
n
y
(3.12)
x
From (2.8) and (3.12) we have
(
) (1 −n f ) Y 2C x2 1 + (1 − 2 )(21 − 2 ) 1 + 2C − (1 − 2 )(21 − 2 )
(3.13)
(1 − 2 )(1 − 2 )  0

1 + 2C −

2
(3.14)
MSE( y P ) − MSE Te( , ) =



which is positive if
 (1 − 2 )(1 − 2 )
1 +

2
224
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys
Thus the proposed estimator Te( , ) is more efficient than the classical product estimator
y p if
either − 1 
(1 − 2 )(1 − 2 )  (2C + 1)






2
or (2C + 1) 
(1 − 2 )(1 − 2 )  −1
2
(3.15)
or equivalently,
min.− 1,(2C + 1) 
(1 − 2 )(1 − 2 )  max.− 1,(2C + 1)
2
(3.16)
4. Unbiased Asymptotically Optimum Estimator (AOE)
From (2.6) and (2.11), we can calculate the parameters  and  where our proposed
estimator becomes at least up to first approximation- an unbiased AOE. We obtain a line
1
with  = (recall that on this line our estimator always reduces to the sample mean
2
estimator y )
1
C=0
(4.1)
= ,
2
(
)
or a “curve”   (C ),   (C ),C  R 3 in the parameter space with

1
  (C ) = 1 
2
C  
1
 ,  (C ) =  C (2C − 1).

2
2C − 1 
(4.2)
We mention that the parametric “curve” in (4.2) is only defined for C  0 or C 
1
in
2
fact, this parametric “curve” is three hyperbolas.
Inserting the values of   (C ) and   (C ) from (4.2) in (2.1), the proposed estimator takes
the form:
ye (C ) = T
e    (C ),  (C ) 


=

C 
 2 C (2C − 1)(X − x )
 
exp
 + 1 −

2C − 1 
(X + x )



 
y 
 1+
2 


C 
 2 C (2C − 1)(x − X )

exp

2C − 1 
(X + x )




It can be easily shown to the first degree of approximation that

B y e (C ) = 0,

(
1− f ) 2

2 
MSE y e (C ) =
Sy 1−  
n

(
(4.3)
)
(
)
(
)
(4.4)
Thus, the estimator ye (C ) of (4.3) is an unbiased AOE.
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
225
Housila P. Singh, Anita Yadav
Using (2.11) in (2.5) we get the first degree of approximation of the bias of an AOE
B(Te( , ) ) =
(1 − f ) YC 2 4C (1 − 2C ) + (1 − 2 )2 
8n
(4.5)
x
1
.
2
Otherwise, there is always a positive contribution coming from the term 4C (1 − 2C ) that
does not vanish no matter what we select for   R. Further from (4.5) we note that the
It follows from (4.5) and (2.11) that the bias can only be made zero if C  0 or C 
choice  =
1
always yields the least possible bias. Given (2.11) and unless C=0, we can
2
only assume  be close to
1
and select  accordingly.
2
Remark 5.1: It is important to mention that the optimum choice of  = (1 − 2 )(1 − 2 )
depends upon the value of C =  yxC y C x .Reddy (1978) has shown that the value of ‘C’
is more stable than other population parameters such as, the linear regression coefficient,
over a period of time and least affected by the sampling fluctuations. So its value can be
quite accurately guessed from the past data or a pilot survey or experienced gathered in
due course of time. However, if the value of ‘C’ is not known, one may estimate it on the
basis of sample observations without much loss of efficiency, for instance, see Singh et al
(1994, p. 216).
6. Empirical Study
To illustrate the performance of the proposed class of estimators Te( , ) over sample
mean estimator y , Bahl and Tuteja’s (1991) ratio-type ( y Re ) and product-type ( y Pe )
exponential estimators, classical ratio estimator y R and product estimator y P we have
two natural population data sets.
Population-I [Source: Kadilar and Cingi (2003)]
y: Apple production amount.
x: Number of apples trees.
N = 106 ,
C y = 5.22 ,
n = 20 ,
Y = 2212.59 , X = 27421.70 ,
 = 0.86
C x = 2.10 ,
Population-II [Source: Gupta and Kothwala (1990)]
y: Population of irrigated area
x: Area under crop gram and mixture during 1983-1984 in a village of Rajasthan
N = 400 ,
C y = 0.9928 ,
226
n = 100
,
C x = 0.9617 ,
Y = 36.7183 ,
X = 6.5638 ,
 = −0.4020
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys
We have computed the percent relative efficiency (PRE) of the proposed class of
estimators Te( , ) with respect to y , y Re , y Pe , y R and y P , using the following formulae:
(
)
PRE Te( , ), y =
(
C + (C 4)(1 − 2 ) (1 − 2 )
)
2
y

(
  100

+ C x2 (1 − 2C )
2
y
(6.1)
− 4C (1 − 2 )(1 − 2 )
C + (C 4)(1 − 2 ) (1 − 2 ) − 4C(1 − 2 )(1 − 2 )  100
C + C (1 − 2C )
)=
C + (C 4)(1 − 2 ) (1 − 2 ) − 4C(1 − 2 )(1 − 2 )  100
(6.3)
+
(
C x2
)
4 (1 − 2 ) (1 − 2 ) − 4C (1 − 2 )(1 − 2 )
2
2
C + (C 4)(1 − 4C )
2
y
)
(
C
2
(6.2)
C y2
PRE Te( , ), y Re =
PRE Te( , ), y P
2
2
x
  100
PRE Te( , ), y R =
(
C y2
2
y
2
x
2
2
x
2
2
y
2
y
)
PRE Te( , ), y Pe =
2
x
2
2
x
C + (C 4)(1 + 4C )
2
y
2
x
C + (C 4)(1 − 2 ) (1 − 2 )
2
y
2
2
x
(6.4)
2
2
  100
(6.5)
− 4C (1 − 2 )(1 − 2 )
Note 1: We have computed the values of PRE (Te( , ), y ), PRE (Te( , ), y R ), and
PRE (Te( , ), y Re ) for population I as the correlation coefficient between study variate y
and auxiliary variate x is positive.
Note 2: We
have
computed
the
values
of PRE (Te( , ), y ), PRE (Te( , ), y P )
and
PRE (Te( , ), y Pe ) for population II as the correlation between the study variable y and
variable x is negative.
Findings are shown in tables 6.1 and 6.2.
Table 6.1: PREs of the proposed class of estimators Te( , ), with respect to the
yR
usual unbiased estimator y , ratio estimator
exponential estimator y Re for population I
(
( ,  )
PRE Te ( , ) , y
(-2.00,0.00)
(-2.00,0.25)
(-1.75,0.00)
(-1.75,0.25)
(-1.50,-0.25)
(-1.50,-0.00)
(-1.50,0.25)
355.06
257.77
381.04
234.53
262.65
379.55
212.82
)
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
(
PRE Te ( , ) , y R
246.58
179.02
264.63
162.88
182.41
263.59
147.80
)
and ratio-type
(
PRE Te ( , ) , y Re
)
166.84
121.12
179.05
110.20
123.42
178.35
100.00
Table 6.1 continued…
227
Housila P. Singh, Anita Yadav
( ,  )
(-1.25,-0.25)
(-1.25,0.00)
(-1.00,0.50)
(-1.00,0.25)
(-1.00,0.00)
(-0.75,-0.75)
(-0.75,-0.50)
(-0.75,-0.25)
(-0.75,0.00)
(-0.50,-1.00)
(-0.50,-0.75)
(-0.50,-0.50)
(-0.50,-0.25)
(-0.50,-0.00)
(-0.25,-1.50)
(-0.25,-1.25)
(-0.25,-1.00)
(-0.25,-0.75)
(-0.25,-0.50)
(-0.25,-0.25)
(0.00,-2.00)
(0.00,-1.75)
(0.00,-1.50)
(0.00,-1.25)
(0.00,-1.00)
(0.00,-0.75)
(0.00,-0.50)
(0.25,-2.00)
(0.25,-1.75)
(0.25,-1.50)
(1.25,1.25)
(1.25,1.50)
(1.25,1.75)
(1.25,2.00)
(1.25,2.25)
(1.50,1.00)
(1.50,1.25)
(1.50,1.50)
(1.50,1.75)
(1.50,2.00)
(1.75,1.00)
(1.75,1.25)
(
PRE Te ( , ) , y
334.64
351.21
262.65
381.04
306.54
239.15
355.06
368.23
257.77
262.65
355.06
379.55
306.54
212.82
262.65
334.64
381.04
368.23
306.54
234.53
355.06
381.04
379.55
351.21
306.54
257.77
212.82
257.77
234.53
212.82
234.53
306.54
368.23
381.04
334.64
212.82
306.54
379.55
355.06
262.65
257.77
368.23
)
(
PRE Te ( , ) , y R
232.40
243.91
182.41
264.63
212.89
166.08
246.58
255.73
179.02
182.41
246.58
263.59
212.89
147.80
182.41
232.40
264.63
255.73
212.89
162.88
246.58
264.63
263.59
243.91
212.89
179.02
147.80
179.02
162.88
147.80
162.88
212.89
255.73
264.63
232.40
147.80
212.89
263.59
246.58
182.41
179.02
255.73
)
(
PRE Te ( , ) , y Re
157.24
165.03
123.42
179.05
144.04
112.37
166.84
173.03
121.12
123.42
166.84
178.35
144.04
100.00
123.42
157.24
179.03
173.03
144.04
110.20
166.84
179.05
178.35
165.03
144.04
121.12
100.00
121.12
110.20
100.00
110.20
144.04
173.03
179.05
157.24
100.00
144.04
178.35
166.84
123.42
121.12
173.03
)
Table 6.1 continued…
228
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys
( ,  )
(1.75,1.50)
(1.75,1.75)
(2.00,1.00)
(2.00,1.25)
(2.00,1.50)
(2.25,1.00)
(2.25,1.25)
(2.50,0.75)
(2.50,1.00)
(2.50,1.25)
opt (,)
(
PRE Te ( , ) , y
355.06
239.15
306.54
381.04
262.65
351.21
334.64
212.82
379.55
262.65
384.02
)
(
PRE Te ( , ) , y R
246.58
166.08
212.89
264.63
182.41
243.91
232.40
147.80
263.59
182.41
266.70
)
(
PRE Te ( , ) , y Re
166.84
112.37
144.04
179.05
123.42
165.03
157.24
100.00
178.35
123.42
180.45
)
Table 6.2: Percent relative efficiencies (PREs) of the proposed class of estimators
Te( , ) with respect to usual unbiased estimator y , product estimator y P
and Product- type exponential estimator y Pe for population II
( ,  )
(-2.00,0.75)
(-2.00,1.00)
(-1.75,0.75)
(-1.75,1.00)
(-1.75,1.25)
(-1.50,0.75)
(-1.50,1.00)
(-1.50,1.25)
(-1.25,1.00)
(-1.25,1.25)
(-1.25,1.50)
(-1.00,1.00)
(-1.00,1.25)
(-1.00,1.50)
(-0.75,1.00)
(-0.75,1.25)
(-0.75,1.50)
(-0.75,1.75)
(-0.50,1.00)
(-0.50,1.25)
(-0.50,1.50)
(-0.50,1.75)
(-0.50,2.00)
(-0.50,2.25)
(
PRE Te ( , ) , y
163.25
197.16
156.60
197.74
166.29
149.82
193.99
183.80
186.38
195.24
159.72
175.78
197.74
183.80
163.25
190.62
197.16
178.46
149.82
175.78
193.99
197.16
183.80
159.72
)
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
(
PRE Te ( , ) , y p
132.43
159.93
127.02
160.40
134.89
121.53
157.36
149.09
151.18
158.37
129.56
142.58
160.40
149.09
132.43
154.62
159.93
144.76
121.53
142.58
157.36
159.93
149.09
129.56
)
(
)
PRE Te ( , ) , y pe
108.97
131.60
104.52
131.99
110.99
100.00
129.48
122.68
124.40
130.31
106.61
117.32
131.99
122.68
108.97
127.23
131.60
119.12
100.00
117.32
129.48
131.60
122.68
106.61
Table 6.2 continued…
229
Housila P. Singh, Anita Yadav
( ,  )
(-0.25,1.25)
(-0.25,1.50)
(-0.25,1.75)
(-0.25,2.00)
(-0.25,2.25)
(0.75,-2.00)
(0.75,-1.75)
(0.75,-1.50)
(1.00,-2.00)
(1.00,-1.75)
(1.00,-1.50)
(1.00,-1.25)
(1.00,-1.00)
(1.00,-0.75)
(1.00,-0.50)
(1.25,-1.75)
(-1.75,-1.50)
(-1.25,-1.25)
(-1.25,-1.00)
(-1.25,-0.75)
(-0.75,-0.50)
(-0.50,-0.25)
(1.50,-1.25)
(1.50,-1.00)
(1.50,-0.75)
(1.50,-0.50)
(1.50,-0.25)
(1.50,0.00)
(1.75,-0.75)
(1.75,-0.50)
(1.75,-0.25)
(1.75,0.00)
(2.00,-0.50)
(2.00,-0.25)
(2.00,0.00)
(2.25,-0.50)
(2.25,-0.25)
(2.25,0.00)
(2.50,-0.25)
(2.50,0.00)
(2.50,0.25)
opt ( ,  )
230
(
PRE Te ( , ) , y
156.60
175.78
190.62
197.74
195.24
163.25
156.60
149.82
197.16
197.74
193.99
186.38
175.78
163.25
149.82
166.29
183.80
195.24
197.47
190.62
175.78
156.60
159.72
183.80
197.16
193.99
175.78
149.82
178.46
197.16
190.62
163.25
183.80
197.74
175.78
159.72
195.24
186.38
183.80
193.99
149.82
198.04
)
(
PRE Te ( , ) , y p
127.02
142.58
154.62
160.40
158.37
132.43
127.02
121.53
159.93
160.40
157.36
151.18
142.58
132.43
121.53
134.89
149.09
158.37
160.40
154.62
142.58
127.02
129.56
149.09
159.93
157.36
142.58
121.53
144.76
159.93
154.62
132.43
149.09
160.40
142.58
129.56
158.37
151.18
149.09
157.36
121.53
160.64
)
(
PRE Te ( , ) , y pe
104.52
117.32
127.23
131.99
130.31
108.97
104.52
100.00
131.60
131.99
129.48
124.40
117.32
108.97
100.00
110.99
122.68
130.31
131.99
127.23
117.32
104.52
106.61
122.68
131.60
129.48
117.32
100.00
119.12
131.60
127.23
108.97
122.68
131.99
117.32
106.61
130.31
124.40
122.68
129.48
100.00
)
132.18
Pak.j.stat.oper.res. Vol.XIV No.2 2018 pp215-232
A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys
Tables 6.1 and 6.2 exhibit that
(i)
there is considerable gain in efficiency by using the proposed class of estimators
Te( , ) over ( y , y R , y Re ) for population I and ( y , y P , y Pe ) for population II;
(ii)
(iii)
(iv)
larger gain in efficiency is seen by using Te( , ) over y as compared to y R ( y P )
and y Re ( y Pe ) .
largest gain in efficiency is observed at optimum ( ,  ) ;
there is considerable gain in efficiency by using the proposed class of estimators
Te( , ) over ( y , y R , y Re ) for population I and ( y , y P , y Pe ) for population II even
when the values of ( ,  ) deviates from their optimum values of ( ,  ) .Thus there
is enough scope of choosing the values of scalars ( ,  ) for obtaining estimators
better than ( y , y R , y Re , y P , y Pe ) from the proposed class of estimators Te( ,  ) .
Acknowledgement
Authors are thankful to the referee for his valuable suggestions regarding improvement of
the paper.
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