Abstract Using auxiliary information, a family of difference-cum-exponentia

Jhajj H S

Lata K

CiteSeerX

 Research Journal of Mathematical and Statistical Sciences ________________________________ISSN 2320–6047 
 Vol. 3(4), 10-14, April (2015) Res. J. Mathematical and Statistical Sci. 
 
 International Science Congress Association        10 
A Family of Difference-cum- Exponential Type Estimators for Estimating the 
Population Variance Using Auxiliary Information in Sample Surveys 
Jhajj H.S.
1
 and Lata K.
2 
1Dept. of Statistics, Punjabi University Patiala, INDIA 
2Swami Devi Dyal College, INDIA 
Available online at: www.isca.in, www.isca.me 
Received 3rd February 2015, revised 31st March 2015, accepted 8th April 2015 
 
 
 
Abstract 
Using auxiliary information, a family of difference-cum-exponential type estimators for estimating the population variance of 
variable under study have been proposed under double sampling design. Expressions for bias, mean squared error and its 
minimum values have been obtained. The comparisons have been made with the regression-type estimator by using simple 
random sampling at both occasions in double sampling design. It has also been shown that better estimators can be obtained 
from the proposed family of estimators which are more efficient than the linear regression type estimator. Results have also 
been illustrated numerically as well as graphically. 
 
Keywords: Regression-type estimator, mean squared error, double sampling, auxiliary variable, efficiency. 
 
Introduction 
In survey sampling, taking the advantage of the correlation 
between the auxiliary variable x and study variable y, the 
construction of efficient ratio and product type estimators for 
the population variance has widely been made by many 
statisticians to assess the variations in population. Because it is 
well known that the use of auxiliary information in the 
construction of estimators can increase the efficiency of the 
estimators of parameter of interest. When the population 
variance 2S x  of auxiliary variable x, which is highly correlated 
with the study variable y, is known in advance, several 
estimators have been defined by different authors such as Das 
and Tripathi
1
, Isaki
2
, Ahmed et al.
3
, Jhajj et al.
4
, Kadilar and 
Cingi
5
, Pradhan et al.
6 
in the literature for estimating the 
population variance of study variable y. Sometimes information 
on population variance 2S x  of auxiliary variable x is not known 
in advance then we generally use two phase (double) sampling 
design. In the two- phase sampling design, a large preliminary 
random sample (called first phase sample) is drawn and 
auxiliary information are taken on sample units which are used 
to estimate the value of unknown population variance 2S x  of 
auxiliary variable x. Then second phase sample is drawn either 
from the first phase sample or independently from the 
population and observations are taken on study as well as 
auxiliary variables. 
 
In present paper we have proposed a family of Difference-
Cum-Exponential type estimators for estimating the population 
variance of study variable using auxiliary information under 
double sampling design. The efforts have been made to 
compare the proposed family of estimators under simple 
random sampling design as a special case. 
Notations and Results 
A preliminary large random sample (first phase sample) of size 
n′  is drawn from a finite population of size  N  and both 
auxiliary variable  x  and study variable  y  are measured on it. 
Then second phase random sample of size ( )n n′<  is drawn 
from the first phase sample. 
Let Yi  and Xi  denote the respective values of variables y and 
x on the  ( )1, 2,....,thi i N= unit      of the population and the 
corresponding small letters denote the values in the samples. 
Denoting 
( )
12 2
1
1 ( )
n
y i
i
s n y y
−
=
= − −∑ ( )
12 2
1
1 ( )
n
y i
i
s n y y
′
−
=
′ ′= − −∑
 
( )
12 2
1
1 ( )
n
x i
i
s n x x
−
=
= − −∑
 
( )
12 2
1
1 ( )
n
x i
i
s n x x
′
−
=
′ ′= − −∑
 
( ) ( )
212
1
1
N
Y i
i
S N Y Y
−
=
= − −∑
 
( ) ( )
212
1
1
N
X i
i
S N X X
−
=
= − −∑
  
( ) ( ) ( )1
1
1
N
XY i i
i
S N Y Y X X
−
=
= − − −∑
 
( )
1
1
( )
N
ba
ab i i
i
X X Y Y
N
µ
=
= − −∑
2 2
20 02
ab
ab a b
µ
λ
µ µ
=
  
 
where 
2
sx′ ,
2
sx  and  
2
sy′ ,
2
sy   are the sample variances of 
variable x and y based on the sampling units of first and second 
phase samples of size n′  and n  respectively. 
 
Research Journal of Mathematical and Statistical Sciences ___________________________________________ISSN 2320–6047 
Vol. 3(4), 10-14, April (2015)       Res. J. Mathematical and Statistical Sci. 
 International Science Congress Association           11 
Defining  
2
0 2
1
y
y
s
S
∈ = −
       
2
1 2
1
y
y
s
S
′
∈ = −
 
2
2 2
1x
x
s
S
∈ = −
        
2
3 2
1x
x
s
S
′
∈ = −
  
 
Assuming that 
( ) ( ) ( ) ( )0 1 2 3 0E E E E∈ = ∈ = ∈ = ∈ =                                
( ) ( )
( ) ( )
( ) ( )
( )
2 2
var( ) var( )2 2
                      
0 14 4
2 2
var( ) var( )2 2
                       
2 34 4
2 2 2 2cov( , ) cov( , )
            
0 1 2 34 4
2 2
cov( , )
         
0 2 2 2
s sy y
E E
S Sy y
s sx x
E E
S Sx x
s s s sy y x xE E
S Sy x
s sx y
E
S Sx y
ε ε
ε ε
ε ε ε ε
ε ε
′
= =
′
= =
′ ′
= =
= ( )
( ) ( )
2 2
cov( , )
    
1 2 2 2
2 2 2 2
cov( , ) cov( , )
              
0 3 1 32 2 2 2
s sx y
E
S Sx y
s s s sx y x y
E E
S S S Sx y x y
ε ε
ε ε ε ε
′
=
′ ′ ′
= =


















  (1)  
 
The Proposed Family of Estimators and Its Results 
When we don’t have an information about any parameter of the 
auxiliary variable, then we propose a family of estimators of the 
population variance  
2
y
S
 of the study variable y under the 
sampling design defined in section 2 as 
( )
( )2 2 2 22 2 2 2ˆ 1 exp 2
2 2 2
s sx x s sx x
S s s sy y y y
s s sx x x
α
θ
θ
θα
′ − ′ −
′= + − −
′ ′ +
                       
(2)  
Where  0 and  θ α> are any constants 
To obtain the bias and mean square error of estimator 2Ŝ yθα
, up 
to first order of approximation, we expand 2Ŝ yθα
 in terms of 
' sε  and retaining terms up to second degree of approximation.   
( ) ( )( ) ( )( )
( )( ) ( ) ( )
( ) ( ) ( ) ( )
2 2ˆ 1 1 1
0 1 0 3 2 1 3 1 2 0 3 0 2
21 1 2 2
1
0 3 0 2 3 2 2 3
2 2
21 22
 1  
2 3 3 3 2
2
S Sy y ε θ ε ε α θ ε ε αθ θ ε ε ε ε ε ε ε εθα
α θ ε ε ε ε α θ ε ε ε ε
α α
αθ ε ε ε θ ε ε
= + + − + − − + − − − +
+ − − + − − − −
−
− − + − −


  
  
  


 (3) 
Noting that expectation exists and finite, we take expectation in 
(3) and using the results of section 2, we have 
( ) ( ){ }
( ) ( ) ( )
( ) ( )
( ) ( )2 2 2 2 2 2cov , cov , 12 22 2 2ˆ 1 + 1 1   
2 2 42
s s s s V s V sx y x y x x
E S S Sy y y
S S Sx y x
α α
α θ θ α θ
θα
′ ′− −−
= + − − + −
    
                  
which implies that 
( ) ( ){ }
( ) ( )
( )
( ) ( )
( ) ( )
2 2 2 2
cov , cov ,22 2ˆ 1
2 2
2 2
1 2
+ 1 1  
42
s s s sx y x y
Bias S Sy y
S Sx y
V s V sx x
Sx
α θ
θα
α α
θ α θ
′ −
= −
′−−
− + −
  
      
        
(4) 
 
Upto first order of approximation the mean square error (MSE) 
of estimator 2Ŝ yθα
 is obtained from (3) and using the results of 
section 2, as 
 
( )
2
2 2 2ˆ ˆ
y y y
MSE S E S S
θα θα
 = −   
( ) ( ) ( ) ( ){ }
( ) ( ) ( ){ }
( ) ( ) ( ){ }
2 2 2 2
2
4
S2 y2 2 2
   1  
4
Sx
2
S2 y 2 2 2 2
    - 2 1 cov , cov ,
2
Sx
V s V s V sy y y
V s V sx x
s s s sx y x y
θ θ
α θ
α θ
′= + − −
′+ − −
′− −
      (5) 
The expression (5) depends upon two unknown constants
 and α θ , so keeping the value of  θ  fixed, we differentiate (5) 
w.r.t  α and equating to zero, we get  
( ) ( ) ( ){ } ( ) ( ) ( ){ }
4 2
S S2 2y y2 2 2 2 2 2
   2 1  -2 1 cov , cov , 0
4 2
S Sx x
V s V s s s s sx x x y x yα θ θ′ ′− − − − =
 
After solving the above equation, we get the optimum value of  α  
( )
( ) ( ){ }
( ) ( ){ }
2 2 2 2 2
S cov , cov ,x
   =
opt 2 2 2
S  y
s s s sx y x y
V s V sx x
α
′−
′−
          (6) 
 
Substituting the optimum value of  α from (6) in (5), we get 
minimum mean square error, ( )2ˆ. MSEMin S yθα  as 
Research Journal of Mathematical and Statistical Sciences ___________________________________________ISSN 2320–6047 
Vol. 3(4), 10-14, April (2015)       Res. J. Mathematical and Statistical Sci. 
 International Science Congress Association           12 
( ) ( ) ( ) ( ) ( ){ }
( )
( ) ( ){ }
( ) ( ){ }
2 2 2 2 2ˆ. MSE 2
2 2 2 2
cov , cov ,
2
 - 1
2 2
Min S V s V s V sy y y y
s s s sx y x y
V s V sx x
θ θ
θα
θ
′= + − −
′−
−
′−
   (7) 
 
Theorem1: Up to first order of approximation, the bias of 
estimator 
2
Ŝ yθα
 is 
( ) ( ){ } ( ) ( )
( )
( ) ( )
( ) ( )
2 2 2 2
cov , cov ,22 2ˆ 1
2 2
2 2
1 2
                          + 1 1          
42
s s s sx y x y
Bias S Sy y
S Sx y
V s V sx x
Sx
α θ
θα
α α
θ α θ
′ −
= −
′−−
− + −
  
      
 
       
 
 
And its mean square error is  
( ) ( ) ( ) ( ) ( ){ }
( ) ( ) ( ){ }
( ) ( ) ( ){ }
2 2 2 2 2ˆ 2
4
S2 y2 2 2
                   1  
4
S x
2
S2 y 2 2 2 2
                   - 2 1 c o v , co v ,
2
S x
M S E S V s V s V sy y y y
V s V sx x
s s s sx y x y
θ θ
θ α
α θ
α θ
′= + − −
′+ − −
′− −
 
Theorem 2:  Up to first order of approximation, the MSE of 
estimator 
2
Ŝ yθα
 is minimized for  
( )
( ) ( ){ }
( ) ( ){ }
2 2 2 2 2
S cov , cov ,x
   =
opt 2 2 2
S  y
s s s sx y x y
V s V sx x
α
′−
′−
 
And its minimum value is given by 
 
( ) ( ) ( ) ( ) ( ){ }
( )
( ) ( ){ }
( ) ( ){ }
2 2 2 2 2ˆ. MSE 2
2 2 2 2
cov , cov ,
2
 - 1
2 2
Min S V s V s V sy y y y
s s s sx y x y
V s V sx x
θ θ
θα
θ
′= + − −
′−
−
′−
 
For comparing the proposed family of estimator with the 
existing ones we consider the special case. 
 
Special case: when simple random sampling is applied for 
selection of samples in both phases of double sampling design 
we have  
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
1 1 1 12 4 2 4
var 1                     var 1
40 40
1 1 1 12 4 2 4
var 1                     var 1
04 04
1 12 2 2 2 2
cov , 1          cov ,
22
s S s Sx x x x
n N n N
s S s Sy y y y
n N n N
s s S S s sx y x y x
n N
λ λ
λ λ
λ
′= − − = − −
′
′= − − = − −
′
′ ′= − −
   
   
   
   
   
   
 
 
 
( ) ( )1 12 2 2 122S Sy x y
n N
λ= − −
′






 
  
  
 (8) 
 
Substituting results of (8) in (4) to (7) respectively, we have
( ) ( ) ( ) ( ) ( ) ( ) ( )11 1 2 22 2ˆ 1 1 1 1 140 222Bias S Sy yn n
α α
θ α θ λ α θ λ
θα
−
= − − + − − − − −
′
   
    
    
(9) 
( ) ( )
( )( ) ( ) ( ) ( ) ( ){ }
4
1 12 4ˆ 1
04
2 22 2
2 1 1 1 2 1 1
04 40 22
S y
MSE S Sy y
n n n
λ
θα
θ θ λ α θ λ α θ λ
= − + −
′
− − + − − − − −
 
 
 
 
(10) 
 
( )
1
22
1
40
opt
λ
α
λ
−
=
−
                                                    (11) 
 
( ) ( )( )( )( )
( )
( )
( ) ( )( )
22
1 1 11 1 22 2 22 22 40 22ˆ 1 1 1
22
2 1 1
40 40
Bias S Sy y
n nopt
λ λ λ θ λ
θ λ θ
θα λ λ
− − − −
= − − − + − −
′ − −
 
   
    
 
( ) ( ) ( )( ) ( )( ) ( )
24
11 1 22 4 2 22ˆ 1 2 1 1
04 04min
1
40
Sy
MSE S Sy y
n n n
λ
λ θ θ λ θ
θα λ
−
= − + − − − − −
′ −
 
   
    
 
(12) 
From (12), we see that ( )2ˆ. M S EM in S yθα will vary with the 
change in variation of θ  so range of variation of θ  can be 
obtained at which proposed family of estimator is better than 
the existing ones.  
 
Comparison 
For comparing the proposed family of estimators with the linear 
regression type estimator
2
s
lrd
, considered by  Mishra
8 
( )
( )
( )
2 2
cov ,
2 2 2 2
2
var
s sy x
s s s sy x xlrd
sx
′= + −  
 
we first obtain expressions of its bias and mean square error, 
under double sampling design using simple random sampling at 
both phases, up to first order of approximation,  
( ) ( )
( )
( )
2
11 1 12 4 4 22
1
04
1
40
MSE s S Sy ylrd n n n
λ
λ
λ
−
= − − −
′ −
 
 
 
  (13) 
 
Using (12) and (13) and after some algebra, we obtain   
( ) ( ) ( )( )( )1 12 2 4 2 2ˆ 1 2 1 004minMSE s MSE S Sy y vlrd n n λ θ θ ρθα− = − − − − ≥′
 
 
 
  (14) 
Where ( )
( )( )
1
2 2
1 1
0 4 4 0
v
λ
ρ
λ λ
−
=
− −
 
( ) ( )2 2ˆ     if      0 < < 2minM SE S M SE sy lrd θθα⇒ ≤          (15) 
 
 
Research Journal of Mathematical and Statistical Sciences ___________________________________________ISSN 2320–6047 
Vol. 3(4), 10-14, April (2015)       Res. J. Mathematical and Statistical Sci. 
 International Science Congress Association           13 
Numerical Illustration 
For comparing the efficiency of the proposed estimator ( 2Ŝ yθ α
) with Regression-type estimate ( 2s
lrd
) under double sampling 
design, we take the empirical population considered in 
literature (source: Institute of Statistics, Republic of Turkey). 
This empirical data concerns the level of apple production (1 
unit=100 tonnes) as the variate of interest and number of apple 
trees (1 unit=100 trees) as the auxiliary variate in 106 villages 
in the Marmarian Region respectively in 1999. The values of 
population parameters obtained are given in Table 1. The 
biases, mean squared error and relative efficiency of proposed 
estimator ( 2Ŝ yθα
) w.r.t regression-type estimator ( 2s
lrd
) are 
given for some different values of θ  in table 2. 
 
Table-1 
Value of population parameters 
N  n′  n  yS  04
λ
 40
λ
 22
λ
 31
λ
 
106 48 20 64.25 80.13 25.71 33.30 29.67 
 
Table-2 
Mean Squared Error and Relative Efficiency 
Values of 
θ  
Mean Squared Error Efficiency 
2
lrd
s
 
2ˆ
y
S
θα  
2
lrd
s
 
2ˆ
y
S
θα  
0 46483968.52 46483968.5 100 100 
0.5 46483968.52 32757889.7 100 142 
1.0 46483968.52 28182530.1 100 165 
1.5 46483968.52 32757889.7 100 142 
2.0 46483968.52 46483968.5 100 100 
-3 -2 -1 0 1 2 3
0.00E+000
5.00E+007
1.00E+008
1.50E+008
2.00E+008
2.50E+008
3.00E+008
3.50E+008
M
S
E
ϑ
 Proposed estimator
 Regression- type estimator
 
Figure-1 
Graphical Representation of MSE of Estimators 
 
Research Journal of Mathematical and Statistical Sciences ___________________________________________ISSN 2320–6047 
Vol. 3(4), 10-14, April (2015)       Res. J. Mathematical and Statistical Sci. 
 International Science Congress Association           14 
Conclusion 
From table 2, we observe that in double sampling for 0 < < 2θ  
the efficiency of proposed estimator ( 2Ŝ yθ α
) got increased as 
compared to the Regression type estimator ( 2s
l r d
). The 
graphical representation also predicts that there is gain in 
efficiency of proposed estimator ( 2Ŝ yθ α
) over the Regression–
type estimator ( 2s
lrd
) for 0 < < 2θ . Hence we conclude that 
the proposed estimator will always be better than existing 
Regression-type estimator under double sampling design for 
0 < < 2θ . 
 
References 
1. A.K. Das and B. Tripathi, Use of auxiliary information 
in estimating the finite population variance, Sankhya, 
(40), 139-148 (1978) 
2. Isaki C.T., Variance estimation using auxiliary 
information, Jour. Amer. Stat. Assoc., 78, 117-123 
(1983) 
3. Ahmed M.S., Walid A.D. and Ahmed A.O.H., Some 
estimators for finite population variance under two-phase 
sampling, Statistics in Transtition, 6(1), 143-150 (2003) 
4. Jhajj H.S., Sharma M.K. and Grover L.K., An efficient 
class of chain estimators of population variance under 
sub-sampling scheme, Journal of  Japan Statistical 
Society, 35(2), 273-286 (2005) 
5. Kadilar C. and Cingi H., Ratio estimator for the 
population variance in simple and stratified random 
sampling, Applied Mathematics and Computation, 173, 
1047-1059, (2006) 
6. Pradhan B.K., Regression type estimators of finite 
population variance under multiphase sampling, Bull. 
Malays. Math. Sci. Soc., 33(2), 345-353 (2010) 
7. Jhajj H.S. and Walia G.S., A generalized difference-
cum-ratio type estimator for the population variance in 
double sampling, International journal of Applied 
Mathematics, 41(4), 30-40 (2011) 
8. Mishra G., On estimation of finite population variance 
and coefficient of variation using auxiliary information, 
Ph.D. Thesis, Utkal University, Bhubaneswar, India, 
(1991
 


A Family of Difference-cum-Exponential Type Estimators for Estimating the Population Variance Using Auxiliary Information in Sample Surveys

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=B1157419613E9E285B5C2DEEC073A6EC?doi=10.1.1.1073.6676&rep=rep1&type=pdf

A Family of Difference-cum-Exponential Type Estimators for Estimating the Population Variance Using Auxiliary Information in Sample Surveys

Abstract

Similar works

Full text

Available Versions

CiteSeerX