International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 3 Issue 8 || December. 2015 || PP-23-28
www.ijmsi.org 23 | Page
On Estimation of Population Variance Using Auxiliary
Information
Peeyush Misra
Department of Statistics, D.A.V.(P.G.) College, Dehradun- 248001, Uttarakhand, India
Abstract: A generalized class of estimator representing a class of estimators using auxiliary information in the
form of mean and variance is proposed. The expression for bias and mean square error are found and it is
shown that the proposed generalized class of estimator is more efficient than few of the estimators available in
the literature. An empirical study is also included as an illustration.
Keywords : Auxiliary information, Bias, Mean square error and Taylor's Series Expansion.
I. INTRODUCTION
It is well known that the use of auxiliary information in sample surveys results in substantial improvement in the
precision of the population parameters. By using the auxiliary information in different forms, estimators for
population parameters mainly population mean and variance are studied and are available in the literature.
Consider a finite population U with N units (U1, U2, . . . , UN ) for each of which the information is available on
auxiliary variable X, Y being the study variable.
Let us denote by



N
i
i
Y
N
Y
1
1
= population mean of study variable Y



N
i
i
X
N
X
1
1
= population mean of auxiliary variable X
 




N
i
iY
YY
N
S
1
2
2
1
1
= population variance of study variable Y
 




N
i
iX
XX
N
S
1
2
2
1
1
= population variance of auxiliary variable X
and    


N
i
s
i
r
irs
XXYY
N 1
1
 .
Also, let a sample of size n be drawn with simple random sample without replacement to estimate the population
variance of the study variable Y.
Let 


n
i
i
y
n
y
1
1
= sample mean of study variable Y



n
i
i
x
n
x
1
1
= sample mean of auxiliary variable X
 




n
i
iy
yy
n
s
1
2
2
1
1
= sample variance of study variable Y
 




n
i
ix
xx
n
s
1
2
2
1
1
= sample variance of auxiliary variable X.
For simplicity, we assume that N is large enough as compared to n so that the finite population correction terms
are ignored.

2. On Estimation of Population Variance Using Auxiliary …
www.ijmsi.org 24 | Page
In order to have an estimate of population variance of the study variable Y, assuming the knowledge of mean
and variance of auxiliary character X, proposed generalized class of estimator is given by
 22
,,ˆ
xg
sxygd   (1.1)
where 


n
i
i
y
n 1
21ˆ satisfying the validity conditions of Taylor’s series expansion is a bounded function of
 22
,, X
SXY such that
(i) g  
2
22
,, YSXY X
 (1.2)
(ii) first order partial differential coefficient of  22
,, x
sxyg with respect to
2
y at T =  22
,, X
SXY is
unity, that is
 
  1,,
22
20












T
x
sxyg
y
g (1.3)
(iii) second order partial differential coefficient of  22
,, x
sxyg with respect to
2
y at T =  22
,, X
SXY
is zero, that is
 
  0,,
22
22
2
00












T
x
sxyg
y
g (1.4)
II. BIAS AND MEAN SQUARE ERROR OF THE PROPOSED ESTIMATOR
In order to obtain bias and mean square error, let us denote by
0
eYy 
1
eXx 
2
22
eSs Xx

3
ˆ e  where 


N
i
i
Y
N 1
21
 (2.1)
with E( 0
e ) = E( 1
e ) = E( 2
e ) = E( 3
e ) = 0 (2.2)
 
n
eE
202
0


 
n
eE
022
1


  )1( 2
2
022
2
 

n
eE
  )44(
1 2
2020
2
3040
2
3
  YY
n
eE
 
n
eeE
11
10


 
n
eeE
12
20



3. On Estimation of Population Variance Using Auxiliary …
www.ijmsi.org 25 | Page
 
n
eeE
03
21


  )2(
1
203030
 Y
n
eeE 
  )2(
1
112131
 Y
n
eeE 
  )2(
1
2002122232
  Y
n
eeE (2.3)
Now expanding  22
,, x
sxygt  in the third order Taylor's series about the point T =  22
,, X
SXY , we have
        2
22
10
2
22
2
,, gSsgXxgYySXYgt XxX

         01
2
2
22
22
11
2
00
22
2
2
!2
1
gXxYygSsgXxgYy Xx





       12
22
02
22
2
2
22 gSsXxgSsYy XxXx

 
 
     2***2
3
2
22
2
22
,,
!3
1
x
x
Xx
sxyg
s
Ss
x
Xx
y
Yy















where 0
g , 00
g are already defined earlier and
 
T
x
sxyg
x
g 








22
1
,,
 
T
x
x
sxyg
s
g











22
22
,,
 
T
x
sxyg
x
g











22
2
2
11
,,
 
 
T
x
x
sxyg
s
g











22
22
2
22
,,
 
 
T
x
sxyg
xy
g











22
2
2
01
,,
 
 
T
x
x
sxyg
sy
g











22
22
2
02
,,
 
T
x
x
sxyg
sx
g











22
2
2
12
,,
and  2
2
2*2
YyhYy 
 XxhXx 
*
 222*2
XxXx
SshSs  for 0 < h < 1.
Now using the conditions given in (1.2), (1.3) and (1.4), we have to the first degree of approximation
 12210220011022
2
211
2
122110
2
0
2
244
!2
1
2 geegeeYgeeYgegegegeeYeYt 
(2.4)
Now using (2.4) in (1.1), we have

4. On Estimation of Population Variance Using Auxiliary …
www.ijmsi.org 26 | Page
 22110
2
0
2
3
1
2
2
1
gegeeYeYeY
N
d
N
i
ig
 

 


 12210220011022
2
211
2
1
244
!2
1
geegeeYgeeYgege
2
0221103
2
2 egegeeYed Yg
 
 12210220011022
2
211
2
1
244
2
1
geegeeYgeeYgege  (2.5)
Now taking expectation on both the sides of (2.5), the bias in g
d to the first degree of approximation is given
by
Bias in   2
Ygg
dEd 
         2
0221103
2 eEgeEgeEeEYeE 
           12210220011022
2
211
2
1
244
2
1
geeEgeeEYgeeEYgeEgeE 
using the values of the expectation given in (2.2) and (2.3), we have
Bias in g
d   0312120211012
2
021102
20
2441
2
1


ggYgYg
nn
 (2.6)
Now squaring (2.5) on both the sides and then taking expectation, the mean square error to the first degree of
approximation is given by
MSE    
22
Ygg
dEd   
2
221103
2 gegeeYeE 
           31130
2
2
2
2
2
1
2
1
2
0
22
3
244 eeEgeeEYeEgeEgeEYeE 
       2121202101322
2442 eeEggeeEgYeeEgYeeEg 
using values of the expectation given in (2.2) and (2.3), we have
MSE  g
d    1444
1
2
2
022
2
022
1
2022
2020
2
3040
 


n
g
n
g
n
YYY
n
     200212222112112030
2
1
22
1
22
1
4   Y
n
gY
n
gY
n
Y
n
gg
n
gY
n
gY
03
21
12
2
11
1
244


MSE  g
d = MSE  2
y
s   2
22
2
02121
2
120
12
1
ggg
n
 
21032222002
2)(2 ggg   (2.7)
For minimizing (2.7) in two unknowns 1
g and 2
g , the normal equations after differentiating (2.7) partially
with respect to 1
g and 2
g are
021203102
  gg and (2.8)
    01 22200222
2
02103
  gg . (2.9)
Solving (2.8) and (2.9) for 1
g and 2
g , we get the minimizing optimum values to be
   
  3
0212
222002032
2
0221*
1
1
1




g and (2.10)

5. On Estimation of Population Variance Using Auxiliary …
www.ijmsi.org 27 | Page
 
  3
0212
222002020321*
2
1 



g (2.11)

 
    
2
222002032
2
02215
02
2
12
2
102
1
1
1


 

g (2.12)
 
 
 
  
2
2220020203214
02
2
12
22
22
2
02
1
1
1 


 


 g (2.13)
and
 
    

 222002032
2
02216
02
2
12
03
2103
1
1
2
2 


 gg
  222002020321
  (2.14)
Now adding (2.12), (2.13) and (2.14), we get
  2103
2
22
2
02
2
102
21 gggg  
 
    
2
1
2
1
02212220022
021202
2
21
1
1





 (2.15)
also
 
    222002032
2
02213
0212
21
121
1
1
2
2 


 

 g (2.16)
 
 
 
  2220020203213
0212
222002
2222002
1
2
2 


 


 g (2.17)
on adding (2.16) and (2.17), we get
  2222002121
22 gg  
 
    
2
1
2
1
02212220022
021202
2
21
1
2
2 





(2.18)
putting (2.15) and (2.18) in (2.7), we get
MSE  ming
d = MSE  2
y
s
 
 
2
1
2
1
02
21
22
21
2002
2
0212
2
21
02
2
21
1










 








nn
(2.19)
III. EFFICIENCY COMPARISON WITH THE AVAILABLE ESTIMATORS
For comparing the efficiency of the proposed generalized estimator, let us consider the following
(i) Usual Conventional unbiased Estimator of Population Variance in case of SRSWOR
 y
1
1ˆ
n
1i
2
i
2
1 



 y
n
sd y
   2
20401
1ˆwith  
n
dMSE (3.1)
from (3.1) and (2.19), it is clear that the proposed generalized class of estimator has mean square error lesser
than the usual conventional unbiased estimator.
(ii) Estimator of Population Variance given by Peeyush Misra and R. Karan Singh
 ,.ˆˆ
2
xyfyd   and  ˆˆ 2
3
yufd  
with      
02
2
212
204032
1ˆˆ



nn
dMSEdMSE  (3.2)

6. On Estimation of Population Variance Using Auxiliary …
www.ijmsi.org 28 | Page
from (3.2) and (2.19), it is clear that the proposed generalized class of estimator has mean square error lesser
than the mean square error of the estimator of population variance given by Peeyush Misra and R. Karan
Singh.
IV. EMPIRICAL STUDY
For comparing efficiency of the proposed generalized class of estimator, let us consider the data given in,
William G. Cochran (1977), Sampling Techniques, 3rd
Edition, John Wiley and Sons, New York, dealing with
Paralytic Polio Cases ‘Placebo’ (Y) group, Paralytic Polio Cases in not inoculated group (X), we have
n = 34
Y = 2.58
X = 8370.6
20
 = 9.8894
30
 = 47.015235
40
 = 421.96088
21
 = 93.464705 × 103
11
 = 19.34352945 × 103
02
 = 7.1865882 × 107
03
 = 1.4510955 × 1012
04
 = 4.5961952 × 1016
12
 = 3.443287 × 108
22
 = 3.0156658 × 109
.
We have  1
ˆdMSE = 9.534136695.
 2
ˆdMSE =  3
ˆdMSE = 5.958992179.
 g
dMSE = 5.512540843.
Table 4.1: PRE of the Proposed Estimator over the Estimators Described Above
V. CONCLUSION
The comparative study and empirical study of the proposed generalized sampling estimator of population
variance establishes its superiority in the sense of having minimum mean square error over the usual
conventional unbiased estimator of population variance in case of SRSWOR and the estimator of population
variance given by Peeyush Misra and R. Karan Singh.
REFERENCES
[1] Cochran, W.G. (1977): Sampling Techniques, 3rd edition, John Wiley and Sons, New York.
[2] Das, A. K. and Tripathi, T. P (1978): Use of auxiliary information in estimating the finite population variance, Sankhya, C, 40,
139-148.
[3] Liu, T. P (1974): A general unbiased estimator for the variance of a finite population, Sankhya, C, 36, 23-32.
[4] Murthy, M. N. (1967): Sampling Theory and Methods, 1st
edition, Statistical Publishing Society, Calcutta (India).
[5] Peeyush Misra and R. Karan Singh (2015): Ratio-Product-Difference type estimators for finite population variance using
auxiliary information, Accepted for publication in ‘Progress in Mathematics’.
[6] R. P. Gupta (1983): Unbiased estimation of finite population variance using auxiliary information, Statistics and Probability
Letters, 1, 121-124.
[7] Sukhatme, P.V. and Sukhatme, B.V. (1970): Sampling theory of surveys with applications, 3rd
revised edition, IOWA State
University Press, Ames, U.S.A.
[8] Srivastava, S.K. and Jhajj, H. S. (1980): A class of estimators using auxiliary information for estimating finite population
variance, Sankhya, C, 42, 87-96.
[9] Singh, R. K., Zaidi, S. M. H. and Rizvi, S. A. H. (1995): Estimation of finite population variance using auxiliary information,
journal of Statistical Studies, Volume 15, 17-28.