This document presents a class of estimators proposed by Abu-Dayyeh et al. to estimate the mean of a variable of interest (Y) in a finite population using information from two auxiliary variables (X1 and X2). The estimators are shown to be biased. The authors then propose using a jackknife technique to create an unbiased class of estimators from the original estimators. They derive expressions for the bias and mean square error of the original and proposed jackknifed estimators. Finally, they compare the proposed jackknifed estimators to other existing estimators such as the sample mean, ratio estimator, and product estimator in terms of bias and mean square error.

1. International Journal Of Mathematics And Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
Www.Ijmsi.Org || Volume 2 Issue 09 || September. 2014 || PP-32-37
www.ijmsi.org 32 | P a g e
Some Unbiased Classes of Estimators of Finite Population
Mean
Praveen Kumar Misra 1
and Shashi Bhushan 2
1. Department of Statistics, Amity University, Lucknow, India.
2. Department of Applied Statistics,Babasaheb Bhimrao Ambedkar University,
Lucknow, India
ABSTRACT: In this paper, a class of biased multivariate estimators proposed by Abu – Dayyeh et al is
reconsidered under the jack – knife technique of Gray and Schucany and Sukhatme et al. The proposed classes
of estimators are shown to be unbiased while retaining the optimum mean square error. In order to enhance the
practical utility Abu Dayyeh estimators, the class of estimators utilizing the estimated optimum values is also
proposed. The comparative study shows that the proposed class of jack – knifed estimators and the proposed
class of estimators based on the estimated optimum values are better than the Abu Dayyeh classes of estimators
in the sense of unbiasedness and practical utility respectively.
KEYWORDS: Unbiasedness, generalized class of estimators and mean square error.
I. INTRODUCTION
W.G.Cochran and R.J.Jessen[3,4] discussed the use of auxiliary information to increase the precision of
estimators. The ratio estimator among the most commonly used estimator of the population mean or total of
some variable of interest of a finite population with the help of a auxiliary variable when the correlation
coefficient between the two variables is positive. In case of negative correlation, product estimator is used.
These estimators are more efficient i.e. has smaller variance than the usual estimator of the population mean
based on the sample mean of a simple random sample.
In this paper, mutivariate class of estimators given by Abu – Dayyeh et al are reconsidered under the
jack – knife technique of Gray and Schucany and Sukhatme et al in section 2. In a finite population, let y denote
the variable whose population mean Y is to be estimated by using information of two auxiliary variables 1x and
2x .assuming that the population means 1X and 2X of the auxiliary variable are known the following
multivariate class of estimators of the population mean were given by Abu – Dayyeh et al(2003)
1
ia
p
i
h
i i
x
y y
X
 
  
 
 (1)
where ; 1,2,..., are suitably chosen real numbers.ia i p
1.1 Definitions and results
Assume that the population means iX of the auxiliary variables are known. Let , iy x and respectively
denote the sample means of the variables , iy x and based on a simple random sample without replacement
(SRSWOR) of size n drawn from the population
Define:
0 , i i
i
i
x Xy Y
e e
Y X

  ;i=1,2,…………p
Then
0( ) ( ) 0iE e E e 

2. Some Unbiased Classes of Estimators of Finite Population Mean
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2 2 2 2
0( ) , ( ) iy i x
f f
E e C E e C
n n
 
0( ) i ii yx y x
f
E e e C C
n
 , ( ) i j i ji j x x x x
f
E e e C C
n

Where
   
2 22
12 2 2
2 2
1
, , , 1 ,
N
y x
y x y i
i
S SN n
f C C S N Y Y
N Y X



     
    1
2 2 1
1
N
yx
yx yx i i
i
y x
S
S N Y Y X X
S S



    
Also, iyx and
i jx x denote the correlation coefficient between Y and iX , iX and jX respectively and
2
yC ,
2
ixC denote the coefficient of variation of Y , iX respectively.
1.2 Properties of the estimator hy
Bias of the Abu Dayyeh classes of estimators
 
 
1 1
2
1
1
( )
1
2
i i i j i j
i
p p
h i yx y x i j x x x x
i i j
p
i i
x
i
f
Bias y Y a C C a a C C
n
a a
C
 
  

 
 

 
 

 

(2)
MSE of the Abu Dayyeh class of estimators
 2 2 2 2
1 1 1
The MSE of given by
1
( ) 2 2i i i i j i j
h
p p p
h y i x i yx y x i j x x x x
i i i j
y is
f
MSE y Y C a C a C C a a C C
n
 
   
 
    
 
  
 2 2
1 1 1
1
2i j i j i i
p p p
y i j x x x x i yx y x
i j i
f
Y C a a C C a C C
n
 
  
 
   
 
 
 
 21
= 1 ' 2 'y
f
S b Ab b e
n

  (3)

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1X 12
1X 1 2
1
2
min
where ( ) , ' ( , , )
' ( , ,..., ), /
The optimum value minimizing the MSE is ' .
The minimum MSE of the class of estimators is given by
1
( )
i j
ij ij p p
y
p p i yi i y
h
C C
A a b a a
C
e e e e e C C
b e A
MSE y Y



 
    
 
 
 



 
 2 2
.12...
2
.12...
1
where is the multiple correlation coefficient.
y y p
y p
f
C
n



II. THE PROPOSED JACK-KNIFE ESTIMATOR hjy
Let a simple random sample of size n=2m is drawn without replacement from the population of size N. this
sample of size n=2m is then split up at random in to two sub samples each of size m
1
ia
p
i
h
i i
x
y y
X
 
  
 

(4)
Let us define  01y Y e  ,  1i i ix X e 
Substituting these values and simplifying we get
    2
1 1 1
1 1
( )
2i i i j i j i
p p p
i i
h i yx y x i j x x x x x
i i j i
f a a
Bias y Y a C C a a C C C
n
 
   
  
   

   = B1(say) (5)
3
Using the jack - knife technique we propose to estimate the population mean by
'
1
h h
hJ
y Ry
y
R



(6)
   31
1 2
2
where ; and 'h h
B
R B Bias y B Bias y
B
   (7)
3
1 2
estimator based on sample of size 2
' pooled estimator based on sub samples of size each
2
h
h h
h
y n m
y y
y m
 

 
(8)
Now,
( ) ( )
( )
1
h h
hj
E y RE y
E y
R


 (9)
Y (i.e hjy is an unbiased estimator of population mean Y ) (10)
2.1 Mean square error of ajy
2
( ) ( )
1
h h
hj hj
y Ry
MSE y E y Y E Y
R
 
    
 

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2
( ) ( )
1
h hy Y R y Y
E
R
   
  
 
=
2 2
2
( ) ( ) 2 ( )( )
(1 )
h h h hE y Y R E y Y R y Y y Y
R
      

(11)
2
( ) ( )h hMSE y E y Y  =
    2
1 1 1
1 1
2i i i j i j i
p p p
i i
i yx y x i j x x x x x
i i j i
f a a
Y a C C a a C C C
n
 
   
  
  

  
2 f
Y A
n
 (12)
whereA=
    2
1 1 1
1 1
2i i i j i j i
p p p
i i
i yx y x i j x x x x x
i i j i
f a a
Y a C C a a C C C
n
 
   
  
  

  
 
 2
The MSE of the proposed class of jack-knifed estimators is given by
1
( ) 1 ' 2 '
which can again be minimized so that the minimum MSE is
h y
f
MSE y S b Ab b e
n

  
 
 2 2 2
* min .12...
min
1
( ) 1
= ( )
hJ y y p
h
f
MSE y Y C
n
MSE y


 
(13)
1X 1 1X 1 22
1
where ( ) , ' ( , , ), ' ( , ,..., ), /
The optimum value minimizing the MSE is ' .
The minimum MSE of the class of estimators is given by
i j
ij ij p p p p i yi i y
y
C C
A a b a a e e e e e C C
C
b e A
 

 
      
 
 

 
 2 2 2
min .12...
2
.12...
1
( ) 1
where is the multiple correlation coefficient.
h y y p
y p
f
MSE y Y C
n



 
(14)
III. COMPARISON OF THE ESTIMATORS
In this section, we compare the proposed estimators with some known estimator. The comparison will
be in terms of the bias and the mean square error up to the order of n-1
. We consider the following known
estimators.
1. The SRS mean y . It is unbiased estimator and its variance is given by
Var( y )=
f
n
2
yS
2. The ratio estimator
R
X
Y y
x


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Where x may be chosen as x1 or x2.
The bias and the mean square error of this estimator are respectively given by
( ) ( )R x x y
f
B y Y C C C
n
 
2 2 2
( ) ( 2 )R y x y x
f
MSE y Y C C C C
n
  
3. Product estimator
R
x
Y y
X

The bias and the mean square error of this estimator x1 or x2 where x may be chosen as are respectively given by
( )P y x
f
B y Y C C
n

4. The estimator
a
a
x
y y
X
 
  
 
where x may be chosen as x1 or x2.
5. If a1= a2=1, then the estimator reduces to
1 2
1 2
x x
y y
X X

6. If a1= a2=-1 then the estimator (1) reduces to
1 2
1 2
X X
y y
x x

The proposed jack-knife estimators hjy are unbiased and mean square error less or equal as compared with the
above estimators.
IV. NUMERICAL ILLUSTRATIONS
The comparison among these estimators is given by using a real data set.
The data for this illustration has been taken from [1], District Handbook of Aligarh, India. The population that
we like to study contains 332 villages. We consider the variables Y, X1, X2 where Y is the number of cultivators,
X1 is the area of villages and X2 is the number of household in a village. Throughout this study, two types of
calculations are used. The first type depends on the population data, and the second type depends on simulation
study of 30,000 repeated samples without replacement of sizes 80 from the data. We compute the bias and the
MSE for all estimators.
Numerical computation
The following values were obtained using the whole data set:
Y =1093.1, 1X =181.57, 2X =143.31
Cy =0.7625, 1xC =0.7684, 2xC =0.7616
1yx =0.973, 2yx =0.862, 2 1x x =0.842

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Using the above results we calculated the MSE, bias, and the efficiency for all the estimators in section-4.
In the Table1 , we present the MSE, efficiency, and the bias all for the estimators given in Section 4. As we can
see that our suggested estimators hjy and dominate all other estimators with efficiency 24.51 . They are superior
over all other estimators and are unbiased.
Table1: Bias and MSE of estimators under study based on population data
Estimator Auxiliary variable MSE Efficiecy Bias
y none 11807.6 1 0
*
a
y 1 2,x x 482.337 24.5098 0.265
*
w
y 1 2,x x 620.5 18.1818 -0.40110
ay

1x 3118.804 3.7878 0.78127
ay

2x 1629.45 7.2463 0.63602
Ry 1x 6143.05 1.9223 0.3753
Ry 2x 3254.39 3.6284 1.4736
Py 1x 46942.5 0.2520 10.589
Py 2x 43910.4 0.2689 9.2983
ajy 1 2,x x 99050.6 0.1192 0
wjy 1 2,x x 12095.1 0.9765 0
MRy 1 2,x x 655.2 18.0214 9.2983
REFERENCES
[1] W.A.Abu-Dayyeh et al. ,Some estimators of a finite population mean using auxiliary information, Applied Mathematics and
Computation 139 (2003) 287-298.
[2] Jack – knife technique of Gray and Schucany and Sukhatme et al.
[3] W.G.Cochran, Sampling Techniques, third ed. John Wiley &sons, New York, 1977.
[4] R.S.Jessen, Statistical Survey Techniques, John Wiley &sons, New York, 1978.
[5] D.Singh,F.S.Chaudhary, Theory and Analysis of sample Survey Design, New Age Publication, New Delhi, India 1986.
[6] S.K.Srivastava, An estimator using auxiliary information in sample surveys, Calcutta Statistical Association Bulletin 16(1967) 121-
132.