REVISTA INVESTIGACION OPERACIONAL
VOL. 35, NO. 1, 49-57, 2014
ON AN IMPROVED RATIO TYPE ESTIMATOR OF
FINITE POPULATION MEAN IN SAMPLE SURVEYS
A K P C Swain
Former Professor of Statistics, Utkal University, Bhubaneswar-751004, India
ABSTRACT
In this paper an alternative ratio type exponential estimator is suggested and is compared with Bahl and Tuteja’s ratio
type exponential estimator and classical ratio estimator as regards bias and mean square error with large sample
approximations both theoretically and with numerical illustration.
KEYWORDS: Simple random sampling, ratio type estimators, Bias, Mean square error.
MSC: 62D05
RESUMEN
En este trabajo se sugiere un estimador del tipo razón- exponencial alternativo y se compara con el de Bahl y Tuteja
el de razón clásico, considerando sesgo y error cuadrático medio, bajo aproximaciones basadas en muestras grandes y
haciendo ilustraciones teóricas y numéricas.
1. INTRODUCTION:
The use of auxiliary information dates back to year 1934 , when Neyman used it for stratification of the finite
population. Cochran(1940) used auxiliary information in estimation procedure and proposed ratio method of
estimation to provide more efficient estimator of the population mean or total compared to the simple mean per unit
estimator under certain conditions when the auxiliary variable has positive correlation with the study variable in
question.
Let U = (U1, U2… UN) be the finite population of size N. To each unit Ui (i=1, 2…, N) in the population paired values
(yi, xi) corresponding to study variable y and an auxiliary variable x, correlated with y are attached.
Now, define the population means of the study variable y and auxiliary variable x as
1 N
1 N
Y   yi , X   xi
N i 1
N i 1
Thus, the population ratio is defined as R  Y
X
Further, define the finite population variances of y and x and their covariance as S y2 
1 N
 yi  Y 2 ,
N 1
i 1
2
N
1 N
 xi  X  , and S yx  1   yi  Y xi  X  , respectively.

N  1 i 1
N  1 i 1
Thus, the squared coefficients of variation of y and x and their coefficient of covariation are respectively defined by
C20, C02 and C11 ,where
S x2 
Crs 
1 N
( yi  Y )r ( xi  X ) s , r, s  0,1,2,...

N i 1
Define  as the correlation coefficient between
y
and
x
in the population.
A simple random sample ‘s’ of size n is selected from U without replacement and the values (yi, xi), i=1, 2 … n are
observed on the sampled units. Assume that y and x are positively correlated. The simple mean y is an unbiased
estimator of the population mean Y and variance of
y
is given by
Var ( y )   Y 2C20
The classical ratio estimator of the population mean
Y
,using auxiliary information on
49
x
is given by
yR 
where
y
X,
x
(1.1)
y and x
are sample means of y and x respectively, defined by
yR
The ratio estimator
to
O(1/ n)
envisages advance knowledge of
X
y
1 n
1 n
yi and x   xi .

n i 1
n i 1
. The bias( B ) and mean square error (
MSE )of yR
are given by(see Sukhatme and Sukhatme,1970)
B( yR )  Y (C02  C11)
(1.2)
MSE ( yR )  Y 2 (C20  C02  2C11)
(1.3)
where

N n
( N  1)n
The ratio estimator
In large samples
or if

1
2
Crs 
,
yR
yR
1 N
( yi  Y )r ( xi  X )s , r , s  0,1,2

N i 1
is a biased estimator and the bias decreases with increase in sample size.
is more efficient than the simple mean per unit estimator
when
y
if
k   C20  1
C02
2
C20  C02 .
Bahl and Tuteja(1991) have suggested a ratio type exponential estimator given by
yBTR  y exp(
To
O(1/ n)
X x
)
X x
,the bias and mean square error of
(1.4)
yBTR
are given by
3
1
B( yBTR )   Y ( C02  C11)
8
2
1
MSE ( yBTR )   Y 2 (C20  C02  C11)
4
1
3
yBTR is more efficient than yR and y if  k 
4
4
(1.5)
(1.6)
, (Bahl and Tuteja,1991):
O(1 / n2 ) ,Bahl and Tuteja’s(1991) ratio type
exponential estimator is more efficient than the classical ratio estimator y R when y and x follow bivariate
normal distribution , coefficients of variation of y and x are equal and
3
0   .
4
Upadhyaya,Singh,Chatterjee and Yadav (2011) have shown that to
In the following we suggest a ratio type estimator with square root transformation and compare it with the BahlTuteja’s (1991) ratio type exponential estimator and classical ratio estimator both theoretically and with the help
of some natural and artificial populations.
2. A RATIO TYPE ESTIMATOR USING SQUARE ROOT TRANSFORMATION
Define the ratio type estimator using square root transformation as
50
ySQR  y (
2.1. Bias and Mean square error of
Write
y  Y (1  e0 )
and
X 1/2
)
x
(2.1)
ySQR
x  X (1  e1)
with
E (e0 )  E (e1)  0 , V (e0 )   C20 ,
V (e1)   C02 and Cov(eo , e1)   C11 .
Expanding ySQR in power series under the assumption that e1  1 for all possible samples and retaining terms
up to fourth degree in
e0
and e1 ,we have
1
3
15
105 4 1
3
15
ySQR  Y  Y (e0  e1  e12  e13 
e1  e0e1  e0e12  e0e13 )
2
8
48
384
2
8
48
1
( ySQR  Y )2  Y 2  (e02  e12  e0e1) e02e12 +
4
5
33
3
87 4
 e0e12  e02e1  e0e13  e13 
e1 
4
24
8
192
(2.2)
(2.3)
Thus,
1
3
15
105
B( ySQR )  Y  E (e0 )  E (e1)  E (e12 )  E (e13 ) 
E (e14 )
2
8
48
384
1
3
15
 E (e0e1)  E (e0e12 )  E (e0e13 ) 
2
8
48
1


MSE ( ySQR )  E ( ySQR  Y )2  Y 2  E (e02 )  E (e12 )  E (e0e1) 
4


5 2
33
3
87
2
2 2
2
3
3
4 
E
(
e
e
)

e
e

E
(
e
e
)

E
(
e
e
)

E
(
e
)

E
(
e
)
+Y
0
1
0
1
0
1
0
1
1
1

4
24
8
192

(2.4)
(2.5)
Now,
E (e14 ) 
N  n  N 2  N  6nN  6n2
3N ( N  n  1)(n  1) 2 
C

C02 

04
( N  1)( N  2)( N  3)
n3  ( N  1)( N  2)( N  3)

E (e0e13 ) 

N  n  N 2  N  6nN  6n2
3N ( N  n  1)(n  1)
C

C
C

13
11 02 
( N  1)( N  2)( N  3)
n3  ( N  1)( N  2)( N  3)

E (e02e12 ) 
N  n  N 2  N  6nN  6n 2
N ( N  n  1)(n  1)
2
C

(
C
C

2
C

22
20 02
11 
( N  1)( N  2)( N  3)
n3  ( N  1)( N  2)( N  3)

( N  n)( N  2n)
C03
( N  1)( N  2)n2
( N  n)( N  2n)
E (e0e12 ) 
C12
( N  1)( N  2)n2
E (e13 ) 
E (e02e1 ) 
( N  n)( N  2n)
C21
( N  1)( N  2)n2
where
Crs 
1 N
( yi  Y )r ( xi  X ) s / Y r X s

N i 1
,
(r , s  1,2,3,4) .
51
(See Sukhatme ,Sukhatme and Asok (1984))
Hence,
N n 1 3
1

(
C

C
)
02
11

N  1 n  8
2
N n 1A 3
15
B 105
15
Y
( C12  C03 )  2 (
C04  C13 )

N 1 n  n 8
48
48
n 384
3D 105 2 15
 2(
C02  C11C02 ) 
48
n 384
N n 1
1
1 5
3

MSE ( ySQR )  Y 2
(C20  C02  C11)  A( C12  C21  C03 ) 

N 1 n 
4
n 4
8

B( ySQR )  Y
Y 2
(2.6)
(2.7)
N n 1 1
87
33
1
33
87 2 
2
B(
C04  C22  C13 )  2 D(C20C02  2C11
 C11C02  C02
)
2

N 1 n  n
192
24
8
64
n

where
N 2  N  6nN  6n2
B
( N  2)( N  3)
2.2. Bias and Mean square error of yBTR
N  2n
A
N 2
Expanding
yBTR
D
N ( N  n  1)(n  1)
( N  2)( N  3)
in power series with the same assumptions used in expanding
and including degree four in
e0
ySQR
and retaining terms up to
and e1 ,we have
e1 3 2 13 3 73 4 e0e1 3 2 13 3
 e1  e1 
e1 
 e0e1  e0e1 )
2 8
48
384
2
8
48
5
3
79 4 2 2 31 3
( yBTR  Y )2  Y 2 (e02  1 e12  e0e1  e0e12  e02e1  e13 
e1  e0 e1  e0e1 )
4
4
8
192
24
N n 1 3
1

(
C

C11 ) 
B( yBTR )  Y
02

N 1 n  8
2

yBTR  Y  Y (e0 
Y
N n 1A 3
13
B 73
13
(
C

C
)

(
C

C13 )
12
03
04
N  1 n  n 8
48
48
n2 384
D 73 2 13
 2(
C02  C02C11) 
16
n 128
MSE ( yBTR )  E ( yBTR  Y )  Y
2
2
E (e02

1 2
e1
4
(2.8)
 e0e1)
N n 1
1
1 5
3
5
3
79 4 2 2 31 3
Y 2 E ( e0e12  e02e1  e13 
e1  e0 e1  e0e1 )  Y 2
(C20  C02  C11 )  A( C12  C21  C03 ) +

4
8
192
24
N 1 n 
4
n 4
8
1
79
31
1
31
79 2
2
B(
C04  C22  C13 )  2 D(C20C02  2C11
 C11C02  C02
)
2
192
24
8
64
n
n
2.3. Bias and Mean Square Error of
yR
52

(2.9))
We find (see Sukhatme , Sukhatme and Asok,1984) to second order approximation
B( yR )  Y
N n 1
N n 1 1
1
1 2

(C02  C11)  Y
A (C12  C03 )  B 2 (C04  C13 )  3D 2 (C02
 C02C11) 
N 1 n
N  1 n  n
n
n

MSE ( yR )  Y 2
Y 2
(2.10)
N n 1
2

(C20  C02  2C11)  A(2C12  C21  C03 ) 

N 1 n 
n

N n 1 3
3
2
2 
B(C04  C22  2C13 )  2 D(C20C02  2C11
 6C11C02  3C02
)
2

N 1 n  n
n

Under the assumption of bivariate normality of
( y, x)
C20  C02
it may be shown that
MSE ( yR )  Y 2
C02 
C

2(1   )  02 (12  18  6  2 ) 

n 
n

(2.11)
and equality of coefficients variation of
y
and
x ,i.e.
(2.12)
C  5
C 143 31

MSE ( yBTR )  Y 02 (   )  02 (
   2 2 )
n  4
n 64 8

(2.13)
C  5
C 151 33

MSE ( ySQR )  Y 02 (   )  02 (
   2 2 )
n  4
n 64 8

(2.14)
2
2
3. COMPARISON OF EFFICIENCY
(i) Considering terms up to O(1 / n ) ,
MSE ( yR )  Y 2 (C20  C02  2C11)
1
MSE ( ySQR )  MSE ( yBTR )  Y 2 (C20  C02  C11)
4
B = B( yR )  Y  (C02  C11)
B1 = B( ySQR )  B( yBTR )  Y  ( 3 C02  1 C11)
8
2
Now,
MSE ( yR )  MSE ( ySQR )
 MSE ( yR )  MSE ( yBTR )
3
 Y 2 ( C02  C11 )
4
Thus, to
O(1/ n) , yBTR
and
ySQR
are equally efficient. They are more efficient than
yR
if C11  3
C02
4
or k  3 ,
4
and more efficient than
Hence ,
ySQR
Further, to
and
y
if
yBTR
O(1/ n)
k
1
.
4
are more efficient than both
the biases of
ySQR
and
yBTR
yR
and
y
if
.
are equal and further they are less biased than
is either greater than 5/4 or less than 11/12..
2
(ii) To O(1 / n ) ,for large N,
53
yR
if
k
87 79
33 31

4
2
MSE ( ySQR )  MSE( yBTR )  (

)
E
(
e
)

(
 ) E (e0e13 )   Y 2 1 ( 1 C02
 C11C02 )
1
 192 192
24 24
4n 2 2

Hence,
ySQR
yBTR to O(1 / n2 )
will be more efficient than
if 1 C 2  C C  0
02
11 02
2
C11 1

C02 2
or
or   1 when C20  C02
2
Under bivariate normality of
between
y
and
x
equal to
y

and
x
with the assumption that
C20  C02 and the correlation coefficient
we have


1
n
3
4
MSE ( yBTR ) MSE ( yR ) = Y 2C02 (   ) 

(see Upadhyaya et al,2011 and correct the coefficient of
1
n

as -113/8 instead of -103/8 )


3
4
MSE ( ySQR ) MSE ( yR ) = Y C02 (   ) 

2
C02 625 113

(

  4 2 )
n 64
8

C02 617 111

(

  4 2 )
n 64
8

3
, both MSE ( yBTR )  MSE ( yR ) and MSE ( ySQR )  MSE ( yR ) are negative.Hence,
4
O(1 / n2 ) the sufficient condition that both yBTR and ySQR are more efficient than yR is that
If

0  
Further ,
ySQR
will be more efficient than
yBTR
ySQR
will be more efficient than
yBTR
3
4
if

Hence,
to
and
1
2
yR
if
y
and
1
3

2
4
To sum up under the assumptions of bivariate normality of
y
and
x ,the preferred estimators for ranges of 
x
and the equality of coefficients of variation of
re shown below in Table-1.
Table-1 Comparison of Estimators

Preferred
estimator
0
1
4
y
yBTR
1 1

4 2
1 3

2 4
ySQR
54
yR
3
1
4
4. NUMERICAL ILLUSTRATION:
Illustration 1
Consider 12 natural populations described in Table-2 to compare the ratio type estimator with square root
transformation estimator with classical ratio estimator as regards bias and mean square error.Bias and mean square
error have been computed excepting the common constants(Table-3):
Table-2 Natural Populations
Pop
no.
Description
N
1
Sampford (1962)
17
Acreage under oats in
1957
2
Singh and
Chaudhary(1986)
16
3
4
Konijn(1973)
Murthy(1967)
5
y
 yx
Cx
Cy
Acreage of crops and
grass in 1947
0.4
0.22
0.45
Area under wheat !979-80
Total cultivated area
during 1978-79
0.96
0.74
0.69
16
16
Food expenditure
Area under winter
paddy(in acres)
Total expenditure
Geographical area (in
acres)
0.95
0.29
0.08
0.09
0.11
0.47
Singh and
Chaudhary(1986)
17
No. of milch animals in
survey1977-78
No. of milch animals in
census1976
0.72
0.02
0.01
6
Murthy(1967)
16
Output for
factories(000Rs)
Fixed capital(000Rs.)
0.84
0.15
0.09
7
Panse and
Sukhatme(1967)
16
Progeny mean(mm)
Parental plant
value(mm)
0.68
0.07
0.06
8
Panse and
Sukhatme(1967)
Singh and
Chaudhary((1986)
Panse and
Sukhatme(1967)
20
Parental plot mean(mm)
0.0.56
0.07
0.04
15
Area under wheat in 1973
0.23
0.82
0.67
25
Parental plot mean(mm)
Parental plant
value(mm)
Area under wheat in
1971
Parental plant
value(mm)
0.54
0.07
0.03
11
Panse and
Sukhatme(1967)
20
Progeny mean(mm)
Parental plant
value(mm)
0.68
0.07
0.05
12
Swain(2003)
19
No. of milch cows in
1957
No. of milch cows
census 1956
0.72
1.14
1.12
9
10
x
Table 3 Comparison of Bias and Mean square Error
Pop. no
Bias( ySQR )
Y
1
2
3
4
5
6
7
0.34
0.07
0.28
0.38
0.20
0.12
0.09
Bias( yR
Y
0.18
0.11
0.31
0.51
0.64
0.50
0.42
MSE ( ySQR )
Y 2
0.174912
0.125636
0.005316
0.210694
0.000056
0.002475
0.001983
55
MSE ( yR )
Y 2
MSE ( yR )
100
MSE ( ySQR )
0.171524
0.048972
0.001732
0.204538
0.000212
0.0081
0.002816
98.06
38.98
32.58
97.08
378.57
327.27
142.01
8
9
10
11
12
0.04
0.28
0.26
0.13
0.02
0.33
0.81
0.77
0.51
0.37
0.001257
0.489244
0.000998
0.001324
0.654131
0.003364
0.865788
0.003546
0.002598
0.703661
267.62
176.96
355.31
196.22
107.57
Comments : For the populations under consideration,the ratio type estimator with square root transformation is less
biased than the classical ratio estimator.Further, For populations 5-12 , the ratio type estimator with square root
transformation is more efficient than the classical ratio estimator in the sense of having lesser mean square error.
Illustration 2
Consider a hypothetical population of size
y:
x
N =5 as follows:
3 5 8 4 5
: 5 2 6 4 3
For a simple random sample (without replacement) of size
ySQR
and
yBTR
n =2,the exact biases and mean square errors of yR ,
are given in Table-4.
Table 4: Comparison of Biases and Mean square errors
Estimators
Absolute Bias
M. S .E
0
1.05
Relative Efficiency
100
y
yR
ySQR
0.1903
1.8667
56.25
0.0592
0.9996
105.04
yBTR
0.0569
0.9922
105.82
Comments: As per Illustration 2
(i)The ratio type estimator with square root transformation and ratio type exponential estimator have
nearly equal bias and efficiency and further they are less biased and more efficient than the classical ratio estimator..
(ii) The classical ratio estimator is less efficient than the simple mean per unit estimator y ,where as
the ratio type estimator with square root transformation and the ratio type exponential estimator are more efficient
than y .
5. CONCLUSIONS:
(i)To
O(1/ n) ySQR
and
yBTR
are equally efficient and more efficient than both
1
3
 k  . To same order of approximation both ySQR
4
4
11
5
greater than
or less than
.
12
4
(ii) Under the assumption of bivariate normality of
and
x ,to O(1 / n2 ) ySQR
and
yBTR
( y, x)
is more efficient than both y R and
1
3
   . However , for the values of 
2
4
yBTR .
in the range
are less biased than
yR
if
and
k
y
if
is either
and equality of coefficients of variation of
yBTR
y
for
3
   1, yR
4
56
yR
is to be preferred over
y , ySQR
and
(iii) Numerical illustration with the help of twelve natural populations shows that there might arise situations
when
ySQR
is less biased and more efficient than y R under large sample approximations. Also, for the artificial
population under consideration
ySQR
happens to be less biased and more efficient than y R . However , the
difference between ySQR and
yBTR
as regards bias and mean square error is only marginal.
RECEIVED APRIL, 2013
REVISED NOVEMBER, 2014
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