1. IJAMSS Efficient Family Of Exponential And Dual Estimators Of Finite Population Mean In Ranked Set Sampling 1

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by  © IASET

International Journal of Applied Mathematics
& Statistical Sciences (IJAMSS)
ISSN (P): 2319-3972; ISSN (E): 2319-3980
Vol. 9, Issue 1, Dec–Jan 2020; 1-10
© IASET

EFFICIENT FAMILY OF EXPONENTIAL AND DUAL ESTIMATORS OF FINITE
POPULATION MEAN IN RANKED SET SAMPLING
Nitu Mehta (Ranka)1 & V. L. Mandowara2
1Assistant Statistician, CCPC, Department of Agricultural Economics & Management, Rajasthan College of Agriculture,
Maharana Pratap University of Agriculture and Technology, Udaipur, Rajasthan, India
2Retired Professor, Department of Mathematics & Statistics, University College of Science, M. L. Sukhadia University,
Udaipur, Rajasthan, India

ABSTRACT
This study proposed improved family of exponential estimators and dual type ratio estimator of finite population mean
using some known population parameters of the auxiliary variable in Ranked Set Sampling (RSS). It has been shown that
this method is highly beneficial to the estimation based on Simple Random Sampling (SRS). The bias and mean squared
error of the proposed estimators with first degree approximation are derived. Theoretically, it is shown that the suggested
estimators are more efficient than the estimators in simple random sampling. It is also shown that the suggested dual
estimator is more efficient than the usual ratio estimator in Ranked set sampling.
KEYWORDS: Exponential Estimators, Dual Estimator, Ratio Estimator, Ranked Set Sampling, Population Mean,
Auxiliary Variable, Bias, Mean Squared Error

Article History
Received: 30 Nov 2019 | Revised: 17 Dec 2019 | Accepted: 27 Dec 2019

1. INTRODUCTION
The literature on Ranked set sampling describes a great variety of techniques for using auxiliary information to obtain
more efficient estimators. Ranked set sampling was first suggested by McIntyre (1952) to increase the efficiency of
estimator of population mean. Kadilar et al. (2009) used this technique to improve ratio estimator given by Prasad (1989).
Mehta and Mandowara (2013) suggested a modified ratio-cum-product estimator of finite population mean using ranked
set sampling. Here, we propose improved exponential family of ratio type estimators and dual estimator for the population
mean using some known parameters of the auxiliary variable in ranked set sampling.
LetU={U ,U ,.........,U } be the finite population of size and let y and , respectively, be the study and auxiliary
1 2 N
variables. A sample of size n is drawn, using simple random sampling without replacement, to estimate the population
mean Y = 1 ∑N y of study variable y.
N i
i=1
The classical ratio estimator given by Cochran (1940) for estimating the population mean , respectively for SRS,
is defined as

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2 Nitu Mehta (Ranka) & V. L. Mandowara
 X 
y = y  (1.1)
R
 x 
Bahl and Tuteja(1991) was the first to suggest an exponential ratio type estimator as
X - x
t = yexp  (1.2)
1 X + x
Following Kadilar and Cingi (2006) and Khoshnevisan (2007), Singh Rajesh et al.(2007) define modified exponential
estimator for estimating Y as
. (1.3)
Where a(„ 0),bare either real numbers or the functions of the known parameters of the auxiliary variable xsuch as
coefficient of variation (C ) and coefficient of kurtosisb (x)and correlation coefficient (r ).
x 2
Srivenkataramana and Tarcy (1980) considered the following dual estimator of the population mean Y based on the use of
the mean value of the non-sampled information of the auxiliary variable defined as
NX - nx
y = y( )  (1.4)
nsu  N - n X 
 *
x
Or y = y 
nsu
X 
 
N X - nx 1 N- n
= = ∑
x x
where ( - ) - i denotes the mean of non sampled units of the auxiliary variable.
N n N n
i=1
1 n 1 n
y = ∑y and x = ∑x are sample means of y and x respectively based on sample size n. Here, it is
n i n i
i=1 i=1
1 N
assumed that X = ∑x , population mean of auxiliary variable, is known.
N i
i=1
To the first degree of approximation, the bias and mean squared error (MSE) of the estimators y and y are given as
e nsu
B( y ) = gY(q 2C2 +rq C C ) (1.5)
e x x y
[ ]
MSE(y )=gY2 C2 +q 2C2 - 2rq C C (1.6)
e y x yx y x
and

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Efficient Family of Exponential and Dual Estimators of Finite Population Mean in Ranked Set Sampling 3
yr C C S
B( y )=E( y )- Y =- xy x y= - xy (1.7)
nsu nsu N NX
[ ]
MSE(y )=gY2 C2 +g2C2 - 2gr C C (1.8)
nsu y x yx y x
S S S 1 n ∑N (y - Y)2
where C = y , C = x , r = yx , g = ( ignoring f = ) , S2 = i=1 i ,
y Y x X yx S S n N y N - 1
y x
∑N (x - X)2 ∑N (y - Y)(x - X) aX mr
S2 = i=1 i , S = i=1 i i ,q = and g =
x N - 1 yx N - 1 2(aX +b) N - mr
2. RATIO ESTIMATORS IN RANKED SET SAMPLING
In Ranked set sampling (RSS), mindependent random sets are chosen, each of size m and units in each set are selected
with equal probability and without replacement from the population. The members of each random set are ranked with
respect to the characteristic of the study variable or auxiliary variable. Then, the smallest unit is selected from the first
ordered set and the second smallest unit is selected from the second ordered set. By this way, this procedure is continued
until the unit with the largest rank is chosen from the mth set. This cycle may be repeated r times, so mr (=n) units
have been measured during this process.
When we rank on the auxiliary variable, let (y ,x ) denote the ith judgment ordering for the study variable
[i] (i)
and ith perfect ordering for the auxiliary variable in the ith set, where i =1,2,3...........,m.
Swami (1996) defined the ratio estimator for the population mean in ranked set sampling as
 X 
y = y   , (2.1)
R,RSS [n]
x 
(n)
1 n 1 n
where y = ∑y , x = ∑x are the ranked set sample means for variables y and x
[n] n [i] (n) n (i)
i=1 i=1
respectively.
To the first degree of approximation the mean squared errors (MSE’s) of the estimators y is given as
R,RSS
[ ]
MSE( y ) =Y2 q{C2 +C2 - 2r C C }- {W - W }2 (2.2)
R,RSS y x yx y x y[i] x(i)
3. MODIFIED EXPONENTIAL ESTIMATOR USING RANKED SET SAMPLING
Motivated by Singh Rajesh et al (2007), We proposed modified exponential estimator for Y using Ranked set sampling as
(aX +b)- (ax +b)
y = y exp (n)  (3.1)
e,RSS [n] (aX +b)+(ax +b)
(n)

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4 Nitu Mehta (Ranka) & V. L. Mandowara
1 n 1 n
Here y = ∑y , x = ∑x are the ranked set sample means for variables y and x respectively.
[n] n [i] (n) n (i)
i=1 i=1
y
The Bias and MSE of can be found as follows-
e,RSS
B( y )=E( y )- Y
e,RSS e,RSS
 aX - aX(1+e ) 
Here y = Y(1+e )exp 1 
e,RSS 0 aX +2b+aX(1+e )
1
[ ]
aX
=Y(1+e )exp- eq (1+eq )- 1 where q =
0 1 1 2(aX +b)
e
Expanding the right hand side of eq. and retaining terms up to second power of ’s , we have
( )
E( y ) =YE1+e - eq +q 2e 2 +eq e
e,RSS 0 1 1 0 1
[ ]
So that B( y ) = Y q 2E(e )+qEe( e ) , becauseE(e ) = E(e ) = 0
e,RSS 1 0 1 0 1
[ { } { }]
⇒ B( y ) =Y g q 2C2 +rq C C - q 2W2 +qW (3.2)
e,RSS x x y x(i) yx(i)
Now MSE( y ) = E( y - Y)2
e,RSS e,RSS
=Y2E[e - qe ]2
0 1
[ ]
= Y2Ee 2 +q 2e 2 - 2qe e
0 1 0 1
[ ( )]
= Y2 gC2 - W2 +q 2(gC2 - W2 )- 2q rg C C - W
y y[i] x x(i) yx y x yx(i)
[ ]
⇒MSE( y ) =Y2 g{C2 +q 2C2 - 2rq C C }- {W - qW }2 (3.3)
e,RSS y x yx y x y[i] x(i)
The following table shows some other estimators of the population mean which can be obtained by putting
different values of constants a and b.

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Estimator Different values of
a b
y = y 0 0
e1,RSS [n]
X - x  1 1
y = y exp (n)
e2,RSS [n] X + x 
(n)
 X - x  1 b (x)
y = y exp (n)  2
e3,RSS [n] X +x(n) +2b 2(x)
 X - x  1 C
y = y exp (n)  x
e4,RSS [n] X +x(n) +2Cx
r
 X - x  1
y = y exp (n) 
e5,RSS [n] X +x +2r 
(n)
 b (x)(X - x )  b (x) C
y = y exp 2 (n)  2 x
e6,RSS [n] b 2(x)(X +x(n))+2Cx 
 C (X - x )  C b (x)
y = y exp x (n)  x 2
e7,RSS [n] Cx(X +x(n))+2b 2(x)
r
 C (X - x )  C
y = y exp x (n)  x
e8,RSS [n] Cx(X +x(n))+2r 
r
 r (X - x )  C
y = y exp (n)  x
e9,RSS [n] r (X +x(n))+2Cx 
r
 b (x)(X - x )  b (x)
y = y exp 2 (n)  2
e10,RSS [n] b (x)(X +x )+2r 
2 (n)
r
 r (X - x )  b (x)
y = y exp (n)  2
e11,RSS [n] r (X +x )+2b (x)
(n) 2

It is cleared that bias and MSE of the above estimators given in the table can be obtained by substituting the
values of a and b in (3.2) and (3.3) respectively.
4. MODIFIED DUAL RATIO ESTIMATOR IN RANKED SET SAMPLING
Motivated by Srivenkataramana and Tarcy (1980), we propose dual ratio-type estimator for using Ranked set sampling as
NX - mrx 
y = y  ( ) (n) (4.1)
nsu,RSS (n) N - mr X 
x*  * = N X - mrx(n)
or ynsu,RSS = y(n) X(n)  , where x(n) (N - mr)
 

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6 Nitu Mehta (Ranka) & V. L. Mandowara
y
The Bias and MSE of can be found as follows
nsu,RSS
B( y )=E( y )- Y
nsu,RSS nsu,RSS
( )
 - +e 
( ) N X mr X 1
y = Y 1+e  ( ) 1 
nsu,RSS 0  N - mr X 
( ) mr 
=Y 1+e 1- e 
0  N - mr 1

 mr mr 
=Y1+e - e - e e 
 0 N - mr 1 N - mr 0 1
Taking expected values on both sides, we have
( )
 ( ) mr ( ) mr ( )
E y =Y 1+E e - E e - E e e
 
nsu,RSS  0 N - mr 1 N - mr 0 1 
mr ( )
=Y - Y Ee e
N - mr 0 1
Now B( y )=E( y )
nsu,RSS nsu,RSS
 m 
1 1 1 ∑
- S - t 
= N - mr X  yx m i=1 yx(i)
mr { }
= - N - mrY rg xyCxCy - Wyx(i)
{ } mr
⇒B( ynsu,RSS ) = - gY rg xyCxCy - Wyx(i) , where g = N - mr (4.2)
Now MSE( y ) = E( y - Y)2
nsu,RSS nsu,RSS
2
  mr mr  
= EY1+e0 - N - mre1- N - mre0e1- Y
=Y2Ee0 - Nm- rmre12

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=Y2{gC2 - W2 }+ mr 2{gC2 - W2 }- 2 mr {gr C C - W }
 y y(i) N- mr x x(i) N- mr y x yx(i) 
 
[ { } { }]
=Y2 g C2 + g2C2 - 2gr C C - W2 + g2W2 - 2gW
y x xy x y y(i) x(i) yx(i)

[ ]
⇒MSE( y )=Y2 g{C2 +g2C2 - 2gr C C }- {W - gW }2 (4.3)
nsu,RSS y x yx y x y[i] x(i)
5. EFFICIENCY COMPARISON
On comparing (1.6) and (1.8) with (3.3) and (4.3) respectively, we obtain
MSE( y ) - MSE( y ) = A1 ‡ 0, where A1= [W - qW ]2
e e,RSS y[i] x(i)
⇒ MSE( y ) £ MSE( y )
e,RSS e
[ ]
MSE( y ) - MSE( y ) = A ‡ 0, where A = W - gW 2
nsu nsu,RSS 2 2 y[i] x(i)
⇒ MSE( y ) £ MSE( y )
nsu,RSS nsu
It is easily seen that the MSE of the suggested estimators given in (3.3) and (4.3) are always smaller than the
estimator given in (1.6) and (1.8) respectively, because A and A all are non-negative values. As a result, show that the
1 2
proposed estimators y and y for the population mean using RSS are more efficient than the usual estimators
e,RSS nsu,RSS
y and y respectively.
e nsu
y
Now Comparison between (2.2) and (3.3), we obtain the estimator will be more efficient than the ratio
nsu,RSS
estimator y if
r,RSS
( ) ( )
y < MSE y
MSE
nsu,RSS r,RSS
[ { } { }]
Y2 g C2 + g2C2 - 2grC C - W2 + g2W2 - 2gW
y x y x y(i) x(i) yx(i)

[ { } { }]
<Y2 g C2 +C2 - 2rC C - W2 +W2 - 2W
y x y x y(i) x(i) yx(i)

{( ) } {( ) }
( ) ( )
⇒ g2 - 1gC2 - 2 g- 1gr C C - g2 - 1W2 - 2 g- 1W <0
x y x x(i) yx(i)

( ){ } { }
( )
⇒ g2 - 1 gC2 - W2 - 2 g - 1 gr C C - W < 0
x x(i) y x yx(i)

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8 Nitu Mehta (Ranka) & V. L. Mandowara
[ { } { }]
( )( )
⇒ g- 1 g+1 gC2 - W2 - 2gr C C - W <0 (5.1)
x x(i) y x yx(i)
Now there are two cases -
Case 1: The inequality (5.1) will be satisfied if
[ { } { }]
( )
g- 1<0 and g+1 gC2 - W2 - 2rg C C - W >0
x x(i) y x yx(i)
mr { } { }
or N - mr - 1<0 and (g+1)gCx2 - Wx2(i) >2gr CyCx - Wyx(i)
rg C C - W (g+1)
mr- N+mr y x yx(i) <
or N- mr <0 and gCx2 - Wx2(i) 2
rg C C - W N
y x yx(i) <
mr < N gC2 - W2 2(N - mr)
or 2 and x x(i)
( )
Cov x(n),y[n] XY < N
N ( ) ( )
mr < 2 2 N - mr
V x X
or 2 and (n) (5.2)
As in the case of SRS, it is clear that to the first order of approximation the RSS estimators are unbiased, using
( )
Cov(x(n),y[n])=bV x(n) in (5.2), we obtain
( )
bVar x( ) X N
( n) <

Var x( ) Y N - mr
n
C N
rxy y <
and
C 2(N - mr)
x
N N
for C @ C we have mr < and rxy<
y x 2 2(N - mr)
mr =15 rxy
This condition holds in practice. For example, if N=100 and then is supposed to be less than 0.60.
Case 2: The inequality (3) will be satisfied if
[ { } { }]
( ) ( )
g- 1 >0 and g+1 C2 - W2 - 2rg C C - W2 <0
x x(i) y x yx(i)
N N
Here for Cy @ Cx,we have n> 2 & r xy > 2(N - mr)
This condition will not hold in practice. For example, if N=100 and mr =70then rxyis supposed to be greater
than 1.667, which is not possible.

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REFERENCES
1. Bahl, S. and Tuteja, R.K. (1991). Ratio and Product type exponential estimator, Information and Optimization
sciences, Vol.XII, I, 159-163.
2. Cochran, W.G. (1940). Some properties of estimators based on sampling scheme with varying probabilities.
Austral. J. Statist., 17, 22-28.
3. Kadilar, C. and Cingi, H. (2006(b)).Improvement in estimating the population mean in simple random sampling.
Applied Mathematics Letters, 19, 75-79.
4. Kadilar, C.,Unyazici, Y. and Cingi H.(2009), Ratio estimator for the population mean using ranked set sampling,
Stat. papers,50,301-309.
5. Khoshnevisan, M. Singh, R., Chauhan, P., Sawan, N. and Smarandache, F. (2007): A general family of estimators
for estimating population means using known value of some population parameter(s). Far East journal of
theoretical statistics, 22(2), 181-191.
6. McIntyre, G.A.(1952), A method of unbiased selective sampling using ranked sets, Australian Journal of
Agricultural Research, 3, 385-390.
7. Mehta (Ranka), Nitu and Mandowara, V.L. (2016). A Modified Ratio-Cum-Product Estimator of Finite
Population Mean Using Ranked Set Sampling. Communication and Statistics-Theory and Methods.Vol.45 (2),
267-276.
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surveys. Commun Stat Theory Methods, 18,379–392.
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Estimator in Simple Random Sampling. Renaissance High Press, 33-40.
10. Samawi, H.M., and Muttlak, H.A. (1996): Estimation of ratio using rank set sampling. The Biometrical Journal,
38, 753-764.
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12. Sisodia, B.V.S. and Dwivedi, V.K. (1981). A modified ratio estimator using coefficient of variation of auxiliary
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APPENDIX
To obtain bias and MSE of y , we put y =Y(1+e ) and x = X(1+e )so that E(e ) = E(e ) = 0, and
e,RSS [n] 0 (n) 1 0 1
therefore,

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10 Nitu Mehta (Ranka) & V. L. Mandowara
V(y )
V(e ) = E(e 2) = [n]
0 0 2
Y
1 1  1 m  [ ]
= S2 - ∑t 2 = gC2 - W2
mr Y2  y m i=1 y[i] y y[i]
[ ]
similarly, V(e ) = E(e 2)= gC2 - W2
1 1 x x(i)
( )
Cov y ,x
and Cov(e ,e ) = E(e ,e ) = [n] (n)
0 1 0 1
XY
1 1  1 m  [ ]
= S - ∑t  = gr C C - W
XY mr  yx m yx(i) yx y x yx(i)
i=1
1 y - Y x - X S2 S2 Syx
where g = , e = [n] , e = (n) , C2 = y , C2 = x , C = = r C C
mr 0 Y 1 X y Y2 x X2 yx XY yx y x
W2 = 1 1 ∑m t 2 , W2 = 1 1 ∑m t 2 and W = 1 1 ∑m t . Here we would like to remind that
x(i) m2r X2 i=1 x(i) y[i] m2r Y2 i=1 y[i] yx(i) m2r XY i=1 yx(i)
t = m - X , t = m - Y and t = (m - X) (m - Y).
x(i) x(i) y[i] y[i] yx(i) x(i) y[i]
e e
Further to validate first degree of approximation, we assume that the sample size is large enough to get and
0 1
e e
as small so that the terms involving and or in a degree greater than two will be negligible.
0 1

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