201O International Conference on Computer and Communication Technologies in Agriculture Engineering
Some new ratio estimators using coefficient of variation and Kurtosis of auxiliary variate* Jingli Lu, Zaizai Yan, Changjiang Ding, Zhimin Hong
College o/Sciences Inner Mongolia University o/Technology Hohhot 010051, China
[email protected]
Abstract
-
We propose some ratio estimators of a finite
The classical ratio estimator for the population mean Y of the variate of interest y is defined by
population mean using auxiliary variable and obtain mean square error (MSE) equations for all proposed estimators. We
-X Yr =Y-= X
find theoretical conditions that make each proposed estimator more efficient than the traditional ratio estimators. In addition, we support these theoretical results with the aid of a numerical example.
(1)
where it is assumed that the population mean auxiliary variate x is known. Here
- 1 X=LX;
Index Terms - Ratio estimator; Auxiliary variable; Mean
n
square error; Efficiency.
I.
n
of the
y=-1 LY; n
-
(2)
n ; 1 ; 1 where n is the number of units in the sample[I9]. The MSE of the classical ratio estimator is
INTRODUCTION
=
=
f MSE(Yr)= 1- (S: +R2S; -2RS x)
where
R=
f=
Y
X
n -
N
(3)
Y
n
Ratio estimators take advantage of the correlation between the auxiliary variable, x and the study variable, y. When information is available on the auxiliary variable that is positively correlated with the study variable, the ratio estimator is a suitable estimator to estimate the population mean. Recently, the use of supplementary information provided by auxiliary variables in survey sampling was extensively discussed [1-18]. For ratio estimators in sampling theory, population information of the auxiliary variable, such as the coefficient of variation or the kurtosis, is often used to increase the efficiency of the estimation for a population mean.
; N is the number of units in the population;
is the population ratio;
S;
variance of the auxiliary variate and
S:
is the population is the population
variance of the variate of interest [19]. When the population coefficient of variation of auxiliary variate is known, Sisodia and Dwivedi [20] suggested a
Cx
modified ratio estimator for Y as
- X+C YSD = y X x+Cx
In this study, we proposed some new ratio estimators using coefficient of variation and Kurtosis of auxiliary variate, and obtain mean square error (MSE) equations for all proposed estimators in Section m. We find theoretical conditions that make each proposed estimator more efficient than the traditional ratio estimators in section N. In addition, we support these theoretical results with the aid of a numerical example in section V.
(4)
MSE of this estimator was given as follows:
MSE(YSD) where Y,
II.
and
X
Cy
=
denote
a = _X ; X+Cx
I-f-2 n
--
the
y
C [C:+C;a(a-2p--L) ] Cx
coefficient
of
(5)
variation
of
p denote the correlation coefficient between
YandX.
THE EXISTED ESTIMATORS
Singh and Kakran[2I] proposed ratio estimator for Y as
*This study was supported by Foundation of Inner Mongolia University of Technology (multivariate sampling techniques and application), No.x200832, the National Natural Science Foundation of China,No. 10761004 and the college science research project in Inner Mongolia,NO.NJIO08S.
978-1-4244-6947-5/10/$26.00 ©2010 IEEE
CCTAE 2010
132
-
-x
+fJ2 (x) X+fJ2 ()X
YSK = y
where
f3 (x) 2
(6)
is the population coefficient of Kurtosis of
auxiliary variate. MSE of this estimator was given as follows: 1- f-2
MSE(YSK)='
where 0 =
where
n
--
X
Y
�
= =---
X +Cx
MSE of this estimator was given as follows:
MSE(YprJ) E(YprJ_y)2 y � � E[(y_ )2 +(R +�)2( _X)2 -2(R +�)(y -Y)(x -X)] 1- f [S: +(R +�)2S; -2(R +�)Syx] =
C Y [C;+C;o(o-2p2.) ] Cx
(7)
-
X+ fJ2(X)
=
=---
-
n
(14)
Upadhyaya and Singh [22] suggested ratio estimators for When
Y as
- X+fJ2 (X) Ypr2 = Yr x+fJ2 (x)
(9)
MSE of the two estimators were given as follows: C 1- f-2
MSE(YuS J) =. Y [C;+C;e(e-2p2.)] n Cx 2 C I-fMSE(YusJ =. --Y [C;+C;lJ(lJ-2p2.)] n Cx --
Yr '
the proposed ratio
- X X+fJ2 (X) Y x x+fJ2 (X)
(15)
in (6) is replaced with
estimator is obtained as
(8)
- XCx+fJ/x) YUS2 = Y xCx+fJ2 (x)
y
When Taloy series method is used for this estimator in the same way to obtain its MSE equation, we define
f(y,x) Ypr2 in (13). Therefore, =
(10)
MSE(Ypr ) � 1- f [S: +(R +AJ2S; -2(R +AJSyx] n 2
(11)
(16)
where
A.z
When III. THE SUGGESTED ESTIMATORS
=
Y
Y
X + f3 (x) 2
in (8) and (9) are replaced with
Y
in (4) is replaced with
Yr,
Yr' respectively,
we also propose the ratio estimators as follows:
- XfJ2 (X)+Cx Ypr3 = Yr xfJ2 (X)+Cx -
When
,
the proposed ratio
--
- X XfJ/x)+Cx = Y-=x xfJ2 ()X +Cx
estimator is obtained as
- X+Cx YprJ = Yr x+Cx
(18)
- X X+Cx = Y X x+Cx
(12) Similar to (14) and (16), MSE of these two estimators were given as follows:
MSE of this estimator can be found using Taylor series method defined as
o �-X) (y-Y)+ f(Y,�) =. f(Y, X)+ [ ox (Y,X) ( oy O',X)
0[1
-
=
-
n
(19)
MSE(ypr4) � 1- f [S: +(R + ..14)2S; -2(R + A4)Syx] n
(20) Y
Y
(y -Y) -(= + )(x -X) X X +Cx (y -Y) -(R +�)(x -X)
YprJ =
f(y,x) Yprl
-Y�
MSE(ypr3) � 1- f [S: +(R + ..1,)2S; -2(R + A,)SyJ
__
(13) where
(17)
IV.
133
EFFICIENCY COMPARISONS
From Table n, we understand that the most efficient estimator is the third proposed estimator. We examine the conditions, determined in Section N, for this data set,
If we compare the MSE of the proposed estimators given in Eqs. (14), (16), (19) and (20) with the MSE of the traditional ratio estimator given in Eq.(3) we have the conditions
MSE(Ypr) < MSE(Yr)
1- f [S; +(R+AYS;
�
n
