201O International Conference on Computer and Communication Technologies in Agriculture Engineering
The chain ratio estimator using two auxiliary information* Jingli Lu, Zaizai Yan, Changjiang Ding, Zhimin Hong College o/Sciences Inner Mongolia University o/Technology Hohhot 010051, China
[email protected]
Abstract
-In sample surveys, it is usual to make use of
-x
Yr = Y-= X
auxiliary information to increase the precision of estimators. We propose a new chain ratio estimator of a finite population mean using two auxiliary variables and obtain mean square error
where it is assumed that the population mean auxiliary variate x is known. Here
(MSE) equations for proposed estimator. We find theoretical conditions that make proposed estimator more efficient than estimator of Abu-Dayyeh etc using two auxiliary variables.
-
1 X=-L Xi n i= 1
Index Terms - Chain ratio estimator; Auxiliary variable; Mean square error; Efficiency.
I.
n
and
-
X
of the
1 y=-LYi n i= 1 n
(2)
where n is the number of units in the sample[20]. The MSE of the classical ratio estimator is
MSE(Yr )= 1- f (S; + RZS; - 2RSyx)
INTRODUCTION
(3)
n
where
The use of supplementary information provided by auxiliary variables in survey sampling was extensively discussed [1-19]. The ratio estimator is among the most commonly adopted estimator of the population mean or total of some variable of interest of a finite population with the help of an auxiliary variable when the correlation coefficient between the two variables is positive. It is well known that these estimators are more efficient than the usual estimator of the population mean based on the sample mean of a simple random sampling.
R=
n
f= - ; N
Y
X
N
is the number of units in the population;
S;
is the population ratio;
variance of the auxiliary variate and
S;
is the population is the population
variance of the variate of interest [20]. Kadilar and Qingi[21] proposed the chain ratio estimator using one auxiliary information for -
Y as
-X
Ycr = y(-=-t X
In this study, we proposed a new chain ratio estimator using two auxiliary variates, and obtain mean square error (MSE) equation for this proposed estimator in Section m. We find theoretical conditions that make proposed estimator more efficient than Abu-Dayyeh etc estimator using two auxiliary variables in section N.
II.
(1)
(4)
MSE of this estimator was given as follows:
MSE(ycr )= 1- f (Sz + azRzSz - 2aRS ) n Y
x
(5)
yx
The traditional multivariate ratio estimator [22] using information of two auxiliary variables Xl and Xz to estimate the population mean, Y , as follows:
THE EXISTED ESTIMATORS
(6)
The classical ratio estimator for the popUlation mean Y of the variate of interest y using one auxiliary information is defined by
where
Xi
and
Xi (i=1,2) denote respectively the sample and
the population means of the variable Xi; and
WI' Wz
are the
*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.NJI008S.
978-1-4244-6947-5/10/$26.00 ©2010 IEEE
CCTAE 2010
136
weights that satisfY the condition:
WI +W2 = I.The MSE of
1- I -2 ( 2 +W 2 2 + 2 2 MSE(YMR )=--Y Cy I CX W2 CX
=
_
'
n
=
I
this estimator is given by -
z aZ
C;2 +aZ PYX2 Cy CX2 -al PyX, Cx, Cy -alaZ PX,X2 Cx, Cx2 alzC � +a/C;2-2alaZPX,X2Cx,Cx2 w; l-w; When al= a2 =-a , Eq.(10) becomes as follows: 2 2 2 2 2 2 2 2 "''S'E ( ) 1- f (S2 Yw =-y +wl a RI Sx +w2 a R2 Sx2 n -2wlaRISyx, - 2w2 aR2SYX2 +2WI w2 a 2 RIR2Sx,X2 ) The optimum values of WI and w2 become as follows: + Z ; -a -aZ w' a C 2 pYX2CyCX2 apYX,CX,Cy pX,X2CX,CX2, aZC � +aZC ;2-2azPX,X2Cx,CX2 w; l-w; The minimum MSE of Yw can be shown to be: 1- I (S2 2 2 2 2 2 2 2 2 MSEmin( yw )=-y +W*la RI Sx +W*2 a R2 Sx2 w'
2
l l Vi,
Cy'Cx,
CX2 denote the coefficient of variation of Y, X\ and X2 respectively and Pyx, ,pyx2 'Pxx , 2 denote the
where
and
correlation coefficient between respectively.
Y and X\
,
Y and X2, X\
and X2
WI and w2 are given by 2 C -P C C +p C C -P C C W*I = X2 YX2 2 Y X2 2 yx, Y x, X,X2 x, X2 Cx, +CX2 - 2PX,X2 CX,CX2 W; = l-w; The minimum MSE of YMR can be shown to be: 1 - I -2 ( 2 MSErnin(yMR )=--Y Cy +W*I2Cx2 +W*22C2x The optimum values of
I
'
=
=
1
n
_
'
n
2
-2W*IPyxC , yCx, - 2W* 2 a12PYX2 CyCX2 +2w* 1w* 2PX,X2 CxC , x)
(11) (8)
Abu-Dayyeh etc [23] proposed the ratio estimator using two auxiliary variables for
III. THE SUGGESTED ESTIMATOR
Y as
Based on Eq.(4) and (6), we propose the multivariate ratio estimator using information of two auxiliary variables as follows:
- X - W - XI (9) Yw = I y( t, +W2 y( 2 t2 X2 XI where WI' W2 are the weights that satisfY the condition: WI +W2 = I.The MSE of this estimator is given by 2 MSE(yw ) == 1- I y (Cy2 +wI2 al2Cx,2 +w22 a22Cx2 2 n +2wlaIPyxC , yCX, + 2w2 a2PYX2 CyCX2 +2WIw2 ala2Px,X2 Cx,CX2 ) = 1- 1 (Sy2 +wI2 aI2RI2Sx,2 +w22 a22R/Sx2 2 n +2wlaIRISyx, + 2w2 a2RzSYX2 (10) +2wlw2 ala2RIR2Sxx, 2 ) Y where RI = XI The optimum value of WI and w2 which minimizes MSE( yw ) can be shown to be:
- _ - XC a _ - lUI XI +lU2 X2 a ) Yamr -Y( ) -Y( lUI XI +lU2 X2 Xc where Xc = lUI XI +lU2 X 2 , lUI and lU2 are weights that satisfY the condition: lUI +lU2 = 1. -
----!.-=-----"'="-
(12)
MSE of this estimator can be found using Taylor series method defined as
__
=-,
(13)
137
Yamr-Y==(Y-Y)-
mlaY (XI -XI ) ml XI + mZ XZ
The minimum MSE of
(Yamr) 1-nI[Szy +m*zI azAzSzXI +m*zZazAzSzX2 -2m*laASwr'1- 2m*zaASyx2 + 2m* 1m*zaZAZSxIX2 ]
. MSEmm
mZaY (X -X ) ml XI + mZ XZ Z Z ( y -Y)-mlaA(xl -Xl) -mzaA(xz-Xz) -
-
-
Y
ml XI + mz XZ
-=---=,-
=
(18)
IV.
MSE( y amr) < MSE( yw ) (m *zAZ _ w*zR z)azSZ +(m *zAZ _ w*zR Z )a zSZ I I I X2 XI Z Z Z -2(m*I A -w*IRI )aS}XI - 2(m*zA -w*zRz)aSYX2
MSE(Yamr) E( Yamr -rY == E[(Y _ Y)z +m zI azAz(� I -Xl f +m ZaZAZ(�z-Xz)z- 2mlaA(Y-Y)(� I -XI ) Z - 2mzaA(Y-Y)(Xz-Xz) + 2mlmzazAz(� I-XI )(�z-Xz)] 1- 1 [Sz y +mIzazAZSzXI +mZzazAZSzX2 - 2mIaAS}XI n =
=
-
-
-
V.
__
We develop a population mean theoretically show efficient than the condition.
(14) The optimal values of
MSE( y amr) To minimize
ml
and
under the condition of
MSE(Yamr)
EFFICIENCY COMPARISON
We compare the MSE of the proposed estimators given in Eqs. (18) with the MSE of Abu-Dayyeh etc proposed the ratio estimator given in Eq.(11) as follows:
A
MSE of this estimator was given as follows:
-
=
-
=
where
Yamr can be shown to be:
-
mz is ml + mz
=
is equal to minimize
to minimize
CONCLUSION
new chain ratio estimator of a finite using two auxiliary variables and that the proposed estimator is more Abu-Dayyeh etc estimator in certain
1.
MSE*
REFERENCES
where
MSE* SZy + m IZaZAZSZXI + mZZaZAZSZX2 - 2mIaAS}XI - 2mzaASwr 2 + 2mlmzazAZSxx 12 +8(ml + mz - 1) =
Solved the simultaneous Equations (15),(16) and (17) as follows:
(15)
ml + mz
(16) =
1
(17)
ml and mz -a).,S; +a)"Sxx1 2- Syx1 +SyX OJ = OJ2=1-wl, a).,(S� +S;,-2SX1X,)
We obtain the optimal values of •
I
2
2,
•
•
138
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