Research Journal of Applied Sciences, Engineering and Technology 7(19): 4095-4099, 2014
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2014
Submitted: December 09, 2013
Accepted: January 10, 2014
Published: May 15, 2014
Ratio Type Exponential Estimator for the Estimation of Finite Population
Variance under Two-stage Sampling
1
Muhammad Jabbar, 2Javid Shabbir, 2Zaheer Ahmed and 3Zainab Rehman
1
QEC, The University of Lahore, Lahore, Pakistan
2
Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan
3
Department of Statistics, Arid Agriculture University Rawalpindi, Pakistan
Abstract: In this study we propose an improved exponential ratio type estimator in estimating the finite population
variance in two stage sampling under two cases: (i) when sum of the weights cannot equal to one and (ii) when sum
of the weights are equal to one. The bias and Mean Squared Error (MSE) are derived up to first order of
approximation. The efficiency conditions, under which proposed estimator is more efficient than the usual sample
variance estimator and regression estimator are obtained. Numerical and simulated studies are conducted to support
the superiority of the estimators. Real data set is used to observe the performances of estimator.
Keywords: Bias, efficiency, exponential ratio type estimator, MSE, two-stage sampling
INTRODUCTION
In many situations, information on the auxiliary
variable is required either at the designing stage or
estimation stage or both stages, to increase precision of
the estimators. Ratio, product and regression estimators
are often used when advance knowledge of population
mean of the auxiliary variable is readily available.
In application when sampling frame is not
available, then it is expense or not feasible to obtain the
sampling units directly from the population. Instead one
can use the two stage sampling which is generally
preferable for large scale surveys and at each stage
frame is easily accessible.
Mahalanobis (1967) was the first one, who
introduced the procedure of two-stage sampling which
was further extended by Godambe (1951). A better
approach for multi-stage design was introduced by
Saxena et al. (1984). Mostly research papers appeared
in two-stage sampling for estimation of population
mean or total including Yunusa (2010), Saini (2013)
and Singh et al. (2013). Many researchers like Wolter
(1985), Das and Tripathi (1978), Isaki (1983),
Upadhyaya et al. (2004), Shabbir and Gupta (2007),
Singh and Vishwakarma (2008) and Shabbir and Gupta
(2010) worked in estimating the finite population
variance by using the simple random sampling. To best
of our knowledge very few research papers exist in
estimating the population variance, so in this study an
attempt has been made for estimation of finite
population variance by using the auxiliary information
in two stage sampling.
Consider a finite population 𝑈𝑈 = (𝑈𝑈1 , 𝑈𝑈2 , . . , 𝑈𝑈𝑁𝑁 ) of
N first stage units (psus) such that i-th psu 𝑈𝑈𝑖𝑖 =
(1, 2, . . , 𝑗𝑗, . . , 𝑀𝑀𝑖𝑖 ) consist of 𝑀𝑀𝑖𝑖 second stage units (ssus)
and 𝑀𝑀 = ∑𝑁𝑁
𝑖𝑖=1 𝑀𝑀𝑖𝑖 . Let y ij and x ij be the values of the
study variable (y) and the auxiliary variable (x)
respectively for the j-th ssu of the i-th psu 𝑈𝑈𝑖𝑖 =
(𝑗𝑗 = 1, 2, . . , 𝑀𝑀𝑖𝑖 ; 𝑖𝑖 = 1, 2, . . , 𝑁𝑁). To estimate 𝑆𝑆𝑦𝑦2 under
two-stage sampling scheme, it is assumed that 𝑆𝑆𝑥𝑥2
known.
Let we define the following notations and symbols:
𝑛𝑛
𝑀𝑀 2 1
1
1
; 𝑓𝑓 = ; 𝛾𝛾2𝑖𝑖 = � �𝑖𝑖 �
� − �;
𝑀𝑀
𝑛𝑛𝑛𝑛 𝑚𝑚 𝑖𝑖
𝑛𝑛
𝑁𝑁
𝑀𝑀𝑖𝑖
1
1
𝑀𝑀𝑖𝑖
𝑃𝑃� = ∑𝑁𝑁
𝑃𝑃� , where 𝑃𝑃�𝑖𝑖 = ∑𝑗𝑗 =1
𝑃𝑃𝑖𝑖𝑖𝑖 ,
𝑁𝑁 𝑖𝑖=1 𝑖𝑖
𝑀𝑀𝑖𝑖
2
1
𝑀𝑀
2
�𝑖𝑖 − 𝑃𝑃�� ;
∑𝑁𝑁 � �𝑖𝑖 𝑃𝑃
=
𝑆𝑆1𝑝𝑝
𝑁𝑁−1 𝑖𝑖=1 𝑀𝑀
1
𝑀𝑀
�𝑖𝑖 − 𝑃𝑃�� �𝑀𝑀𝑖𝑖 𝑄𝑄�𝑖𝑖 − 𝑄𝑄� �;
∑𝑁𝑁 � �𝑖𝑖 𝑃𝑃
𝑆𝑆1𝑝𝑝𝑝𝑝 =
�
𝑀𝑀
𝑁𝑁−1 𝑖𝑖=1 𝑀𝑀
2
1
𝑀𝑀𝑖𝑖
2
�
∑ �𝑃𝑃 − 𝑃𝑃𝑖𝑖 � ;
𝑆𝑆2𝑝𝑝𝑝𝑝 =
𝑀𝑀𝑖𝑖 −1 𝑗𝑗 =1 𝑖𝑖𝑖𝑖
1
𝑀𝑀
∑ 𝑖𝑖 �𝑃𝑃 − 𝑃𝑃�𝑖𝑖 ��𝑄𝑄𝑖𝑖𝑖𝑖 − 𝑄𝑄�𝑖𝑖 �
𝑆𝑆2𝑝𝑝𝑝𝑝𝑝𝑝 =
𝑀𝑀𝑖𝑖 −1 𝑗𝑗 =1 𝑖𝑖𝑖𝑖
𝛾𝛾 =
1−𝑓𝑓
2
2
and 𝑆𝑆2𝑝𝑝𝑝𝑝
are the
where, 𝑃𝑃, 𝑄𝑄 = 𝑋𝑋, 𝑌𝑌; and 𝑃𝑃 ≠ 𝑄𝑄; 𝑆𝑆1𝑝𝑝
variances among psus means and variances among
subunits for the i-th psus, while 𝑆𝑆1𝑝𝑝𝑝𝑝 and 𝑆𝑆2𝑝𝑝𝑝𝑝𝑝𝑝 are their
corresponding covariances.
Assume that a sample of n psus is drawn from U
and then a sample of 𝑚𝑚𝑖𝑖 from 𝑀𝑀𝑖𝑖 ssus units i.e., from
the i-th selected psu using simple random sampling
without replacement at both stages.
We define the following relative error terms.
Let 𝑒𝑒0 =
𝑠𝑠𝑦𝑦2 (2𝑠𝑠) −𝑆𝑆𝑦𝑦2
𝑆𝑆𝑦𝑦2
𝐸𝐸(𝑒𝑒0 ) = 𝐸𝐸(𝑒𝑒1 ) = 0.
and 𝑒𝑒1 =
Corresponding Author: Muhammad Jabbar, QEC, The University of Lahore, Lahore, Pakistan
4095
2
𝑠𝑠𝑥𝑥(2𝑠𝑠)
−𝑆𝑆𝑥𝑥2
𝑆𝑆𝑥𝑥2
such that
Res. J. Appl. Sci. Eng. Technol., 7(19): 4095-4099, 2014
∗
∗
∗
𝑁𝑁
∗
2
𝐸𝐸(𝑒𝑒02 ) = 𝛾𝛾𝛽𝛽2(1𝑦𝑦)
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) ; 𝐸𝐸(𝑒𝑒1 ) = 𝛾𝛾𝛽𝛽2(1𝑥𝑥) + ∑𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) ;
∗
∗
∗
𝐸𝐸(𝑒𝑒0 𝑒𝑒1 ) = 𝛾𝛾 𝛾𝛾22(1)
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛾𝛾2(1) 𝛾𝛾22(2𝑖𝑖)
where,
𝜇𝜇 40(1)
𝛽𝛽2(1𝑦𝑦) =
; 𝛽𝛽2(1𝑥𝑥) =
𝜇𝜇 04(1)
; 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) =
𝜇𝜇 40(2𝑖𝑖)
; 𝛽𝛽
=
𝜇𝜇 04(2𝑖𝑖)
;
2(2𝑥𝑥𝑥𝑥 )
2
2
𝜇𝜇 20(2𝑖𝑖)
𝜇𝜇 02(2𝑖𝑖)
∗
∗
∗
𝛽𝛽2(1𝑦𝑦) = 𝛽𝛽2(1𝑦𝑦) − 1; 𝛽𝛽2(1𝑥𝑥) = 𝛽𝛽2(1𝑥𝑥) − 1; 𝛾𝛾22(1) = 𝛾𝛾22(1) − 1;
∗
∗
∗
𝛽𝛽2(2𝑦𝑦𝑦𝑦
) = 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) − 1 ; 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) = 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) − 1; 𝛾𝛾22(2𝑖𝑖) = 𝛾𝛾22(2𝑖𝑖) − 1;
𝛾𝛾𝑟𝑟𝑟𝑟(1) =
2
𝜇𝜇 20(1)
𝜇𝜇 𝑟𝑟𝑟𝑟 (1)
𝑟𝑟�
𝛾𝛾𝑟𝑟𝑟𝑟(2𝑖𝑖) =
𝑠𝑠�
2 𝜇𝜇 2
𝜇𝜇 20(1)
02(1)
2
𝜇𝜇 02(1)
; 𝜇𝜇𝑟𝑟𝑟𝑟(1) =
𝜇𝜇 𝑟𝑟𝑟𝑟 (2𝑖𝑖)
𝑠𝑠�
𝑟𝑟�
2
2
𝜇𝜇 20(2𝑖𝑖) 𝜇𝜇 02(2𝑖𝑖)
𝑚𝑚
𝑀𝑀
𝑟𝑟 𝑚𝑚
𝑀𝑀
𝑠𝑠
∑� ���𝑖𝑖 𝑦𝑦 𝑖𝑖 −𝑌𝑌� � � ���𝑖𝑖 𝑥𝑥 𝑖𝑖 −𝑋𝑋� �
; 𝜇𝜇𝑟𝑟𝑟𝑟(2𝑖𝑖) =
𝑀𝑀
𝑁𝑁
𝑟𝑟
;
𝑠𝑠
𝑖𝑖 �𝑦𝑦 −𝑌𝑌
∑𝑗𝑗 =1
𝑖𝑖𝑖𝑖 � 𝑖𝑖 � �𝑥𝑥 𝑖𝑖𝑖𝑖 −𝑋𝑋� 𝑖𝑖 �
𝑁𝑁
The usual variance estimator for population variance in two-stage sampling is given by:
2
2
= 𝑠𝑠𝑦𝑦(2𝑠𝑠)
𝑆𝑆̂0(2𝑠𝑠)
(1)
2
The MSE of 𝑆𝑆̂0(2𝑠𝑠)
is given by:
2
∗
∗
� ≅ 𝑆𝑆𝑦𝑦4 �𝛾𝛾𝛽𝛽2(1𝑦𝑦)
+ ∑𝑁𝑁
𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂0(2𝑠𝑠)
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) �
(2)
The traditional regression estimator for population variance under two-stage sampling is given by:
2
2
2
2
𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟
(2𝑠𝑠) = 𝑠𝑠𝑦𝑦(2𝑠𝑠) + 𝑏𝑏�𝑆𝑆𝑥𝑥(2𝑠𝑠) − 𝑠𝑠𝑥𝑥(2𝑠𝑠) �
(3)
where, b is the sample regression coefficient in two stage sampling.
2
The MSE of 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟
(2𝑠𝑠) is given by:
∗
2
∗
𝑁𝑁
2
2
4
𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟
(2𝑠𝑠) � ≅ 𝑆𝑆𝑦𝑦 �𝛾𝛾𝛽𝛽2(1𝑦𝑦) (1 − 𝜌𝜌1 ) + ∑𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) (1 − 𝜌𝜌2𝑖𝑖 )�
where, 𝜌𝜌12 =
∗2
𝛾𝛾22(2𝑖𝑖)
∗
∗
𝛽𝛽 2(1𝑦𝑦
) 𝛽𝛽 2(1𝑥𝑥)
2
and 𝜌𝜌2𝑖𝑖
=
∗2
𝛾𝛾22(2𝑖𝑖)
∗
∗
𝛽𝛽 2(2𝑦𝑦𝑦𝑦
) 𝛽𝛽 2(2𝑥𝑥𝑥𝑥 )
(4)
.
PROPOSED ESTIMATOR
On the lines of Gupta and Shabbir (2008), we propose the following an improved exponential ratio type
estimator for population variance (𝑆𝑆𝑦𝑦2 ) in two-stage sampling, given by:
2
2
2
𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
= �𝑘𝑘1 𝑠𝑠𝑦𝑦(2𝑠𝑠)
+ 𝑘𝑘2 (𝑆𝑆𝑥𝑥2 − 𝑠𝑠𝑥𝑥(2𝑠𝑠)
)�𝑒𝑒𝑒𝑒𝑒𝑒 �
2
𝑆𝑆𝑥𝑥2 −𝑠𝑠𝑥𝑥(2𝑠𝑠)
2
𝑆𝑆𝑥𝑥2 +𝑠𝑠𝑥𝑥(2𝑠𝑠)
�
where, 𝑘𝑘1 and 𝑘𝑘2 are suitably chosen constants. We discuss the two cases:
•
•
When 𝑘𝑘1 + 𝑘𝑘2 ≠ 1
When 𝑘𝑘1 + 𝑘𝑘2 = 1
2
), we consider the following two cases.
To find the properties of our proposed estimator (𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
Case 1: When 𝑘𝑘1 + 𝑘𝑘2 ≠ 1.
Using notations from above section, we have:
𝑒𝑒
3
2
≅ �𝑘𝑘1 𝑆𝑆𝑦𝑦2 (1 + 𝑒𝑒0 ) − 𝑘𝑘2 𝑆𝑆𝑥𝑥2 𝑒𝑒1 � �1 − 0 + 𝑒𝑒12 −. . . �
𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
2
8
4096
(5)
Res. J. Appl. Sci. Eng. Technol., 7(19): 4095-4099, 2014
To first order of approximation, we have:
𝑒𝑒
3
1
1
2
− 𝑆𝑆𝑦𝑦2 � ≅ (𝑘𝑘1 − 1)𝑆𝑆𝑦𝑦2 + 𝑘𝑘1 𝑆𝑆𝑦𝑦2 �𝑒𝑒0 − 1 + 𝑒𝑒12 − 𝑒𝑒0 𝑒𝑒1 � − 𝑘𝑘2 𝑆𝑆𝑥𝑥2 (𝑒𝑒1 − 𝑒𝑒12 )
�𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
2
8
2
2
to first order of approximation, is given by:
Using (6), the bias of 𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
(6)
2
3 ∗
1 ∗
2
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵�𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
� ≅ �(𝑘𝑘1 − 1)𝑆𝑆𝑦𝑦2 + 𝑘𝑘1 𝑆𝑆𝑦𝑦2 �𝛾𝛾 � 𝛽𝛽2(1𝑥𝑥)
− 𝛾𝛾22(1)
�
3
1
8
1
2
∗
∗
∗
𝑁𝑁
∗
2
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 � 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) − 𝛾𝛾22(2𝑖𝑖) �� + 𝑘𝑘2 𝑆𝑆𝑥𝑥 �𝛾𝛾𝛽𝛽2(1𝑥𝑥) + ∑𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) ��
8
2
2
(7)
2
to first order of approximation, is given by:
Using (6), the MSE of 𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
∗
∗
2
∗
𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
� ≅ 𝑆𝑆𝑦𝑦4 �1 + 𝑘𝑘12 �1 + 𝛾𝛾�𝛽𝛽2(1𝑦𝑦)
+ 𝛽𝛽2(1𝑥𝑥)
− 2𝛾𝛾22(1)
�
1
3
∗
∗
∗
∗
𝑁𝑁
∗
∗
2 2
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 �𝛽𝛽2(2𝑦𝑦)𝑖𝑖 + 𝛽𝛽2(2𝑥𝑥)𝑖𝑖 − 2𝛾𝛾22(2𝑖𝑖) ��+𝑘𝑘2 𝜑𝜑 �𝛾𝛾𝛽𝛽2(1𝑥𝑥) + ∑𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑥𝑥)𝑖𝑖 � −2𝑘𝑘1 �1 + 2 �𝛾𝛾 �4 𝛽𝛽2(1𝑥𝑥) −
𝛾𝛾22(1)∗+𝑖𝑖=1𝑁𝑁𝛾𝛾2𝑖𝑖34𝛽𝛽2(2𝑥𝑥)𝑖𝑖∗−𝛾𝛾22(2𝑖𝑖)∗−22𝑘𝑘2𝜑𝜑𝛾𝛾𝛽𝛽2(1𝑥𝑥)∗+𝑖𝑖=1𝑁𝑁𝛾𝛾2𝑖𝑖𝛽𝛽2(2𝑥𝑥)𝑖𝑖∗+2𝑘𝑘1𝑘𝑘2𝜑𝜑𝛾𝛾(𝛽𝛽2(2𝑥𝑥)∗−𝛾𝛾22(
1)∗)
∗
∗
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 �𝛽𝛽2(2𝑥𝑥)𝑖𝑖 − 𝛾𝛾22(2𝑖𝑖) ���
where, 𝜑𝜑 =
or:
𝑆𝑆𝑥𝑥2
𝑆𝑆𝑦𝑦2
2
) ≅ 𝑆𝑆𝑦𝑦4 (1 + 𝑘𝑘12 𝐴𝐴1∗ + 𝑘𝑘22 𝐵𝐵1∗ + 2𝑘𝑘1 𝑘𝑘2 𝐶𝐶1∗ − 2𝑘𝑘1 𝐷𝐷1∗ − 2𝑘𝑘2 𝐸𝐸1∗ )
𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
(8)
where,
∗
∗
∗
∗
∗
∗
+ 𝛽𝛽2(1𝑥𝑥)
− 2𝛾𝛾22(1)
) + ∑𝑁𝑁
𝐴𝐴1∗ = 1 + 𝛾𝛾(𝛽𝛽2(1𝑦𝑦)
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 �𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) + 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) − 2𝛾𝛾22(2𝑖𝑖) �
∗
∗
𝐵𝐵1∗ = 𝜑𝜑 2 �𝛾𝛾𝛽𝛽2(1𝑥𝑥)
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) �
∗
∗
∗
∗
∗
𝐶𝐶1 = 𝜑𝜑�𝛾𝛾(𝛽𝛽2(1𝑥𝑥) − 𝛾𝛾22(1) ) + ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 �𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) − 𝛾𝛾22(2𝑖𝑖) ��
1
3
3
∗
∗
∗
∗
− 𝛾𝛾22(1)
� + ∑𝑁𝑁
𝐷𝐷1∗ = 1 + �𝛾𝛾 � 𝛽𝛽2(1𝑥𝑥)
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 � 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) − 𝛾𝛾22(2𝑖𝑖) ��
1
2
4
4
∗
∗
+ ∑𝑁𝑁
𝐸𝐸1∗ = 𝜑𝜑�𝛾𝛾𝛽𝛽2(1𝑥𝑥)
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) �
2
Now differentiating (8) with respect to 𝑘𝑘1 and 𝑘𝑘2 , we get:
𝑘𝑘1𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐵𝐵1∗ 𝐷𝐷1∗ −𝐶𝐶1∗ 𝐸𝐸1∗
𝐴𝐴∗1 𝐵𝐵1∗ −𝐶𝐶1∗2
and 𝑘𝑘2𝑜𝑜𝑜𝑜𝑜𝑜 =
𝐴𝐴∗1 𝐸𝐸1∗ −𝐶𝐶1∗ 𝐷𝐷1∗
𝐴𝐴∗1 𝐵𝐵1∗ −𝐶𝐶1∗2
2
, given by:
Substituting the optimum values of 𝑘𝑘1 and 𝑘𝑘2 in (8), we get the minimum MSE of 𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
∗ ∗2 +𝐵𝐵 ∗ 𝐷𝐷 ∗2 −2𝐶𝐶 ∗ 𝐷𝐷 ∗ 𝐸𝐸 ∗ )
1 1
1 1 1
𝐴𝐴∗1 𝐵𝐵1∗ −𝐶𝐶1∗2
(𝐴𝐴 𝐸𝐸
2
𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
)𝑚𝑚𝑚𝑚𝑚𝑚 ≅ 𝑆𝑆𝑦𝑦4 �1 − 1 1
�
Case 2: When 𝑘𝑘1 + 𝑘𝑘2 = 1
Under Case 2, the proposed estimator becomes:
∗2
2
2
= �𝑘𝑘1 𝑠𝑠𝑦𝑦(2𝑠𝑠)
+ (1 − 𝑘𝑘1 )(𝑆𝑆𝑥𝑥2 − 𝑠𝑠𝑥𝑥(2𝑠𝑠)
)�𝑒𝑒𝑒𝑒𝑒𝑒 �
𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
2
𝑆𝑆𝑥𝑥2 −𝑠𝑠𝑥𝑥(2𝑠𝑠)
2
𝑆𝑆𝑥𝑥2 +𝑠𝑠𝑥𝑥(2𝑠𝑠)
�
(9)
(10)
∗2
respectively as given by:
Putting 𝑘𝑘2 = 1 − 𝑘𝑘1 in (7) and (8), we get the bias and MSE of 𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
1
3
∗2
∗
∗
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵�𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
� ≅ �(𝑘𝑘1 − 1)𝑆𝑆𝑦𝑦2 + 𝑘𝑘1 𝑆𝑆𝑦𝑦2 �𝛾𝛾 �𝛽𝛽2(1𝑥𝑥)
� − 𝜑𝜑� − 𝛾𝛾22(1)
�
1
And
3
2
1
4
∗
∗
∗
𝑁𝑁
∗
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 �𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) � − 𝜑𝜑� − 𝛾𝛾22(2𝑖𝑖) �� + 𝜑𝜑�𝛾𝛾𝛽𝛽2(1𝑥𝑥) + ∑𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑥𝑥𝑥𝑥 ) ��
2
4
2
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Res. J. Appl. Sci. Eng. Technol., 7(19): 4095-4099, 2014
∗2
𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
) ≅ 𝑆𝑆𝑦𝑦4 {1 + 𝑘𝑘12 (𝐴𝐴1∗ + 𝐵𝐵1∗ − 2𝐶𝐶1∗ ) − 2𝑘𝑘1 (𝐵𝐵1∗ − 𝐶𝐶1∗ + 𝐷𝐷1∗ − 𝐸𝐸1∗ )+𝐵𝐵1∗ − 2𝐸𝐸1∗ }
(12)
Now differentiating (12) with respect to 𝑘𝑘1 , we get:
∗
=
𝑘𝑘1𝑜𝑜𝑜𝑜𝑜𝑜
(𝐵𝐵1∗ −𝐶𝐶1∗ +𝐷𝐷1∗ −𝐸𝐸1∗ )
�𝐴𝐴∗1 +𝐵𝐵1∗ −2𝐶𝐶1∗ �
∗2
∗
in (12), we get the minimum MSE of 𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
, given by:
Substituting the optimum value of 𝑘𝑘1 i.e., 𝑘𝑘1𝑜𝑜𝑜𝑜𝑜𝑜
∗2
𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
�
𝑚𝑚𝑚𝑚𝑚𝑚
≅ 𝑆𝑆𝑦𝑦4 �1 + 𝐵𝐵1∗ − 2𝐸𝐸1∗ −
(𝐵𝐵1∗ −𝐶𝐶1∗ +𝐷𝐷1∗ −𝐸𝐸1∗ )2
�𝐴𝐴∗1 +𝐵𝐵1∗ −2𝐶𝐶1∗ �
�
(13)
EFFICIENCY COMPARISONS
We compare the proposed estimator with usual sample variance estimator and regression estimator in two-stage
sampling as follows:
Condition (1): By (2) and (9):
2
2
)𝑚𝑚𝑚𝑚𝑚𝑚 < 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂0(2𝑠𝑠)
� if
𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
∗
∗
𝛾𝛾𝛽𝛽2(1𝑦𝑦)
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) +
�𝐴𝐴∗1 𝐸𝐸1∗2 +𝐵𝐵1∗ 𝐷𝐷1∗2 −2𝐶𝐶1∗ 𝐷𝐷1∗ 𝐸𝐸1∗ �
Condition (2): By (4) and (9):
𝐴𝐴∗1 𝐵𝐵1∗ −𝐶𝐶1∗2
2
2
)𝑚𝑚𝑚𝑚𝑚𝑚 < 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟
𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
(2𝑠𝑠) � if
∗
∗
2
(1 − 𝜌𝜌12 ) + ∑𝑁𝑁
𝛾𝛾𝛽𝛽2(1𝑦𝑦)
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) (1 − 𝜌𝜌2𝑖𝑖 ) +
Condition (3): By (2) and (13):
∗2
2
)𝑚𝑚𝑚𝑚𝑚𝑚 < 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂0(2𝑠𝑠)
� if
𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
∗
∗
∗
∗
𝛾𝛾𝛽𝛽2(1𝑦𝑦)
+ ∑𝑁𝑁
𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) − �1 + 𝐵𝐵1 − 2𝐸𝐸1 −
Condition (4): By (4) and (13):
−1>0
�𝐴𝐴∗1 𝐸𝐸1∗2 +𝐵𝐵1∗ 𝐷𝐷1∗2 −2𝐶𝐶1∗ 𝐷𝐷1∗ 𝐸𝐸1∗ �
𝐴𝐴∗1 𝐵𝐵1∗ −𝐶𝐶1∗2
(𝐵𝐵1∗ −𝐶𝐶1∗ +𝐷𝐷1∗ −𝐸𝐸1∗ )2
�𝐴𝐴∗1 +𝐵𝐵1∗ −2𝐶𝐶1∗ �
−1>0
�>0
∗
∗
∗2
2
𝑁𝑁
2
2
∗
∗
)𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟
𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
(2𝑠𝑠) � if 𝛾𝛾𝛽𝛽2(1𝑦𝑦) (1 − 𝜌𝜌1 ) + ∑𝑖𝑖=1 𝛾𝛾2𝑖𝑖 𝛽𝛽2(2𝑦𝑦𝑦𝑦 ) (1 − 𝜌𝜌2𝑖𝑖 ) − �1 + 𝐵𝐵1 − 2𝐸𝐸1 −
𝐵𝐵1∗−𝐶𝐶1∗+𝐷𝐷1∗−𝐸𝐸1∗2𝐴𝐴1∗+𝐵𝐵1∗−2𝐶𝐶1∗>0
Note: The proposed estimator will be more efficient than the usual variance and regression estimators in two-stage
sampling under two cases, when above Conditions (1)-(4) are satisfied.
Numerical illustration: We use the following real data sets to observe the performances of estimators.
Population: (Sarndal et al., 1992)
y : Revenues from the 1985 municipal taxation
x : 1975 population for M = 284 municipalities (ssus) divided into N = 50 clusters (psus)
We use the following expression to obtain the Percent Relative Efficiency (PRE):
𝑃𝑃𝑃𝑃𝑃𝑃 =
2
�
𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂0(2𝑠𝑠)
(. )
× 100
2
2
̂2
̂ ∗2
where, (.) denote the 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂0(2𝑠𝑠)
�, 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟
(2𝑠𝑠) �, 𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆𝑃𝑃(2𝑠𝑠) )𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑀𝑀𝑀𝑀𝑀𝑀(𝑆𝑆𝑃𝑃(2𝑠𝑠) )𝑚𝑚𝑚𝑚𝑚𝑚
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Res. J. Appl. Sci. Eng. Technol., 7(19): 4095-4099, 2014
2
Table 1: Percent relative efficiency of different estimators with respect to 𝑆𝑆̂0(2𝑠𝑠)
2
Population
𝑚𝑚𝑖𝑖
𝑛𝑛
𝑆𝑆̂0(2𝑠𝑠)
Numerical study
3
10
100
20
100
30
100
4
10
100
20
100
30
100
Simulated study
3
10
100
20
100
30
100
4
10
100
20
100
30
100
For simulation study, we selected 10,000
independent first-stage samples of different sizes from a
population. From every selected psus, a second-stage
sample of different sizes, ssus was again selected. Thus,
we had 10,000 independent samples each of different
sizes. For each sample from 1 to 10,000, values of the
estimators were computed and then on the basis of
these values simulated MSE of different estimators
were calculated. Percentage Relative Efficiency (PRE)
2
for
of different estimators with respect to 𝑆𝑆2𝑠𝑠(0)
different sample sizes are given in Table 1.
CONCLUSION
We proposed an improved exponential ratio type
estimator for population variance in two stage sample
under two cases:
•
•
When sum of the weights cannot equal to one
When sum of the weights are equal to one
Percentage Relative Efficiency (PRE) of different
estimators for different sample sizes are given in
Table 1. Both numerical and simulation studies show
the same behavior of results.
From Table 1, we observed that the proposed
2
) under Case 1 performs better than the
estimator (𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
2
usual sample variance estimator 𝑆𝑆̂0(2𝑠𝑠)
and regression
2
∗2
estimator 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟 (2𝑠𝑠) . The proposed estimator (𝑆𝑆̂𝑃𝑃(2𝑠𝑠)
)
under Case 2 is better than the usual sample variance
2
estimator �𝑆𝑆̂0(2𝑠𝑠)
� but show the weaker performance
2
than the traditional regression estimator 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟
(2𝑠𝑠) . So it
2
is preferable to use the estimator (𝑆𝑆̂𝑃𝑃(2𝑠𝑠) ) under Case 1
for future study.
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𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟
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724.86
777.87
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957.11
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