International Journal of Probability and Statistics 2015, 4(2): 42-50 DOI: 10.5923/j.ijps.20150402.02
A Class of Regression Estimator with Cum-Dual Ratio Estimator as Intercept F. B. Adebola1, N. A. Adegoke1,*, Ridwan A. Sanusi2 1
Department of Statistics, Federal University of Technology Akure, Nigeria Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Saudi Arabia
2
Abstract This paper examines a class of regression estimator with cum-dual ratio estimator as intercept for estimating the
mean of the study variable Y using auxiliary variable X. We obtained the bias and the mean square error of the proposed estimator, also, the asymptotically optimum estimator (AOE) was obtained along with its mean square error. Theoretical and numerical validation of the proposed estimator were done to show itβs superiority over the usual simple random sampling estimator and ratio estimator, product estimator, cum-dual ratio and product estimator. It was found that the estimator while performed better than other competing estimators, performed in almost the same way as the usual regression estimator when compared with the usual simple random estimator for estimating the average sleeping hours of undergraduate students of the department of statistics, Federal University of Technology Akure, Nigeria.
Keywords Difference estimator, Auxiliary variable, Cum-Dual ratio estimator, Bias, Mean square error, Efficiency
1. Introduction Ratio estimation has gained relevance in estimation theory because of its improved precision in estimating the population parameters. It has been widely applied in agriculture to estimate the mean yield of crops in a certain area and in forestry, to estimate with high precision, the mean number of trees or crops in a forest or plantation. Other areas of relevance include economics and Population studies to estimate the ratio of income to family size. Utilizing Information from high resolution satellite data, [1] examined the possibilities of different forms of auxiliary information derived from remote sensing data in two-phase sampling design and suggested an appropriate estimator that would be of broad interest and applications by proposing a new class of regression-cum estimators in two-phase sampling. He found it to be more efficient than the usual regression and ratio estimators. A class of product-cum-dual to product estimators was proposed by [2] for estimating finite population mean of the study variate. The use of auxiliary information at the estimation stage to increase the efficiency of the study variable was proposed by [3]. He used supplementary information on an auxiliary variable X positively correlated with Y to develop the ratio estimator to estimate the population mean or total of the study variable Y. The ratio-estimator is always more * Corresponding author:
[email protected] (N. A. Adegoke) Published online at http://journal.sapub.org/ijps Copyright Β© 2015 Scientific & Academic Publishing. All Rights Reserved
efficient than the normal SRS when the relationship between the study variable Y and the auxiliary variable X is linear through the origin, and Y is proportional to X [4]. Product estimator was proposed by [5]. [6] suggested the use of ratio estimator π¦π¦οΏ½ππ when 1
when β β€ ππ 2
πΆπΆπ¦π¦ πΆπΆπ₯π₯
πππΆπΆπ¦π¦
πΆπΆπ₯π₯ 1
>
1 2
and unbiased estimator π¦π¦οΏ½
β€ , where πΆπΆπ¦π¦ , πΆπΆπ₯π₯ and ππ are coefficient 2
of variation of y, coefficient of variation of x and correlation between y and x respectively. A lots of work have been done using auxiliary information. A ratio-cum-dual to ratio estimator was proposed for finite population mean. It was shown that the proposed estimator is more efficient than the simple mean estimator, usual ratio estimator and dual to ratio estimator under certain given conditions [7]. [8] proposed a modified ratio-cum-product estimator of finite population mean of the study variate Y using known correlation coefficient between two auxiliary characters X1 and X2, while [9] proposed a ratio-cum-product estimator of finite population mean using information on coefficient of variation and coefficient of kurtosis of auxiliary variate and showed that the proposed estimator is more efficient than the sample mean estimator, usual ratio and product estimators and estimators proposed by [10] under certain given conditions. Moreover, two exponential ratio estimators of population mean in simple random sampling without replacement were shown to be more efficient than the regression estimator and some existing estimators under review based on their biases, mean squared errors and also by using analytical and numerical results (at optimal conditions) for comparison [11]. Also, [12] suggested a ratio-cum product estimator of a finite
International Journal of Probability and Statistics 2015, 4(2): 42-50
population mean using information on the coefficient of variation and the coefficient of kurtosis of auxiliary variate in stratified random sampling. Suppose that simple random sampling without replacement (SRSWOR) of n units is drawn from a population of N units to estimate the population mean
Y =
1
N
β yi
N i =1
of the study variable Y. All the sample units
are observed for the variables Y and X. Let ( yi , xi ) where i = 1, 2,3,.., n denotes the set of the observation for the study variable Y and X. Let the sample means ( x, y ) be unbiased of the population means of the auxiliary variable
X and study variable Y based on the n observations. y The usual ratio estimator of Y is given as y R = X x
and
the
usual
regression
estimator
is
given
n
as
n
1 1 y reg = y + Ξ²Λ ( X β x) , where y = β yi , x = β xi and
Ξ²Λ =
sxy sx 2
n i =1
n i =1
is the estimate slope of regression
coefficient of Y and X. [13] obtained dual to ratio-cum β
estimator given as
y dR
β x , where x is the =y X πππποΏ½ βπππ₯π₯Μ
. un-sampled auxiliary variable in X given as π₯π₯Μ
β = ππβππ The use of auxiliary information in sample surveys was π₯π₯Μ
β πποΏ½
β π¦π¦οΏ½π
π
π
π
= π¦π¦οΏ½
43
extensively discussed in well-known classical text books such as [14], [15], [16], [17] and [18] among others. Recent developments in ratio and product methods of estimation along with their variety of modified forms are lucidly described in detail by [19]. In this paper, we proposed a class of difference estimator with dual to ratio cum as the slope of the estimator instead of π¦π¦οΏ½, also, π₯π₯Μ
β was used instead of x in the usual regression estimator. The proposed estimator is used to estimate the average sleeping hours of undergraduate students of the department of statistics, Federal University of Technology Akure, Nigeria. The organization of this article is as follows: In Section 2, we provide the conceptual framework of the proposed class of estimator. We derived its bias, Mean Squared Error (MSE) and the resulting optimum value of the MSE, with their rigorous proofs up to order one. In section 3, we compared the MSE of the proposed estimator π¦π¦οΏ½π
π
π
π
with the MSE of π¦π¦οΏ½ under Simple Random Sampling Scheme, in Section 4, we provide the numerical validation of the proposed estimator by using data on the ages and hours of sleeping by the undergraduate students of the Department of Statistics Federal University of Technology Akure, Ondo State, Nigeria. Finally, Section 5 provides the conclusion of our findings.
2. The Proposed Class of Estimator For estimating population mean πποΏ½, we have proposed a class of difference estimator with dual to Ratio cum as the intercept given as
+ πΌπΌ(πποΏ½ β π₯π₯Μ
β )
(1)
β Where Ξ± is a suitably chosen scalar. The bias and mean square error (MSE) of π¦π¦οΏ½π
π
π
π
to the first order approximation is
obtained by substituting π₯π₯Μ
β =
πππποΏ½ βπππ₯π₯Μ
ππβππ
into equation (1), hence, equation (1) becomes, π¦π¦οΏ½ πππποΏ½ βπππ₯π₯Μ
β π¦π¦οΏ½π
π
π
π
= οΏ½οΏ½ ππ
ππβππ
π¦π¦οΏ½ πππποΏ½ βπππ₯π₯Μ
β οΏ½π¦π¦π
π
π
π
= οΏ½οΏ½
We write,
This implies that
β π¦π¦οΏ½π
π
π
π
=
ππ
π¦π¦οΏ½
ππβππ
ππβππ
ππ0 =
οΏ½
οΏ½ + πΌπΌ οΏ½πποΏ½ β
οΏ½ + πΌπΌ οΏ½
πππποΏ½ βπππ₯π₯Μ
οΏ½ πποΏ½
πππποΏ½ βπππ₯π₯Μ
ππβππ
οΏ½
(ππβππ)πποΏ½ βπππποΏ½ +πππ₯π₯Μ
+ πΌπΌ οΏ½
ππβππ
πππ₯π₯Μ
βπππποΏ½ ππβππ
π¦π¦οΏ½ β πποΏ½ π₯π₯Μ
β πποΏ½ ππππππ ππ1 = οΏ½ ππ πποΏ½
οΏ½
οΏ½
π¦π¦οΏ½ = πποΏ½(1 + ππ0 ) ππππππ π₯π₯Μ
= πποΏ½(1 + ππ1 ), Respectively. Hence, equation (2) becomes, πποΏ½(1 + ππ0 ) (πππποΏ½ β πππποΏ½(1 + ππ1 )) πΌπΌ β π¦π¦οΏ½π
π
π
π
= + (πππποΏ½(1 + ππ1 ) β πππποΏ½) οΏ½ ππ β ππ ππ β ππ ππ πποΏ½(1 + ππ0 ) πΌπΌ β = (ππ β ππ(1 + ππ1 )) + (ππππ1 πποΏ½) π¦π¦οΏ½π
π
π
π
ππ β ππ ππ β ππ
(2)
F. B. Adebola et al.:
44
A Class of Regression Estimator with Cum-Dual Ratio Estimator as Intercept
πποΏ½(1 + ππ0 ) πΌπΌ ((ππ β ππ) β ππππ1 ) + (ππππ1 πποΏ½) ππ β ππ ππ β ππ (ππ β ππ) β = πποΏ½(1 + ππ0 )( β ππππ1 ) + πΌπΌπΌπΌππ1 πποΏ½ π¦π¦οΏ½π
π
π
π
ππ β ππ
β π¦π¦οΏ½π
π
π
π
=
Where ππ =
ππ
ππβππ
β π¦π¦οΏ½π
π
π
π
= πποΏ½(1 + ππ0 )(1 β ππππ1 ) + πΌπΌπΌπΌππ1 πποΏ½ β π¦π¦οΏ½π
π
π
π
= πποΏ½(1 + ππ0 ) β πποΏ½ππππ1 (1 + ππ0 ) + πΌπΌπΌπΌππ1 πποΏ½
β = πποΏ½ + πποΏ½ππ0 β πποΏ½ππππ1 (1 + ππ0 ) + πΌπΌπΌπΌππ1 πποΏ½ π¦π¦οΏ½π
π
π
π
β (π¦π¦οΏ½π
π
π
π
β πποΏ½) = πποΏ½ππ0 β πποΏ½ππππ1 (1 + ππ0 ) + πΌπΌπΌπΌππ1 πποΏ½
By taking the expectation of equation (3) we have β β πποΏ½) = πΈπΈ(πποΏ½ππ0 ) β πποΏ½ππ(πΈπΈ(ππ1 ) + πΈπΈ(ππ0 ππ1 )) + πΌπΌπΌπΌπποΏ½πΈπΈ(ππ1 ) πΈπΈ(π¦π¦οΏ½π
π
π
π
But, πΈπΈ(ππ0 ) =
π¦π¦οΏ½βπποΏ½ πΈπΈ οΏ½ οΏ½ οΏ½ ππ
=
1 πποΏ½
(πποΏ½ β πποΏ½) = 0 ,
πΈπΈ(ππ1 ) = πΈπΈ οΏ½
and
π₯π₯Μ
βπποΏ½ οΏ½ πποΏ½
πΈπΈ(ππ0 ππ1 ) = πΈπΈ οΏ½
Hence, equation (4) becomes,
1 πποΏ½
(πποΏ½ β πποΏ½) = 0
(4)
,
1 β ππ ππππππ π₯π₯Μ
β πποΏ½ π¦π¦οΏ½ β πποΏ½ οΏ½οΏ½ οΏ½=οΏ½ οΏ½ ππ πποΏ½ πποΏ½ πποΏ½
β πΈπΈ(π¦π¦οΏ½π
π
π
π
β πποΏ½) = βπποΏ½ππ
Recall,
=
(3)
(1 β ππ) ππππππ ππ β ππ ππππππ = βππ οΏ½ οΏ½ . ππ ππππ πποΏ½πποΏ½ πποΏ½
ππ ππ β ππ ππ ππ β ππ ππππππ β π΅π΅π΅π΅π΅π΅π΅π΅(π¦π¦οΏ½π
π
π
π
) = βοΏ½ οΏ½οΏ½ οΏ½ ππ β ππ ππππ πποΏ½ ππππππ β π΅π΅π΅π΅π΅π΅π΅π΅(π¦π¦οΏ½π
π
π
π
)=β πππποΏ½ ππ =
The Mean Square Error of the estimator given as ππππππ(π¦π¦οΏ½ππππ ) is obtained by squaring both sides of equation (3) and taking the expectation. We have β (π¦π¦οΏ½π
π
π
π
β πποΏ½)2 = πποΏ½ 2 ππ02 + πποΏ½ 2 ππ2 (ππ12 + ππ12 ππ02 + 2ππ0 ππ12 ) + πΌπΌ 2 ππ2 πποΏ½ 2 ππ12 β 2πποΏ½ 2 ππ(ππ0 ππ1 + ππ1 ππ02 )
+2ππ πΌπΌπποΏ½πποΏ½ππ0 ππ1 β 2 πΌπΌππ2 πποΏ½πποΏ½(ππ12 + ππ12 ππ0 )
Ignoring the higher powers of error greater than or equal to 3, we have. β (π¦π¦οΏ½π
π
π
π
β πποΏ½)2 = πποΏ½ 2 ππ02 + πποΏ½ 2 ππ2 ππ12 + πΌπΌ 2 ππ2 πποΏ½ 2 ππ12 β 2πποΏ½ 2 ππππ0 ππ1 + 2ππ πΌπΌπποΏ½πποΏ½ππ0 ππ1 β 2 πΌπΌππ2 πποΏ½πποΏ½ππ12 β (π¦π¦οΏ½π
π
π
π
β πποΏ½)2 = πποΏ½ 2 ππ02 + ππ12 ππ2 (πποΏ½ 2 β 2 πΌπΌπποΏ½πποΏ½ + πΌπΌ 2 πποΏ½ 2 ) + 2ππππ0 ππ1 ( πΌπΌπποΏ½πποΏ½ β πποΏ½ 2 ) β (π¦π¦οΏ½π
π
π
π
β πποΏ½)2 = πποΏ½ 2 ππ02 + ππ12 ππ2 (πποΏ½ β πΌπΌπποΏ½)2 β 2ππππ0 ππ1 πποΏ½(πποΏ½ β πΌπΌπποΏ½)
(5)
Take the expectation of (5) we have
β β πποΏ½)2 = οΏ½ π₯π₯π₯π₯(π¦π¦οΏ½π
π
π
π
β ) = οΏ½ ππππππ(π¦π¦οΏ½π
π
π
π
Where πΆπΆπ¦π¦ =
πππ¦π¦ , πποΏ½
πππππ₯π₯ πππ¦π¦ πππ₯π₯2 1 β ππ (πποΏ½ β πΌπΌπποΏ½) + ππ2 2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½ οΏ½ οΏ½πππ¦π¦2 β 2πππποΏ½ ππ πποΏ½πποΏ½ πποΏ½
β ) = οΏ½ ππππππ(π¦π¦οΏ½π
π
π
π
and πΆπΆπ₯π₯ =
πππ₯π₯ πποΏ½
.
πππ₯π₯π₯π₯ πππ¦π¦2 πππ₯π₯2 1 β ππ (πποΏ½ β πΌπΌπποΏ½) + ππ2 2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½ οΏ½ οΏ½πποΏ½ 2 2 β 2πππποΏ½ ππ πποΏ½ πποΏ½πποΏ½ πποΏ½
1βππ ππ
οΏ½ οΏ½πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½
(6)
International Journal of Probability and Statistics 2015, 4(2): 42-50
45
The optimum value of the πππππποΏ½π¦π¦οΏ½ππππ οΏ½ is given as
ππ 1 β ππ ππ β ) ππππππ(π¦π¦οΏ½π
π
π
π
οΏ½ οΏ½πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½ = οΏ½ ππππ ππππ ππ ππ
ππππ
ππππππ(π¦π¦οΏ½π
π
π
π
) = οΏ½
Set equation (7) to zero; we have
Where π½π½ =
πππ₯π₯π₯π₯
and π
π
=
πππ₯π₯2
1βππ ππ
ππ οΏ½ οΏ½2πππποΏ½ οΏ½π₯π₯π₯π₯οΏ½ (πποΏ½) β 2ππ2 πΆπΆπ₯π₯2 πποΏ½(πποΏ½ β πΌπΌπποΏ½)οΏ½ ππ ππ
2πππππ₯π₯π₯π₯ β 2ππ2 πΆπΆπ₯π₯2 πποΏ½(πποΏ½ β πΌπΌπποΏ½) = 0 πποΏ½ π½π½ πΌπΌ = β πποΏ½ ππ πΌπΌ = π
π
β
πποΏ½ πποΏ½
β is given as Thus, the resulting OPTIMUM MSE of π¦π¦οΏ½ππππ βππππππ
MSE (π¦π¦οΏ½π
π
π
π
) = οΏ½
(7)
π½π½ ππ
2 πππ₯π₯ πππ¦π¦ πππ₯π₯2 1 β ππ π½π½ π½π½ οΏ½ οΏ½πππ¦π¦2 β 2ππππ οΏ½πποΏ½ β οΏ½π
π
β οΏ½ πποΏ½οΏ½ + ππ2 2 οΏ½πποΏ½ β οΏ½π
π
β οΏ½ πποΏ½οΏ½ οΏ½ ππ ππ ππ πποΏ½ πποΏ½
By substituting πΌπΌ in equation (6) MSE
Where π
π
πποΏ½ = πποΏ½,
βππππππ (π¦π¦οΏ½π
π
π
π
)
2 πππ₯π₯ πππ¦π¦ ππ 2 1 β ππ π½π½πποΏ½ π½π½πποΏ½ 2 2 π₯π₯ οΏ½ οΏ½ οΏ½ οΏ½ =οΏ½ οΏ½ οΏ½πππ¦π¦ β 2ππππ οΏ½ππ β π
π
ππ + οΏ½ + ππ 2 οΏ½ππ β π
π
ππ + οΏ½ οΏ½ ππ ππ ππ πποΏ½ πποΏ½ βππππππ
MSE (π¦π¦οΏ½π
π
π
π
) = οΏ½ βππππππ
MSE (π¦π¦οΏ½π
π
π
π
) = οΏ½ βππππππ
MSE (π¦π¦οΏ½π
π
π
π
) = οΏ½ MSE
βππππππ (π¦π¦οΏ½π
π
π
π
) βππππππ
1 β ππ οΏ½ οΏ½πππ¦π¦2 β 2πππ₯π₯π₯π₯ π½π½ + πππ₯π₯2 π½π½2 οΏ½ ππ
πππ₯π₯π₯π₯ πππ₯π₯π₯π₯ 2 1 β ππ οΏ½ οΏ½πππ¦π¦2 β 2πππ₯π₯π₯π₯ οΏ½ 2 οΏ½ + πππ₯π₯2 οΏ½ 2 οΏ½ οΏ½ πππ₯π₯ πππ₯π₯ ππ 2
2
2οΏ½πππ₯π₯π₯π₯ οΏ½ οΏ½πππ₯π₯π₯π₯ οΏ½ 1 β ππ =οΏ½ οΏ½ οΏ½πππ¦π¦2 β + οΏ½ 2 πππ₯π₯ πππ₯π₯2 ππ
MSE (π¦π¦οΏ½π
π
π
π
) = οΏ½ βππππππ
1βππ
2 πππ₯π₯ πππ¦π¦ π½π½πποΏ½ πππ₯π₯2 π½π½πποΏ½ 1 β ππ οΏ½ οΏ½πππ¦π¦2 β 2ππππ οΏ½ οΏ½ + ππ2 2 οΏ½ οΏ½ οΏ½ ππ ππ πποΏ½πποΏ½ ππ πποΏ½
2
οΏ½πππ₯π₯π₯π₯ οΏ½ 1 β ππ οΏ½ οΏ½πππ¦π¦2 β οΏ½ πππ₯π₯2 ππ
MSE (π¦π¦οΏ½π
π
π
π
) = οΏ½
1βππ ππ
οΏ½ πππ¦π¦2 οΏ½1 β οΏ½
πππ₯π₯π₯π₯
πππ₯π₯ πππ¦π¦
2
οΏ½ οΏ½
(7)
βππππππ βππππππ MSE (π¦π¦οΏ½π
π
π
π
) = οΏ½ οΏ½ πππ¦π¦2 (1 β ππ2 ) Equation (7) shows that the MSE (π¦π¦οΏ½π
π
π
π
) is the same as the MSE Regression ππ estimator. Remark β β ) The Bias of π¦π¦οΏ½π
π
π
π
is the same as Bias of the dual ratio estimator π¦π¦οΏ½π
π
β and when Ξ± = 0, ππππππ(π¦π¦οΏ½π
π
π
π
becomes ππππππ(π¦π¦οΏ½π
π
β ) of β dual to ratio estimator π¦π¦οΏ½π
π
proposed by [13]. The bias of π¦π¦οΏ½π
π
β is given as πππ₯π₯π₯π₯ π΅π΅π΅π΅π΅π΅π΅π΅(π¦π¦οΏ½π
π
β ) = β πππποΏ½ The MSE (π¦π¦οΏ½π
π
β ) is given as
ππππππ(π¦π¦οΏ½π
π
β ) = οΏ½
1βππ ππ
οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ππ2 πΆπΆπ₯π₯2 οΏ½
(7a)
F. B. Adebola et al.:
46
A Class of Regression Estimator with Cum-Dual Ratio Estimator as Intercept
3. Efficiency Comparisons In this section, we compared the MSE of the proposed estimator π¦π¦οΏ½π
π
π
π
with the MSE of π¦π¦οΏ½ under Simple Random Sampling Scheme given as, MSE(π¦π¦οΏ½) = οΏ½
1βππ ππ
οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2
(8)
β ) From equations (5) and (8), the proposed estimator is better than that the usual estimator π¦π¦οΏ½ if, MSE (π¦π¦οΏ½π
π
π
π
< MSE (π¦π¦οΏ½). That is, 1 β ππ 1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½ < οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 οΏ½ ππ ππ β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 < 0
(πποΏ½ β πΌπΌπποΏ½) οΏ½β 2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)οΏ½ < 0
This holds if and only if, Case (1) πποΏ½ β πΌπΌπποΏ½ < 0 and β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½) > 0 Or
Case (2) πποΏ½ β πΌπΌπποΏ½ > 0 and β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½) < 0
β is more efficient than π¦π¦οΏ½ is given as, The Range of Ξ± under which the proposed estimator π¦π¦οΏ½ππππ 2πππΆπΆπ¦π¦ 2πππΆπΆπ¦π¦ ππππππ οΏ½π
π
, π
π
οΏ½1 β οΏ½οΏ½ , ππππππ οΏ½π
π
, π
π
οΏ½1 β οΏ½οΏ½. πππΆπΆπ₯π₯ πππΆπΆπ₯π₯ β with the usual ratio estimator π¦π¦οΏ½π
π
. The MSE of the π¦π¦οΏ½π
π
is given as We also compared the proposed estimator π¦π¦οΏ½π
π
π
π
1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ 2 πΆπΆπ₯π₯2 οΏ½ ππππππ( π¦π¦οΏ½π
π
) = οΏ½ ππ β β It is found that the proposed estimator π¦π¦οΏ½π
π
π
π
will be more efficient than the usual ratio estimator π¦π¦οΏ½π
π
if MSE (π¦π¦οΏ½π
π
π
π
) < MSE (π¦π¦οΏ½π
π
). That is, 1 β ππ 1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½ β€ οΏ½ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ 2 πΆπΆπ₯π₯2 οΏ½ οΏ½ ππ ππ β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + 2πππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 β πποΏ½ 2 πΆπΆπ₯π₯2 < 0 β2πππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (ππ(πποΏ½ β πΌπΌπποΏ½) β πποΏ½) + πΆπΆπ₯π₯2 ((πποΏ½ β πΌπΌπποΏ½)2 β πποΏ½ 2 ) < 0
(ππ(πποΏ½ β πΌπΌπποΏ½) β πποΏ½) οΏ½β2πππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πΆπΆπ₯π₯2 (ππ(πποΏ½ β πΌπΌπποΏ½) + πποΏ½)οΏ½ < 0
This holds if the following two conditions are satisfied (1). (ππ(πποΏ½ β πΌπΌπποΏ½) β πποΏ½) < 0 And β2πππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πΆπΆπ₯π₯2 (ππ(πποΏ½ β πΌπΌπποΏ½) + πποΏ½) > 0. Or
(2). (ππ(πποΏ½ β πΌπΌπποΏ½) β πποΏ½) > 0 And β2πππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πΆπΆπ₯π₯2 (ππ(πποΏ½ β πΌπΌπποΏ½) + πποΏ½) < 0. 2ππβππ
This condition holds if πΌπΌ > π
π
οΏ½
ππ
ππ
οΏ½ and πΌπΌ < π
π
οΏ½ β ππ
2πππΆπΆπ¦π¦ πππΆπΆπ₯π₯
οΏ½ or πΌπΌ < π
π
οΏ½
2ππβππ ππ
ππ
οΏ½ and πΌπΌ > π
π
οΏ½ β ππ
2πππΆπΆπ¦π¦ πππΆπΆπ₯π₯
οΏ½
ππ 2πππΆπΆπ¦π¦ 2ππ β ππ ππ 2πππΆπΆπ¦π¦ 2ππ β ππ οΏ½ , π
π
οΏ½ β οΏ½ οΏ½ , max οΏ½π
π
οΏ½ οΏ½ , π
π
οΏ½ β οΏ½ οΏ½. πππΆπΆπ₯π₯ πππΆπΆπ₯π₯ ππ ππ ππ ππ β with the usual product estimator π¦π¦οΏ½ππ . The MSE of the π¦π¦οΏ½ππ is given as We also compared the proposed estimator π¦π¦οΏ½π
π
π
π
1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 + 2πππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ 2 πΆπΆπ₯π₯2 οΏ½ ππππππ( π¦π¦οΏ½π
π
) = οΏ½ ππ β β It is found that the proposed estimator π¦π¦οΏ½π
π
π
π
will be more efficient than the usual ratio estimator π¦π¦οΏ½ππ if MSE (π¦π¦οΏ½π
π
π
π
) < MSE (π¦π¦οΏ½ππ ). That is, 1 β ππ 1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½ β€ οΏ½ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 + 2πππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ 2 πΆπΆπ₯π₯2 οΏ½ οΏ½ ππ ππ β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) β 2πππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 β πποΏ½ 2 πΆπΆπ₯π₯2 < 0 β2πππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (ππ(πποΏ½ β πΌπΌπποΏ½) + πποΏ½) + πΆπΆπ₯π₯2 ((πποΏ½ β πΌπΌπποΏ½)2 β πποΏ½ 2 ) < 0 min οΏ½π
π
οΏ½
International Journal of Probability and Statistics 2015, 4(2): 42-50
47
(ππ(πποΏ½ β πΌπΌπποΏ½) + πποΏ½) οΏ½β2πππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πΆπΆπ₯π₯2 (ππ(πποΏ½ β πΌπΌπποΏ½) β πποΏ½)οΏ½ < 0
This holds if the following two conditions are satisfied
(1). (ππ(πποΏ½ β πΌπΌπποΏ½) + πποΏ½) < 0 and β2πππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πΆπΆπ₯π₯2 (ππ(πποΏ½ β πΌπΌπποΏ½) β πποΏ½) > 0. Or
(2). (ππ(πποΏ½ β πΌπΌπποΏ½) + πποΏ½) > 0 and β2πππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πΆπΆπ₯π₯2 (ππ(πποΏ½ β πΌπΌπποΏ½) β πποΏ½) < 0. ππ
This condition holds if πΌπΌ > π
π
οΏ½ οΏ½ and πΌπΌ < π
π
οΏ½ ππ
2ππβππ ππ
β
2πππΆπΆπ¦π¦ πππΆπΆπ₯π₯
ππ
οΏ½ or πΌπΌ < π
π
οΏ½ οΏ½ and πΌπΌ > π
π
οΏ½ ππ
2ππβππ ππ
ππ 2ππ β ππ 2πππΆπΆπ¦π¦ ππ 2ππ β ππ 2πππΆπΆπ¦π¦ min οΏ½π
π
οΏ½ οΏ½ , π
π
οΏ½ β οΏ½ οΏ½ , max οΏ½π
π
οΏ½ οΏ½ , π
π
οΏ½ β οΏ½ οΏ½. πππΆπΆπ₯π₯ πππΆπΆπ₯π₯ ππ ππ ππ ππ
β
2πππΆπΆπ¦π¦ πππΆπΆπ₯π₯
οΏ½
We also compared the MSE of the proposed estimator with MSE of dual product estimator π¦π¦οΏ½ππβ proposed by [20]. The MSE proposed by [20] (1980) is given as
(π¦π¦οΏ½ππβ )
ππππππ(π¦π¦οΏ½ππβ ) = οΏ½
1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 + 2πππππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ 2 ππ2 πΆπΆπ₯π₯2 οΏ½ ππ
β β It is found that the proposed estimator π¦π¦οΏ½ππππ will be more efficient than that of [20] estimator π¦π¦οΏ½ππβ if MSE (π¦π¦οΏ½π
π
π
π
) < MSE (π¦π¦οΏ½ππβ ) That is
οΏ½
1βππ ππ
1βππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½ β€ οΏ½ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 + 2πππππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ 2 ππ2 πΆπΆπ₯π₯2 οΏ½ ππ
That is,
β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 < 2πππππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ 2 ππ2 πΆπΆπ₯π₯2
β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ [(πποΏ½ β πΌπΌπποΏ½) + πποΏ½] + ππ2 πΆπΆπ₯π₯2 [(πποΏ½ β πΌπΌπποΏ½))2 β πποΏ½ 2 ] < 0 οΏ½(πποΏ½ β πΌπΌπποΏ½) + πποΏ½οΏ½ οΏ½β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 [(πποΏ½ β πΌπΌπποΏ½) β πποΏ½]οΏ½ < 0 (2πποΏ½ β πΌπΌπποΏ½) οΏ½β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ β ππ2 πΆπΆπ₯π₯2 [ πΌπΌπποΏ½]οΏ½ < 0
This holds if, 1. 2πποΏ½ β πΌπΌπποΏ½ < 0 and β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ β ππ2 πΆπΆπ₯π₯2 [ πΌπΌπποΏ½] > 0 2. 2πποΏ½ β πΌπΌπποΏ½ > 0 and β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ β ππ2 πΆπΆπ₯π₯2 [ πΌπΌπποΏ½] < 0 This condition holds if 2π
π
> πΌπΌ and
β2π
π
π
π
πΆπΆπ¦π¦ πππΆπΆπ₯π₯
< πΌπΌ or 2π
π
< πΌπΌ and
The range of Ξ± under which the proposed estimator min οΏ½2π
π
,
β π¦π¦οΏ½π
π
π
π
β2π
π
π
π
πΆπΆπ¦π¦ πππΆπΆπ₯π₯
> πΌπΌ
iS more efficient than π¦π¦οΏ½ππβ is
β2π
π
π
π
πΆπΆπ¦π¦ β2π
π
π
π
πΆπΆπ¦π¦ οΏ½ , max οΏ½2π
π
, οΏ½ πππΆπΆπ₯π₯ πππΆπΆπ₯π₯
β Lastly, we compared MSE of the proposed estimator π¦π¦οΏ½π
π
π
π
with that of dual to ratio estimator π¦π¦οΏ½π
π
β proposed [13] given in equation (7a). The proposed estimator will be more efficient than π¦π¦οΏ½π
π
β if β MSE (π¦π¦οΏ½π
π
π
π
) < MSE (π¦π¦οΏ½π
π
β ). That is, 1 β ππ 1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ (πποΏ½ β πΌπΌπποΏ½) + ππ2 πΆπΆπ₯π₯2 (πποΏ½ β πΌπΌπποΏ½)2 οΏ½ β€ οΏ½ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ¦π¦2 β 2πππππποΏ½ 2 πΆπΆπ₯π₯ πΆπΆπ¦π¦ + πποΏ½ 2 ππ2 πΆπΆπ₯π₯2 οΏ½ οΏ½ ππ ππ β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ [(πποΏ½ β πΌπΌπποΏ½) β πποΏ½] + ππ2 πΆπΆπ₯π₯2 [(πποΏ½ β πΌπΌπποΏ½)2 β πποΏ½ 2 ] < 0
οΏ½(πποΏ½ β πΌπΌπποΏ½) β πποΏ½οΏ½ οΏ½β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 [(πποΏ½ β πΌπΌπποΏ½) + πποΏ½]οΏ½ < 0 β πΌπΌπποΏ½οΏ½β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 (2πποΏ½ β πΌπΌπποΏ½)οΏ½ < 0
This holds if 1. β πΌπΌπποΏ½ < 0 and β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 (2πποΏ½ β πΌπΌπποΏ½) < 0 Or
2. β πΌπΌπποΏ½ > 0 and β2πππππποΏ½πΆπΆπ₯π₯ πΆπΆπ¦π¦ + ππ2 πΆπΆπ₯π₯2 (2πποΏ½ β πΌπΌπποΏ½) > 0
48
F. B. Adebola et al.:
A Class of Regression Estimator with Cum-Dual Ratio Estimator as Intercept
This condition holds if πΌπΌ > 0 ππππππ πΌπΌ < 2π
π
οΏ½1 β
πππΆπΆπ¦π¦
πππΆπΆπ₯π₯
οΏ½ ππππ πΌπΌ < 0 ππππππ πΌπΌ > 2π
π
οΏ½1 β
πππΆπΆπ¦π¦
πππΆπΆπ₯π₯
οΏ½
β is more efficient than dual ratio estimator Therefore, the range of Ξ± under which the proposed estimator π¦π¦οΏ½π
π
π
π
ππππππ οΏ½0,2π
π
οΏ½1 β
πππΆπΆπ¦π¦ πππΆπΆπ¦π¦ οΏ½οΏ½ , ππππππ οΏ½0,2π
π
οΏ½1 β οΏ½οΏ½ πππΆπΆπ₯π₯ πππΆπΆπ₯π₯
β Thus it seems from the above that the proposed estimator π¦π¦οΏ½π
π
π
π
may be made better than the usual estimator, ratio estimator, product estimator, dual to product estimator π¦π¦οΏ½ππβ and the dual to ratio estimator π¦π¦οΏ½π
π
β , if the given conditions are satisfied.
οΏ½βπΆπΆπΆπΆπΆπΆ Comparison of βAOEβ to ππ πΉπΉπΉπΉ
βππππππ is more efficient than the other existing estimators π¦π¦οΏ½, the ratio estimator π¦π¦οΏ½π
π
, the product estimator π¦π¦οΏ½ππ , the dual to π¦π¦οΏ½π
π
π
π
ratio estimator π¦π¦οΏ½π
π
β and the dual to product estimator π¦π¦οΏ½ππβ since 1 β ππ βππππππ ) οΏ½ οΏ½ πποΏ½ 2 ππ2 πΆπΆπ¦π¦2 οΏ½ > 0 =οΏ½ ππππππ(π¦π¦οΏ½) β ππππππ(π¦π¦οΏ½π
π
π
π
ππ πππΆπΆπ¦π¦ 2 1 β ππ βππππππ ) ππππππ(π¦π¦οΏ½π
π
) β ππππππ(π¦π¦οΏ½π
π
π
π
οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ₯π₯2 οΏ½1 β οΏ½ οΏ½>0 =οΏ½ πΆπΆπ₯π₯ ππ βππππππ ) =οΏ½ ππππππ(π¦π¦οΏ½ππ ) β ππππππ(π¦π¦οΏ½π
π
π
π
πππΆπΆπ¦π¦ 2 1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ₯π₯2 οΏ½1 + οΏ½ οΏ½>0 πΆπΆπ₯π₯ ππ
βππππππ ) =οΏ½ ππππππ(π¦π¦οΏ½π
π
β ) β ππππππ(π¦π¦οΏ½π
π
π
π
βππππππ ) =οΏ½ πππππποΏ½π¦π¦οΏ½ππβ οΏ½ β ππππππ(π¦π¦οΏ½π
π
π
π
2 πππΆπΆπ¦π¦ 1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ₯π₯2 οΏ½ β πποΏ½ οΏ½ > 0 πΆπΆπ₯π₯ ππ
2 πππΆπΆπ¦π¦ 1 β ππ οΏ½ οΏ½ πποΏ½ 2 πΆπΆπ₯π₯2 οΏ½ + πποΏ½ οΏ½ > 0 πΆπΆπ₯π₯ ππ
β is more efficient than other estimator in case of its optimality. Hence, we conclude that the proposed class of estimator π¦π¦οΏ½π
π
π
π
4. Numerical Validation
To illustrate the efficiency of the proposed estimator over the other estimators π¦π¦οΏ½, π¦π¦οΏ½π
π
, π¦π¦οΏ½ππ , π¦π¦οΏ½π
π
β ππππππ π¦π¦οΏ½ππβ . Data on the ages and hours of sleeping by the undergraduate students of the Department of Statistics Federal University of Technology Akure, Ondo State, Nigeria. A sample of 150 out of 461 students of the department was obtained using simple random sampling without replacement. The information on the age of the students was used as auxiliary information to increase the precision of the estimate of the average sleeping hours. The estimate of the average hours of sleeping of the students were obtained and also the 95% confidence intervals of the average hours of sleeping were obtained for the proposed estimator and the other estimators. Table 1., gives the estimates of the average sleeping hours and the 95% confidence Interval. As shown in Table 1.0, the proposed estimator performed better than the other estimators, the width of the confidence interval of the proposed estimator is smallest than the other competing estimators. Table 1. Average Sleeping Hours and 95% confidence intervals for Different Estimators for the undergraduate Students of Department of Statistics, Federals University of Technology Akure. Nigeria ESTIMATOR
Average Sleeping Hours
LCL
UCL
WIDTH
π¦π¦οΏ½
6.08
5.930386531
6.229613469
0.299226939
π¦π¦οΏ½π
π
6.210472103
6.042844235
6.378099971
0.335255737
π¦π¦οΏ½ππ
5.952268908
5.778821411
6.125716404
0.346894993
π¦π¦οΏ½π
π
β
6.141606636
5.988421023
6.294792249
0.306371226
π¦π¦οΏ½ππβ
6.019011342
5.862732122
6.175290562
0.31255844
οΏ½βπΉπΉπΉπΉ ππ
6.194882566
6.050452623
6.33931251
0.288859888
The proposed estimator performed the same way as the regression estimator when compared with the usual simple random sampling. The average Sleeping Hours and 95% confidence intervals for the proposed estimator and the regression estimator is given below, the two estimators have the same width.
International Journal of Probability and Statistics 2015, 4(2): 42-50
49
Table 2. Average Sleeping Hours and 95% confidence intervals for the proposed estimators and regression estimators for the undergraduate Students of Department of Statistics, Federals University of Technology Akure. Nigeria ESTIMATOR
Average Sleeping Hours
LCL
UCL
WIDTH
οΏ½βπΉπΉπΉπΉ ππ
6.194882566
6.050452623
6.33931251
0.288859888
6.089652737
5.945222793
6.234082681
0.288859888
οΏ½βπΉπΉπΉπΉπΉπΉ ππ
β To examine the gain in the efficiency of the proposed estimator π¦π¦οΏ½π
π
π
π
over the estimator π¦π¦οΏ½, π¦π¦οΏ½π
π
, π¦π¦οΏ½ππ , π¦π¦οΏ½π
π
β ππππππ π¦π¦οΏ½ππ,β we obtained οΏ½ the percentage relative efficiency of different estimator of ππ with respect to the usual unbiased estmator π¦π¦οΏ½ in Table 2. The β proposed estimator π¦π¦οΏ½π
π
π
π
performed better than the other estimators π¦π¦οΏ½, π¦π¦οΏ½π
π
, π¦π¦οΏ½ππ , π¦π¦οΏ½π
π
β ππππππ π¦π¦οΏ½ππβ and perfoirmed exactly the same way as regression estimator.
Table 3. The percentage relative efficiency of different estimator of πποΏ½ with respect to the usual unbiased estimator π¦π¦οΏ½ ESTIMATOR
PERCENATGE RELATIVE FFICIENCY
π¦π¦οΏ½
100
π¦π¦οΏ½π
π
79.66158486
π¦π¦οΏ½π
π
β
95.39056726
οΏ½βπΉπΉπΉπΉπΉπΉ ππ
107.3067159
, π¦π¦οΏ½ππ
74.40554745
π¦π¦οΏ½ππβ
91.65136111
οΏ½βπΉπΉπΉπΉ ππ
107.3067159
5. Conclusions We have proposed a class of regression estimator with cum-dual ratio estimator as intercept for estimating the mean of the study variable Y using auxiliary variable X as in equation (1) and obtained βAOEβ for the proposed estimator. Theoretically, we have demonstrated that proposed estimator is always more efficient than other under the effective ranges of πΌπΌ and its optimum values. Table 1. shows that the proposed estimator performed better than the other estimators as the width of the confidence interval of the proposed estimator is smallest than the other competing estimators. The percentage relative efficiency of different estimator of πποΏ½ with respects to the usual unbiased estimator π¦π¦οΏ½ in Table 2. shows that the proposed estimator β performed better than the other estimators π¦π¦οΏ½π
π
π
π
π¦π¦οΏ½, π¦π¦οΏ½π
π
, π¦π¦οΏ½ππ , π¦π¦οΏ½π
π
β ππππππ π¦π¦οΏ½ππβ and performed exactly the same way as regression estimator. Hence, it is preferred to use the proposed class of estimator in practice.
REFERENCES [1]
B. K. Handique, βA Class of Regression-Cum-Ratio Estimators in Two-Phase Sampling for Utilizing Information From High Resolution Satellite Data,β ISPRS Ann. Photogramm. Remote Sens. Spat. Inf. Sci., vol. Iβ4, pp. 71β76, Jul. 2012.
[2]
S. Choudhury and B. K. Singh, βA Class of Product-cum-dual to Product Estimators of the Population Mean in Survey Sampling Using Auxiliary Information,β Asian J Math Stat, vol. 6, 2012.
[3]
W. G. Cochran, βThe estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce,β J. Agric. Sci., vol. 30, no. 02, pp. 262β275, 1940.
[4]
S. Choudhury and B. K. Singh, βAn efficient class of dual to product-cum-dual to ratio estimators of finite population mean in sample surveys,β Glob. J. Sci. Front. Res., vol. 12, no. 3-F, 2012.
[5]
D. S. Robson, βApplications of multivariate polykays to the theory of unbiased ratio-type estimation,β J. Am. Stat. Assoc., vol. 52, no. 280, pp. 511β522, 1957.
[6]
M. N. Murthy, βProduct method of estimation,β SankhyΔ Indian J. Stat. Ser. A, pp. 69β74, 1964.
[7]
B. Sharma and R. Tailor, βA new ratio-cum-dual to ratio estimator of finite population mean in simple random sampling,β Glob. J. Sci. Front. Res., vol. 10, no. 1, 2010.
[8]
H. P. Singh, βEstimation of finite population mean using known correlation coefficient between auxiliary characters,β Statistica, vol. 65, no. 4, pp. 407β418, 2005.
[9]
R. Tailor and B. K. Sharma, βA modified ratio-cum-product estimator of finite population mean using known coefficient of variation and coefficient of kurtosis,β Stat. Transition-new Ser., vol. 10, p. 1, 2009.
[10] L. N. Upadhyaya and H. P. Singh, βUse of transformed
50
F. B. Adebola et al.:
A Class of Regression Estimator with Cum-Dual Ratio Estimator as Intercept
auxiliary variable in estimating the finite population mean,β Biometrical J., vol. 41, no. 5, pp. 627β636, 1999.
[16] P. V Sukhatme, B. V Sukhatme, S. Sukhatme, and C. Asok, βSampling theory of surveys with applications.,β 1984.
[11] E. J. Ekpenyong and E. I. Enang, βEfficient Exponential Ratio Estimator for Estimating the Population Mean in Simple Random Sampling.β
[17] M. N. Murthy, βSampling theory and methods.,β Sampl. theory methods., 1967.
[12] R. Tailor, B. Sharma, and J.-M. Kim, βA generalized ratio-cum-product estimator of finite population mean in stratified random sampling,β Commun Korea Stat Soc, vol. 18, no. 1, pp. 111β118, 2011. [13] T. Srivenkataramana, βA dual to ratio estimator in sample surveys,β Biometrika, vol. 67, no. 1, pp. 199β204, 1980. [14] W. G. Cochran, βSampling techniques.,β 1977. [15] P. V Sukhatme and B. V Sukhatme, βSampling theory of surveys with applications,β 1970.
[18] F. Yates, βSampling methods for censuses and surveys.,β Griffin Books Stat., 1960. [19] S. Singh, Advanced Sampling Theory With Applications: How Michael Selected Amy, vol. 2. Springer Science & Business Media, 2003. [20] S. Bandyopadhyay, βImproved ratio and product estimators,β Sankhya Ser. C, vol. 42, no. 2, pp. 45β49, 1980.