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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 usingΒ ...
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 it’s 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 𝑦𝑦�𝑝𝑝 when 1

when βˆ’ ≀ 𝜌𝜌 2

𝐢𝐢𝑦𝑦 𝐢𝐢π‘₯π‘₯

πœŒπœŒπΆπΆπ‘¦π‘¦

𝐢𝐢π‘₯π‘₯ 1

>

1 2

and unbiased estimator 𝑦𝑦�

≀ , where 𝐢𝐢𝑦𝑦 , 𝐢𝐢π‘₯π‘₯ and 𝜌𝜌 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

βˆ‘ 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 βˆ’ x) , where y = βˆ‘ yi , x = βˆ‘ 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 βˆ—

estimator given as

y dR

βˆ— x , where x is the =y X 𝑁𝑁𝑋𝑋� βˆ’π‘›π‘›π‘₯π‘₯Μ…

. un-sampled auxiliary variable in X given as π‘₯π‘₯Μ… βˆ— = π‘π‘βˆ’π‘›π‘› The use of auxiliary information in sample surveys was π‘₯π‘₯Μ… βˆ— 𝑋𝑋�

βˆ— 𝑦𝑦�𝑅𝑅𝑅𝑅 = 𝑦𝑦�

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 𝑦𝑦�, also, π‘₯π‘₯Μ… βˆ— 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 𝑦𝑦�𝑅𝑅𝑅𝑅 with the MSE of 𝑦𝑦� 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 π‘Œπ‘ŒοΏ½, we have proposed a class of difference estimator with dual to Ratio cum as the intercept given as

+ 𝛼𝛼(𝑋𝑋� βˆ’ π‘₯π‘₯Μ… βˆ— )

(1)

βˆ— Where Ξ± is a suitably chosen scalar. The bias and mean square error (MSE) of 𝑦𝑦�𝑅𝑅𝑅𝑅 to the first order approximation is

obtained by substituting π‘₯π‘₯Μ… βˆ— =

𝑁𝑁𝑋𝑋� βˆ’π‘›π‘›π‘₯π‘₯Μ… π‘π‘βˆ’π‘›π‘›

into equation (1), hence, equation (1) becomes, 𝑦𝑦� 𝑁𝑁𝑋𝑋� βˆ’π‘›π‘›π‘₯π‘₯Μ…

βˆ— 𝑦𝑦�𝑅𝑅𝑅𝑅 = οΏ½οΏ½ 𝑋𝑋

π‘π‘βˆ’π‘›π‘›

𝑦𝑦� 𝑁𝑁𝑋𝑋� βˆ’π‘›π‘›π‘₯π‘₯Μ…

βˆ— �𝑦𝑦𝑅𝑅𝑅𝑅 = οΏ½οΏ½

We write,

This implies that

βˆ— 𝑦𝑦�𝑅𝑅𝑅𝑅 =

𝑋𝑋

𝑦𝑦�

π‘π‘βˆ’π‘›π‘›

π‘π‘βˆ’π‘›π‘›

𝑒𝑒0 =

οΏ½

οΏ½ + 𝛼𝛼 �𝑋𝑋� βˆ’

οΏ½ + 𝛼𝛼 οΏ½

𝑁𝑁𝑋𝑋� βˆ’π‘›π‘›π‘₯π‘₯Μ… οΏ½ 𝑋𝑋�

𝑁𝑁𝑋𝑋� βˆ’π‘›π‘›π‘₯π‘₯Μ… π‘π‘βˆ’π‘›π‘›

οΏ½

(π‘π‘βˆ’π‘›π‘›)𝑋𝑋� βˆ’π‘π‘π‘‹π‘‹οΏ½ +𝑛𝑛π‘₯π‘₯Μ…

+ 𝛼𝛼 οΏ½

π‘π‘βˆ’π‘›π‘›

𝑛𝑛π‘₯π‘₯Μ… βˆ’π‘›π‘›π‘‹π‘‹οΏ½ π‘π‘βˆ’π‘›π‘›

𝑦𝑦� βˆ’ π‘Œπ‘ŒοΏ½ π‘₯π‘₯Μ… βˆ’ 𝑋𝑋� π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝑒𝑒1 = οΏ½ π‘Œπ‘Œ 𝑋𝑋�

οΏ½

οΏ½

𝑦𝑦� = π‘Œπ‘ŒοΏ½(1 + 𝑒𝑒0 ) π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž π‘₯π‘₯Μ… = 𝑋𝑋�(1 + 𝑒𝑒1 ), Respectively. Hence, equation (2) becomes, π‘Œπ‘ŒοΏ½(1 + 𝑒𝑒0 ) (𝑁𝑁𝑋𝑋� βˆ’ 𝑛𝑛𝑋𝑋�(1 + 𝑒𝑒1 )) 𝛼𝛼 βˆ— 𝑦𝑦�𝑅𝑅𝑅𝑅 = + (𝑛𝑛𝑋𝑋�(1 + 𝑒𝑒1 ) βˆ’ 𝑛𝑛𝑋𝑋�) οΏ½ 𝑁𝑁 βˆ’ 𝑛𝑛 𝑁𝑁 βˆ’ 𝑛𝑛 𝑋𝑋 π‘Œπ‘ŒοΏ½(1 + 𝑒𝑒0 ) 𝛼𝛼 βˆ— = (𝑁𝑁 βˆ’ 𝑛𝑛(1 + 𝑒𝑒1 )) + (𝑛𝑛𝑒𝑒1 𝑋𝑋�) 𝑦𝑦�𝑅𝑅𝑅𝑅 𝑁𝑁 βˆ’ 𝑛𝑛 𝑁𝑁 βˆ’ 𝑛𝑛

(2)

F. B. Adebola et al.:

44

A Class of Regression Estimator with Cum-Dual Ratio Estimator as Intercept

π‘Œπ‘ŒοΏ½(1 + 𝑒𝑒0 ) 𝛼𝛼 ((𝑁𝑁 βˆ’ 𝑛𝑛) βˆ’ 𝑛𝑛𝑒𝑒1 ) + (𝑛𝑛𝑒𝑒1 𝑋𝑋�) 𝑁𝑁 βˆ’ 𝑛𝑛 𝑁𝑁 βˆ’ 𝑛𝑛 (𝑁𝑁 βˆ’ 𝑛𝑛) βˆ— = π‘Œπ‘ŒοΏ½(1 + 𝑒𝑒0 )( βˆ’ 𝑔𝑔𝑒𝑒1 ) + 𝛼𝛼𝛼𝛼𝑒𝑒1 𝑋𝑋� 𝑦𝑦�𝑅𝑅𝑅𝑅 𝑁𝑁 βˆ’ 𝑛𝑛

βˆ— 𝑦𝑦�𝑅𝑅𝑅𝑅 =

Where 𝑔𝑔 =

𝑛𝑛

π‘π‘βˆ’π‘›π‘›

βˆ— 𝑦𝑦�𝑅𝑅𝑅𝑅 = π‘Œπ‘ŒοΏ½(1 + 𝑒𝑒0 )(1 βˆ’ 𝑔𝑔𝑒𝑒1 ) + 𝛼𝛼𝛼𝛼𝑒𝑒1 𝑋𝑋� βˆ— 𝑦𝑦�𝑅𝑅𝑅𝑅 = π‘Œπ‘ŒοΏ½(1 + 𝑒𝑒0 ) βˆ’ π‘Œπ‘ŒοΏ½π‘”π‘”π‘’π‘’1 (1 + 𝑒𝑒0 ) + 𝛼𝛼𝛼𝛼𝑒𝑒1 𝑋𝑋�

βˆ— = π‘Œπ‘ŒοΏ½ + π‘Œπ‘ŒοΏ½π‘’π‘’0 βˆ’ π‘Œπ‘ŒοΏ½π‘”π‘”π‘’π‘’1 (1 + 𝑒𝑒0 ) + 𝛼𝛼𝛼𝛼𝑒𝑒1 𝑋𝑋� 𝑦𝑦�𝑅𝑅𝑅𝑅 βˆ— (𝑦𝑦�𝑅𝑅𝑅𝑅 βˆ’ π‘Œπ‘ŒοΏ½) = π‘Œπ‘ŒοΏ½π‘’π‘’0 βˆ’ π‘Œπ‘ŒοΏ½π‘”π‘”π‘’π‘’1 (1 + 𝑒𝑒0 ) + 𝛼𝛼𝛼𝛼𝑒𝑒1 𝑋𝑋�

By taking the expectation of equation (3) we have βˆ— βˆ’ π‘Œπ‘ŒοΏ½) = 𝐸𝐸(π‘Œπ‘ŒοΏ½π‘’π‘’0 ) βˆ’ π‘Œπ‘ŒοΏ½π‘”π‘”(𝐸𝐸(𝑒𝑒1 ) + 𝐸𝐸(𝑒𝑒0 𝑒𝑒1 )) + 𝛼𝛼𝛼𝛼𝑋𝑋�𝐸𝐸(𝑒𝑒1 ) 𝐸𝐸(𝑦𝑦�𝑅𝑅𝑅𝑅 But, 𝐸𝐸(𝑒𝑒0 ) =

π‘¦π‘¦οΏ½βˆ’π‘Œπ‘ŒοΏ½ 𝐸𝐸 οΏ½ οΏ½ οΏ½ π‘Œπ‘Œ

=

1 π‘Œπ‘ŒοΏ½

(π‘Œπ‘ŒοΏ½ βˆ’ π‘Œπ‘ŒοΏ½) = 0 ,

𝐸𝐸(𝑒𝑒1 ) = 𝐸𝐸 οΏ½

and

π‘₯π‘₯Μ… βˆ’π‘‹π‘‹οΏ½ οΏ½ 𝑋𝑋�

𝐸𝐸(𝑒𝑒0 𝑒𝑒1 ) = 𝐸𝐸 οΏ½

Hence, equation (4) becomes,

1 𝑋𝑋�

(𝑋𝑋� βˆ’ 𝑋𝑋�) = 0

(4)

,

1 βˆ’ 𝑓𝑓 𝑆𝑆𝑋𝑋𝑋𝑋 π‘₯π‘₯Μ… βˆ’ 𝑋𝑋� 𝑦𝑦� βˆ’ π‘Œπ‘ŒοΏ½ οΏ½οΏ½ οΏ½=οΏ½ οΏ½ 𝑛𝑛 𝑋𝑋� π‘Œπ‘ŒοΏ½ 𝑋𝑋�

βˆ— 𝐸𝐸(𝑦𝑦�𝑅𝑅𝑅𝑅 βˆ’ π‘Œπ‘ŒοΏ½) = βˆ’π‘Œπ‘ŒοΏ½π‘”π‘”

Recall,

=

(3)

(1 βˆ’ 𝑓𝑓) 𝑆𝑆𝑋𝑋𝑋𝑋 𝑁𝑁 βˆ’ 𝑛𝑛 𝑆𝑆𝑋𝑋𝑋𝑋 = βˆ’π‘”π‘” οΏ½ οΏ½ . 𝑛𝑛 𝑁𝑁𝑁𝑁 π‘‹π‘‹οΏ½π‘Œπ‘ŒοΏ½ 𝑋𝑋�

𝑛𝑛 𝑁𝑁 βˆ’ 𝑛𝑛 𝑛𝑛 𝑁𝑁 βˆ’ 𝑛𝑛 𝑆𝑆𝑋𝑋𝑋𝑋 βˆ— 𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡(𝑦𝑦�𝑅𝑅𝑅𝑅 ) = βˆ’οΏ½ οΏ½οΏ½ οΏ½ 𝑁𝑁 βˆ’ 𝑛𝑛 𝑁𝑁𝑁𝑁 𝑋𝑋� 𝑆𝑆𝑋𝑋𝑋𝑋 βˆ— 𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡(𝑦𝑦�𝑅𝑅𝑅𝑅 )=βˆ’ 𝑁𝑁𝑋𝑋� 𝑔𝑔 =

The Mean Square Error of the estimator given as 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑝𝑝𝑝𝑝 ) is obtained by squaring both sides of equation (3) and taking the expectation. We have βˆ— (𝑦𝑦�𝑅𝑅𝑅𝑅 βˆ’ π‘Œπ‘ŒοΏ½)2 = π‘Œπ‘ŒοΏ½ 2 𝑒𝑒02 + π‘Œπ‘ŒοΏ½ 2 𝑔𝑔2 (𝑒𝑒12 + 𝑒𝑒12 𝑒𝑒02 + 2𝑒𝑒0 𝑒𝑒12 ) + 𝛼𝛼 2 𝑔𝑔2 𝑋𝑋� 2 𝑒𝑒12 βˆ’ 2π‘Œπ‘ŒοΏ½ 2 𝑔𝑔(𝑒𝑒0 𝑒𝑒1 + 𝑒𝑒1 𝑒𝑒02 )

+2𝑔𝑔 π›Όπ›Όπ‘Œπ‘ŒοΏ½π‘‹π‘‹οΏ½π‘’π‘’0 𝑒𝑒1 βˆ’ 2 𝛼𝛼𝑔𝑔2 π‘Œπ‘ŒοΏ½π‘‹π‘‹οΏ½(𝑒𝑒12 + 𝑒𝑒12 𝑒𝑒0 )

Ignoring the higher powers of error greater than or equal to 3, we have. βˆ— (𝑦𝑦�𝑅𝑅𝑅𝑅 βˆ’ π‘Œπ‘ŒοΏ½)2 = π‘Œπ‘ŒοΏ½ 2 𝑒𝑒02 + π‘Œπ‘ŒοΏ½ 2 𝑔𝑔2 𝑒𝑒12 + 𝛼𝛼 2 𝑔𝑔2 𝑋𝑋� 2 𝑒𝑒12 βˆ’ 2π‘Œπ‘ŒοΏ½ 2 𝑔𝑔𝑒𝑒0 𝑒𝑒1 + 2𝑔𝑔 π›Όπ›Όπ‘Œπ‘ŒοΏ½π‘‹π‘‹οΏ½π‘’π‘’0 𝑒𝑒1 βˆ’ 2 𝛼𝛼𝑔𝑔2 π‘Œπ‘ŒοΏ½π‘‹π‘‹οΏ½π‘’π‘’12 βˆ— (𝑦𝑦�𝑅𝑅𝑅𝑅 βˆ’ π‘Œπ‘ŒοΏ½)2 = π‘Œπ‘ŒοΏ½ 2 𝑒𝑒02 + 𝑒𝑒12 𝑔𝑔2 (π‘Œπ‘ŒοΏ½ 2 βˆ’ 2 π›Όπ›Όπ‘‹π‘‹οΏ½π‘Œπ‘ŒοΏ½ + 𝛼𝛼 2 𝑋𝑋� 2 ) + 2𝑔𝑔𝑒𝑒0 𝑒𝑒1 ( π›Όπ›Όπ‘Œπ‘ŒοΏ½π‘‹π‘‹οΏ½ βˆ’ π‘Œπ‘ŒοΏ½ 2 ) βˆ— (𝑦𝑦�𝑅𝑅𝑅𝑅 βˆ’ π‘Œπ‘ŒοΏ½)2 = π‘Œπ‘ŒοΏ½ 2 𝑒𝑒02 + 𝑒𝑒12 𝑔𝑔2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 βˆ’ 2𝑔𝑔𝑒𝑒0 𝑒𝑒1 π‘Œπ‘ŒοΏ½(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)

(5)

Take the expectation of (5) we have

βˆ— βˆ’ π‘Œπ‘ŒοΏ½)2 = οΏ½ π‘₯π‘₯π‘₯π‘₯(𝑦𝑦�𝑅𝑅𝑅𝑅 βˆ— ) = οΏ½ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅

Where 𝐢𝐢𝑦𝑦 =

𝑆𝑆𝑦𝑦 , π‘Œπ‘ŒοΏ½

𝜌𝜌𝜌𝜌π‘₯π‘₯ 𝑆𝑆𝑦𝑦 𝑆𝑆π‘₯π‘₯2 1 βˆ’ 𝑓𝑓 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½ οΏ½ �𝑆𝑆𝑦𝑦2 βˆ’ 2π‘”π‘”π‘Œπ‘ŒοΏ½ 𝑛𝑛 π‘‹π‘‹οΏ½π‘Œπ‘ŒοΏ½ 𝑋𝑋�

βˆ— ) = οΏ½ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅

and 𝐢𝐢π‘₯π‘₯ =

𝑆𝑆π‘₯π‘₯ 𝑋𝑋�

.

𝑆𝑆π‘₯π‘₯π‘₯π‘₯ 𝑆𝑆𝑦𝑦2 𝑆𝑆π‘₯π‘₯2 1 βˆ’ 𝑓𝑓 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½ οΏ½ οΏ½π‘Œπ‘ŒοΏ½ 2 2 βˆ’ 2π‘”π‘”π‘Œπ‘ŒοΏ½ 𝑛𝑛 π‘Œπ‘ŒοΏ½ π‘‹π‘‹οΏ½π‘Œπ‘ŒοΏ½ 𝑋𝑋�

1βˆ’π‘“π‘“ 𝑛𝑛

οΏ½ οΏ½π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½

(6)

International Journal of Probability and Statistics 2015, 4(2): 42-50

45

The optimum value of the 𝑀𝑀𝑀𝑀𝑀𝑀�𝑦𝑦�𝑝𝑝𝑝𝑝 οΏ½ is given as

πœ•πœ• 1 βˆ’ 𝑓𝑓 πœ•πœ• βˆ— ) 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅 οΏ½ οΏ½π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½ = οΏ½ πœ•πœ•πœ•πœ• πœ•πœ•πœ•πœ• 𝑛𝑛 πœ•πœ•

πœ•πœ•πœ•πœ•

𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅 ) = οΏ½

Set equation (7) to zero; we have

Where 𝛽𝛽 =

𝑆𝑆π‘₯π‘₯π‘₯π‘₯

and 𝑅𝑅 =

𝑆𝑆π‘₯π‘₯2

1βˆ’π‘“π‘“ 𝑛𝑛

𝑆𝑆 οΏ½ οΏ½2π‘”π‘”π‘Œπ‘ŒοΏ½ οΏ½π‘₯π‘₯π‘₯π‘₯οΏ½ (𝑋𝑋�) βˆ’ 2𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 𝑋𝑋�(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)οΏ½ 𝑋𝑋 π‘Œπ‘Œ

2𝑔𝑔𝑆𝑆π‘₯π‘₯π‘₯π‘₯ βˆ’ 2𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 𝑋𝑋�(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) = 0 π‘Œπ‘ŒοΏ½ 𝛽𝛽 𝛼𝛼 = βˆ’ 𝑋𝑋� 𝑔𝑔 𝛼𝛼 = 𝑅𝑅 βˆ’

π‘Œπ‘ŒοΏ½ 𝑋𝑋�

βˆ— is given as Thus, the resulting OPTIMUM MSE of 𝑦𝑦�𝑝𝑝𝑝𝑝 βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ

MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) = οΏ½

(7)

𝛽𝛽 𝑔𝑔

2 𝑆𝑆π‘₯π‘₯ 𝑆𝑆𝑦𝑦 𝑆𝑆π‘₯π‘₯2 1 βˆ’ 𝑓𝑓 𝛽𝛽 𝛽𝛽 οΏ½ �𝑆𝑆𝑦𝑦2 βˆ’ 2𝑔𝑔𝑔𝑔 οΏ½π‘Œπ‘ŒοΏ½ βˆ’ �𝑅𝑅 βˆ’ οΏ½ 𝑋𝑋�� + 𝑔𝑔2 2 οΏ½π‘Œπ‘ŒοΏ½ βˆ’ �𝑅𝑅 βˆ’ οΏ½ 𝑋𝑋�� οΏ½ 𝑛𝑛 𝑔𝑔 𝑔𝑔 𝑋𝑋� 𝑋𝑋�

By substituting 𝛼𝛼 in equation (6) MSE

Where 𝑅𝑅𝑋𝑋� = π‘Œπ‘ŒοΏ½,

βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ (𝑦𝑦�𝑅𝑅𝑅𝑅 )

2 𝑆𝑆π‘₯π‘₯ 𝑆𝑆𝑦𝑦 𝑆𝑆 2 1 βˆ’ 𝑓𝑓 𝛽𝛽𝑋𝑋� 𝛽𝛽𝑋𝑋� 2 2 π‘₯π‘₯ οΏ½ οΏ½ οΏ½ οΏ½ =οΏ½ οΏ½ �𝑆𝑆𝑦𝑦 βˆ’ 2𝑔𝑔𝑔𝑔 οΏ½π‘Œπ‘Œ βˆ’ 𝑅𝑅𝑋𝑋 + οΏ½ + 𝑔𝑔 2 οΏ½π‘Œπ‘Œ βˆ’ 𝑅𝑅𝑋𝑋 + οΏ½ οΏ½ 𝑛𝑛 𝑔𝑔 𝑔𝑔 𝑋𝑋� 𝑋𝑋� βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ

MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) = οΏ½ βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ

MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) = οΏ½ βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ

MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) = οΏ½ MSE

βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ (𝑦𝑦�𝑅𝑅𝑅𝑅 ) βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ

1 βˆ’ 𝑓𝑓 οΏ½ �𝑆𝑆𝑦𝑦2 βˆ’ 2𝑆𝑆π‘₯π‘₯π‘₯π‘₯ 𝛽𝛽 + 𝑆𝑆π‘₯π‘₯2 𝛽𝛽2 οΏ½ 𝑛𝑛

𝑆𝑆π‘₯π‘₯π‘₯π‘₯ 𝑆𝑆π‘₯π‘₯π‘₯π‘₯ 2 1 βˆ’ 𝑓𝑓 οΏ½ �𝑆𝑆𝑦𝑦2 βˆ’ 2𝑆𝑆π‘₯π‘₯π‘₯π‘₯ οΏ½ 2 οΏ½ + 𝑆𝑆π‘₯π‘₯2 οΏ½ 2 οΏ½ οΏ½ 𝑆𝑆π‘₯π‘₯ 𝑆𝑆π‘₯π‘₯ 𝑛𝑛 2

2

2�𝑆𝑆π‘₯π‘₯π‘₯π‘₯ οΏ½ �𝑆𝑆π‘₯π‘₯π‘₯π‘₯ οΏ½ 1 βˆ’ 𝑓𝑓 =οΏ½ οΏ½ �𝑆𝑆𝑦𝑦2 βˆ’ + οΏ½ 2 𝑆𝑆π‘₯π‘₯ 𝑆𝑆π‘₯π‘₯2 𝑛𝑛

MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) = οΏ½ βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ

1βˆ’π‘“π‘“

2 𝑆𝑆π‘₯π‘₯ 𝑆𝑆𝑦𝑦 𝛽𝛽𝑋𝑋� 𝑆𝑆π‘₯π‘₯2 𝛽𝛽𝑋𝑋� 1 βˆ’ 𝑓𝑓 οΏ½ �𝑆𝑆𝑦𝑦2 βˆ’ 2𝑔𝑔𝑔𝑔 οΏ½ οΏ½ + 𝑔𝑔2 2 οΏ½ οΏ½ οΏ½ 𝑛𝑛 𝑔𝑔 π‘‹π‘‹οΏ½π‘Œπ‘ŒοΏ½ 𝑔𝑔 𝑋𝑋�

2

�𝑆𝑆π‘₯π‘₯π‘₯π‘₯ οΏ½ 1 βˆ’ 𝑓𝑓 οΏ½ �𝑆𝑆𝑦𝑦2 βˆ’ οΏ½ 𝑆𝑆π‘₯π‘₯2 𝑛𝑛

MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) = οΏ½

1βˆ’π‘“π‘“ 𝑛𝑛

οΏ½ 𝑆𝑆𝑦𝑦2 οΏ½1 βˆ’ οΏ½

𝑆𝑆π‘₯π‘₯π‘₯π‘₯

𝑆𝑆π‘₯π‘₯ 𝑆𝑆𝑦𝑦

2

οΏ½ οΏ½

(7)

βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ βˆ—π‘œπ‘œπ‘œπ‘œπ‘œπ‘œ MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) = οΏ½ οΏ½ 𝑆𝑆𝑦𝑦2 (1 βˆ’ 𝜌𝜌2 ) Equation (7) shows that the MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) is the same as the MSE Regression 𝑛𝑛 estimator. Remark βˆ— βˆ— ) The Bias of 𝑦𝑦�𝑅𝑅𝑅𝑅 is the same as Bias of the dual ratio estimator π‘¦π‘¦οΏ½π‘…π‘…βˆ— and when Ξ± = 0, 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅 becomes 𝑀𝑀𝑀𝑀𝑀𝑀(π‘¦π‘¦οΏ½π‘…π‘…βˆ— ) of βˆ— dual to ratio estimator 𝑦𝑦�𝑅𝑅 proposed by [13]. The bias of π‘¦π‘¦οΏ½π‘…π‘…βˆ— is given as 𝑆𝑆π‘₯π‘₯π‘₯π‘₯ 𝐡𝐡𝐡𝐡𝐡𝐡𝐡𝐡(π‘¦π‘¦οΏ½π‘…π‘…βˆ— ) = βˆ’ 𝑁𝑁𝑋𝑋� The MSE (π‘¦π‘¦οΏ½π‘…π‘…βˆ— ) is given as

𝑀𝑀𝑀𝑀𝑀𝑀(π‘¦π‘¦οΏ½π‘…π‘…βˆ— ) = οΏ½

1βˆ’π‘“π‘“ 𝑛𝑛

οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2πœŒπœŒπœŒπœŒπ‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½π‘”π‘”2 𝐢𝐢π‘₯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 𝑦𝑦�𝑅𝑅𝑅𝑅 with the MSE of 𝑦𝑦� under Simple Random Sampling Scheme given as, MSE(𝑦𝑦�) = οΏ½

1βˆ’π‘“π‘“ 𝑛𝑛

οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2

(8)

βˆ— ) From equations (5) and (8), the proposed estimator is better than that the usual estimator 𝑦𝑦� if, MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 < MSE (𝑦𝑦�). That is, 1 βˆ’ 𝑓𝑓 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½ < οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 οΏ½ 𝑛𝑛 𝑛𝑛 βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 < 0

(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) οΏ½βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)οΏ½ < 0

This holds if and only if, Case (1) π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋� < 0 and βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) > 0 Or

Case (2) π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋� > 0 and βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) < 0

βˆ— is more efficient than 𝑦𝑦� is given as, The Range of Ξ± under which the proposed estimator 𝑦𝑦�𝑝𝑝𝑝𝑝 2πœŒπœŒπΆπΆπ‘¦π‘¦ 2πœŒπœŒπΆπΆπ‘¦π‘¦ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š �𝑅𝑅, 𝑅𝑅 οΏ½1 βˆ’ οΏ½οΏ½ , π‘šπ‘šπ‘šπ‘šπ‘šπ‘š �𝑅𝑅, 𝑅𝑅 οΏ½1 βˆ’ οΏ½οΏ½. 𝑔𝑔𝐢𝐢π‘₯π‘₯ 𝑔𝑔𝐢𝐢π‘₯π‘₯ βˆ— with the usual ratio estimator 𝑦𝑦�𝑅𝑅 . The MSE of the 𝑦𝑦�𝑅𝑅 is given as We also compared the proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2πœŒπœŒπ‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 οΏ½ 𝑀𝑀𝑀𝑀𝑀𝑀( 𝑦𝑦�𝑅𝑅 ) = οΏ½ 𝑛𝑛 βˆ— βˆ— It is found that the proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅 will be more efficient than the usual ratio estimator 𝑦𝑦�𝑅𝑅 if MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) < MSE (𝑦𝑦�𝑅𝑅 ). That is, 1 βˆ’ 𝑓𝑓 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½ ≀ οΏ½ οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2πœŒπœŒπ‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 οΏ½ οΏ½ 𝑛𝑛 𝑛𝑛 βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 2πœŒπœŒπ‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 βˆ’ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 < 0 βˆ’2πœŒπœŒπ‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½) + 𝐢𝐢π‘₯π‘₯2 ((π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 βˆ’ π‘Œπ‘ŒοΏ½ 2 ) < 0

(𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½) οΏ½βˆ’2πœŒπœŒπ‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝐢𝐢π‘₯π‘₯2 (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½)οΏ½ < 0

This holds if the following two conditions are satisfied (1). (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½) < 0 And βˆ’2πœŒπœŒπ‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝐢𝐢π‘₯π‘₯2 (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½) > 0. Or

(2). (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½) > 0 And βˆ’2πœŒπœŒπ‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝐢𝐢π‘₯π‘₯2 (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½) < 0. 2π‘›π‘›βˆ’π‘π‘

This condition holds if 𝛼𝛼 > 𝑅𝑅 οΏ½

𝑛𝑛

𝑁𝑁

οΏ½ and 𝛼𝛼 < 𝑅𝑅 οΏ½ βˆ’ 𝑛𝑛

2πœŒπœŒπΆπΆπ‘¦π‘¦ 𝑔𝑔𝐢𝐢π‘₯π‘₯

οΏ½ or 𝛼𝛼 < 𝑅𝑅 οΏ½

2π‘›π‘›βˆ’π‘π‘ 𝑛𝑛

𝑁𝑁

οΏ½ and 𝛼𝛼 > 𝑅𝑅 οΏ½ βˆ’ 𝑛𝑛

2πœŒπœŒπΆπΆπ‘¦π‘¦ 𝑔𝑔𝐢𝐢π‘₯π‘₯

οΏ½

𝑁𝑁 2πœŒπœŒπΆπΆπ‘¦π‘¦ 2𝑛𝑛 βˆ’ 𝑁𝑁 𝑁𝑁 2πœŒπœŒπΆπΆπ‘¦π‘¦ 2𝑛𝑛 βˆ’ 𝑁𝑁 οΏ½ , 𝑅𝑅 οΏ½ βˆ’ οΏ½ οΏ½ , max �𝑅𝑅 οΏ½ οΏ½ , 𝑅𝑅 οΏ½ βˆ’ οΏ½ οΏ½. 𝑔𝑔𝐢𝐢π‘₯π‘₯ 𝑔𝑔𝐢𝐢π‘₯π‘₯ 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 βˆ— with the usual product estimator 𝑦𝑦�𝑃𝑃 . The MSE of the 𝑦𝑦�𝑃𝑃 is given as We also compared the proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 + 2πœŒπœŒπ‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 οΏ½ 𝑀𝑀𝑀𝑀𝑀𝑀( 𝑦𝑦�𝑅𝑅 ) = οΏ½ 𝑛𝑛 βˆ— βˆ— It is found that the proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅 will be more efficient than the usual ratio estimator 𝑦𝑦�𝑃𝑃 if MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) < MSE (𝑦𝑦�𝑃𝑃 ). That is, 1 βˆ’ 𝑓𝑓 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½ ≀ οΏ½ οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 + 2πœŒπœŒπ‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 οΏ½ οΏ½ 𝑛𝑛 𝑛𝑛 βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ 2πœŒπœŒπ‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 βˆ’ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 < 0 βˆ’2πœŒπœŒπ‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½) + 𝐢𝐢π‘₯π‘₯2 ((π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 βˆ’ π‘Œπ‘ŒοΏ½ 2 ) < 0 min �𝑅𝑅 οΏ½

International Journal of Probability and Statistics 2015, 4(2): 42-50

47

(𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½) οΏ½βˆ’2πœŒπœŒπ‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝐢𝐢π‘₯π‘₯2 (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½)οΏ½ < 0

This holds if the following two conditions are satisfied

(1). (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½) < 0 and βˆ’2πœŒπœŒπ‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝐢𝐢π‘₯π‘₯2 (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½) > 0. Or

(2). (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½) > 0 and βˆ’2πœŒπœŒπ‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝐢𝐢π‘₯π‘₯2 (𝑔𝑔(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½) < 0. 𝑁𝑁

This condition holds if 𝛼𝛼 > 𝑅𝑅 οΏ½ οΏ½ and 𝛼𝛼 < 𝑅𝑅 οΏ½ 𝑛𝑛

2π‘›π‘›βˆ’π‘π‘ 𝑛𝑛

βˆ’

2πœŒπœŒπΆπΆπ‘¦π‘¦ 𝑔𝑔𝐢𝐢π‘₯π‘₯

𝑁𝑁

οΏ½ or 𝛼𝛼 < 𝑅𝑅 οΏ½ οΏ½ and 𝛼𝛼 > 𝑅𝑅 οΏ½ 𝑛𝑛

2π‘›π‘›βˆ’π‘π‘ 𝑛𝑛

𝑁𝑁 2𝑛𝑛 βˆ’ 𝑁𝑁 2πœŒπœŒπΆπΆπ‘¦π‘¦ 𝑁𝑁 2𝑛𝑛 βˆ’ 𝑁𝑁 2πœŒπœŒπΆπΆπ‘¦π‘¦ min �𝑅𝑅 οΏ½ οΏ½ , 𝑅𝑅 οΏ½ βˆ’ οΏ½ οΏ½ , max �𝑅𝑅 οΏ½ οΏ½ , 𝑅𝑅 οΏ½ βˆ’ οΏ½ οΏ½. 𝑔𝑔𝐢𝐢π‘₯π‘₯ 𝑔𝑔𝐢𝐢π‘₯π‘₯ 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛

βˆ’

2πœŒπœŒπΆπΆπ‘¦π‘¦ 𝑔𝑔𝐢𝐢π‘₯π‘₯

οΏ½

We also compared the MSE of the proposed estimator with MSE of dual product estimator π‘¦π‘¦οΏ½π‘π‘βˆ— proposed by [20]. The MSE proposed by [20] (1980) is given as

(π‘¦π‘¦οΏ½π‘π‘βˆ— )

𝑀𝑀𝑀𝑀𝑀𝑀(π‘¦π‘¦οΏ½π‘π‘βˆ— ) = οΏ½

1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 + 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½ 2 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 οΏ½ 𝑛𝑛

βˆ— βˆ— It is found that the proposed estimator 𝑦𝑦�𝑝𝑝𝑝𝑝 will be more efficient than that of [20] estimator π‘¦π‘¦οΏ½π‘π‘βˆ— if MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) < MSE (π‘¦π‘¦οΏ½π‘π‘βˆ— ) That is

οΏ½

1βˆ’π‘“π‘“ 𝑛𝑛

1βˆ’π‘“π‘“ οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½ ≀ οΏ½ οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 + 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½ 2 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 οΏ½ 𝑛𝑛

That is,

βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 < 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½ 2 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2

βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 [(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½] + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 [(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�))2 βˆ’ π‘Œπ‘ŒοΏ½ 2 ] < 0 οΏ½(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½οΏ½ οΏ½βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 [(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½]οΏ½ < 0 (2π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) οΏ½βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 βˆ’ 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 [ 𝛼𝛼𝑋𝑋�]οΏ½ < 0

This holds if, 1. 2π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋� < 0 and βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 βˆ’ 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 [ 𝛼𝛼𝑋𝑋�] > 0 2. 2π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋� > 0 and βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 βˆ’ 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 [ 𝛼𝛼𝑋𝑋�] < 0 This condition holds if 2𝑅𝑅 > 𝛼𝛼 and

βˆ’2𝑅𝑅𝑅𝑅 𝐢𝐢𝑦𝑦 𝑔𝑔𝐢𝐢π‘₯π‘₯

< 𝛼𝛼 or 2𝑅𝑅 < 𝛼𝛼 and

The range of Ξ± under which the proposed estimator min οΏ½2𝑅𝑅,

βˆ— 𝑦𝑦�𝑅𝑅𝑅𝑅

βˆ’2𝑅𝑅𝑅𝑅 𝐢𝐢𝑦𝑦 𝑔𝑔𝐢𝐢π‘₯π‘₯

> 𝛼𝛼

iS more efficient than π‘¦π‘¦οΏ½π‘π‘βˆ— is

βˆ’2𝑅𝑅𝑅𝑅𝐢𝐢𝑦𝑦 βˆ’2𝑅𝑅𝑅𝑅𝐢𝐢𝑦𝑦 οΏ½ , max οΏ½2𝑅𝑅, οΏ½ 𝑔𝑔𝐢𝐢π‘₯π‘₯ 𝑔𝑔𝐢𝐢π‘₯π‘₯

βˆ— Lastly, we compared MSE of the proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅 with that of dual to ratio estimator π‘¦π‘¦οΏ½π‘…π‘…βˆ— proposed [13] given in equation (7a). The proposed estimator will be more efficient than π‘¦π‘¦οΏ½π‘…π‘…βˆ— if βˆ— MSE (𝑦𝑦�𝑅𝑅𝑅𝑅 ) < MSE (π‘¦π‘¦οΏ½π‘…π‘…βˆ— ). That is, 1 βˆ’ 𝑓𝑓 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 οΏ½ ≀ οΏ½ οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢𝑦𝑦2 βˆ’ 2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯ 𝐢𝐢𝑦𝑦 + π‘Œπ‘ŒοΏ½ 2 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 οΏ½ οΏ½ 𝑛𝑛 𝑛𝑛 βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 [(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½] + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 [(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)2 βˆ’ π‘Œπ‘ŒοΏ½ 2 ] < 0

οΏ½(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) βˆ’ π‘Œπ‘ŒοΏ½οΏ½ οΏ½βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 [(π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) + π‘Œπ‘ŒοΏ½]οΏ½ < 0 βˆ’ π›Όπ›Όπ‘‹π‘‹οΏ½οΏ½βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (2π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�)οΏ½ < 0

This holds if 1. βˆ’ 𝛼𝛼𝑋𝑋� < 0 and βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (2π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) < 0 Or

2. βˆ’ 𝛼𝛼𝑋𝑋� > 0 and βˆ’2π‘”π‘”π‘”π‘”π‘Œπ‘ŒοΏ½πΆπΆπ‘₯π‘₯ 𝐢𝐢𝑦𝑦 + 𝑔𝑔2 𝐢𝐢π‘₯π‘₯2 (2π‘Œπ‘ŒοΏ½ βˆ’ 𝛼𝛼𝑋𝑋�) > 0

48

F. B. Adebola et al.:

A Class of Regression Estimator with Cum-Dual Ratio Estimator as Intercept

This condition holds if 𝛼𝛼 > 0 π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝛼𝛼 < 2𝑅𝑅 οΏ½1 βˆ’

πœŒπœŒπΆπΆπ‘¦π‘¦

𝑔𝑔𝐢𝐢π‘₯π‘₯

οΏ½ π‘œπ‘œπ‘œπ‘œ 𝛼𝛼 < 0 π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝛼𝛼 > 2𝑅𝑅 οΏ½1 βˆ’

πœŒπœŒπΆπΆπ‘¦π‘¦

𝑔𝑔𝐢𝐢π‘₯π‘₯

οΏ½

βˆ— is more efficient than dual ratio estimator Therefore, the range of Ξ± under which the proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅

π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½0,2𝑅𝑅 οΏ½1 βˆ’

πœŒπœŒπΆπΆπ‘¦π‘¦ πœŒπœŒπΆπΆπ‘¦π‘¦ οΏ½οΏ½ , π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½0,2𝑅𝑅 οΏ½1 βˆ’ οΏ½οΏ½ 𝑔𝑔𝐢𝐢π‘₯π‘₯ 𝑔𝑔𝐢𝐢π‘₯π‘₯

βˆ— Thus it seems from the above that the proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅 may be made better than the usual estimator, ratio estimator, product estimator, dual to product estimator π‘¦π‘¦οΏ½π‘π‘βˆ— and the dual to ratio estimator π‘¦π‘¦οΏ½π‘…π‘…βˆ— , if the given conditions are satisfied.

οΏ½βˆ—π‘Άπ‘Άπ‘Άπ‘Άπ‘Άπ‘Ά Comparison of β€˜AOE’ to π’šπ’š 𝑹𝑹𝑹𝑹

βˆ—π‘‚π‘‚π‘‚π‘‚π‘‚π‘‚ is more efficient than the other existing estimators 𝑦𝑦�, the ratio estimator 𝑦𝑦�𝑅𝑅 , the product estimator 𝑦𝑦�𝑝𝑝 , the dual to 𝑦𝑦�𝑅𝑅𝑅𝑅 ratio estimator π‘¦π‘¦οΏ½π‘…π‘…βˆ— and the dual to product estimator π‘¦π‘¦οΏ½π‘π‘βˆ— since 1 βˆ’ 𝑓𝑓 βˆ—π‘‚π‘‚π‘‚π‘‚π‘‚π‘‚ ) οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝜌𝜌2 𝐢𝐢𝑦𝑦2 οΏ½ > 0 =οΏ½ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�) βˆ’ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅 𝑛𝑛 πœŒπœŒπΆπΆπ‘¦π‘¦ 2 1 βˆ’ 𝑓𝑓 βˆ—π‘‚π‘‚π‘‚π‘‚π‘‚π‘‚ ) 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅 ) βˆ’ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 οΏ½1 βˆ’ οΏ½ οΏ½>0 =οΏ½ 𝐢𝐢π‘₯π‘₯ 𝑛𝑛 βˆ—π‘‚π‘‚π‘‚π‘‚π‘‚π‘‚ ) =οΏ½ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑃𝑃 ) βˆ’ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅

πœŒπœŒπΆπΆπ‘¦π‘¦ 2 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 οΏ½1 + οΏ½ οΏ½>0 𝐢𝐢π‘₯π‘₯ 𝑛𝑛

βˆ—π‘‚π‘‚π‘‚π‘‚π‘‚π‘‚ ) =οΏ½ 𝑀𝑀𝑀𝑀𝑀𝑀(π‘¦π‘¦οΏ½π‘…π‘…βˆ— ) βˆ’ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅

βˆ—π‘‚π‘‚π‘‚π‘‚π‘‚π‘‚ ) =οΏ½ π‘€π‘€π‘€π‘€π‘€π‘€οΏ½π‘¦π‘¦οΏ½π‘π‘βˆ— οΏ½ βˆ’ 𝑀𝑀𝑀𝑀𝑀𝑀(𝑦𝑦�𝑅𝑅𝑅𝑅

2 πœŒπœŒπΆπΆπ‘¦π‘¦ 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 οΏ½ βˆ’ 𝑔𝑔� οΏ½ > 0 𝐢𝐢π‘₯π‘₯ 𝑛𝑛

2 πœŒπœŒπΆπΆπ‘¦π‘¦ 1 βˆ’ 𝑓𝑓 οΏ½ οΏ½ π‘Œπ‘ŒοΏ½ 2 𝐢𝐢π‘₯π‘₯2 οΏ½ + 𝑔𝑔� οΏ½ > 0 𝐢𝐢π‘₯π‘₯ 𝑛𝑛

βˆ— is more efficient than other estimator in case of its optimality. Hence, we conclude that the proposed class of estimator 𝑦𝑦�𝑅𝑅𝑅𝑅

4. Numerical Validation

To illustrate the efficiency of the proposed estimator over the other estimators 𝑦𝑦�, 𝑦𝑦�𝑅𝑅 , 𝑦𝑦�𝑝𝑝 , π‘¦π‘¦οΏ½π‘…π‘…βˆ— π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž π‘¦π‘¦οΏ½π‘π‘βˆ— . 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

𝑦𝑦�

6.08

5.930386531

6.229613469

0.299226939

𝑦𝑦�𝑅𝑅

6.210472103

6.042844235

6.378099971

0.335255737

𝑦𝑦�𝑝𝑝

5.952268908

5.778821411

6.125716404

0.346894993

π‘¦π‘¦οΏ½π‘…π‘…βˆ—

6.141606636

5.988421023

6.294792249

0.306371226

π‘¦π‘¦οΏ½π‘π‘βˆ—

6.019011342

5.862732122

6.175290562

0.31255844

οΏ½βˆ—π‘Ήπ‘Ήπ‘Ήπ‘Ή π’šπ’š

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

οΏ½βˆ—π‘Ήπ‘Ήπ‘Ήπ‘Ή π’šπ’š

6.194882566

6.050452623

6.33931251

0.288859888

6.089652737

5.945222793

6.234082681

0.288859888

οΏ½βˆ—π‘Ήπ‘Ήπ‘Ήπ‘Ήπ‘Ήπ‘Ή π’šπ’š

βˆ— To examine the gain in the efficiency of the proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅 over the estimator 𝑦𝑦�, 𝑦𝑦�𝑅𝑅 , 𝑦𝑦�𝑝𝑝 , π‘¦π‘¦οΏ½π‘…π‘…βˆ— π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝑦𝑦�𝑝𝑝,βˆ— we obtained οΏ½ the percentage relative efficiency of different estimator of π‘Œπ‘Œ with respect to the usual unbiased estmator 𝑦𝑦� in Table 2. The βˆ— proposed estimator 𝑦𝑦�𝑅𝑅𝑅𝑅 performed better than the other estimators 𝑦𝑦�, 𝑦𝑦�𝑅𝑅 , 𝑦𝑦�𝑝𝑝 , π‘¦π‘¦οΏ½π‘…π‘…βˆ— π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž π‘¦π‘¦οΏ½π‘π‘βˆ— and perfoirmed exactly the same way as regression estimator.

Table 3. The percentage relative efficiency of different estimator of π‘Œπ‘ŒοΏ½ with respect to the usual unbiased estimator 𝑦𝑦� ESTIMATOR

PERCENATGE RELATIVE FFICIENCY

𝑦𝑦�

100

𝑦𝑦�𝑅𝑅

79.66158486

π‘¦π‘¦οΏ½π‘…π‘…βˆ—

95.39056726

οΏ½βˆ—π‘Ήπ‘Ήπ‘Ήπ‘Ήπ‘Ήπ‘Ή π’šπ’š

107.3067159

, 𝑦𝑦�𝑝𝑝

74.40554745

π‘¦π‘¦οΏ½π‘π‘βˆ—

91.65136111

οΏ½βˆ—π‘Ήπ‘Ήπ‘Ήπ‘Ή π’šπ’š

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 β€˜AOE’ for the proposed estimator. Theoretically, we have demonstrated that proposed estimator is always more efficient than other under the effective ranges of 𝛼𝛼 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 π‘Œπ‘ŒοΏ½ with respects to the usual unbiased estimator 𝑦𝑦� in Table 2. shows that the proposed estimator βˆ— performed better than the other estimators 𝑦𝑦�𝑅𝑅𝑅𝑅 𝑦𝑦�, 𝑦𝑦�𝑅𝑅 , 𝑦𝑦�𝑝𝑝 , π‘¦π‘¦οΏ½π‘…π‘…βˆ— π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž π‘¦π‘¦οΏ½π‘π‘βˆ— and performed exactly the same way as regression estimator. Hence, it is preferred to use the proposed class of estimator in practice.

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