Table of Contents
ISRN Probability and Statistics
VolumeΒ 2012, Article IDΒ 657682, 14 pages
http://dx.doi.org/10.5402/2012/657682
Research Article

An Alternative Estimator for Estimating the Finite Population Mean Using Auxiliary Information in Sample Surveys

School of Studies in Statistics, Vikram University, Ujjain 456010, India

Received 28 February 2012; Accepted 29 March 2012

Academic Editors: J.Β Hu and J.Β LΓ³pez-Fidalgo

Copyright Β© 2012 Ramkrishna S. Solanki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper suggests a class of estimators for estimating the finite population mean π‘Œ of the study variable 𝑦 using known population mean 𝑋 of the auxiliary variable π‘₯. Asymptotic expressions of bias and variance of the suggested class of estimators have been obtained. Asymptotic optimum estimator (AOE) in the class is identified along with its variance formula. It has been shown that the proposed class of estimators is more efficient than usual unbiased, usual ratio, usual product, Bahl and Tuteja (1991), and Kadilar and Cingi (2003) estimators under some realistic conditions. An empirical study is carried out to judge the merits of suggested estimator over other competitors practically.

1. Introduction

The literature on survey sampling describes a great variety of techniques for using auxiliary information to obtain more efficient estimators. Ratio, product, and regression methods of estimation are good examples in this context (see Singh [1]). If the correlation between the study variable 𝑦 and the auxiliary variable π‘₯ is positive (high), the ratio method of estimation is quite effective. On the other hand, if this correlation is negative (high) the product method of estimation envisaged by Robson [2] and rediscovered by Murthy [3] can be employed quite effectively. The classical ratio and product estimators are considered to be the most practicable in many situations, but they have the limitation of having at most the same efficiency as that of the linear regression estimator. In the situation where the relation between the study variable 𝑦 and the auxiliary variable π‘₯ is a straight line and passing through the origin, the ratio and product estimators have efficiencies equal to the usual regression estimator. But in many practical situations, the line does not pass through the origin, and in such circumstances the ratio and product estimators do not perform equally well to the regression estimator. Keeping this fact in view, a large number of authors have paid their attention toward the formulation of modified ratio and product estimators, for instance, see Singh [4, 5], Srivastava [6–8], Reddy [9, 10], Gupta [11], Sahai [12], Sahai and Ray [13], Ray and Sahai [14], Srivenkataramana and Tracy [15, 16], Bandyopadhyay [17], Vos [18], Sisodia and Dwivedi [19], Adhvaryu and Gupta [20], Chaubey et al. [21], Mohanty and Sahoo [22], Singh and Shukla [23], Naik and Gupta [24], Bahl and Tuteja [25], Mohanty and Sahoo [26], Upadhyaya and Singh [27], Singh and Tailor [28], Kadilar and Cingi [29], Singh et al. [30], and others.

In this paper, we have envisaged a new class of estimators for population mean of study variable 𝑦 using information on an auxiliary variable π‘₯ which is highly correlated with the study variable and have shown that the suggested class of estimators is more efficient than some existing estimators.

Consider a finite population π‘ˆ=(𝑒1,𝑒2,…,𝑒𝑁) of size 𝑁 from which a sample of size 𝑛 is drawn according to simple random sampling without replacement (SRSWOR). Let 𝑦 and π‘₯ be the sample mean estimators of the population means π‘Œ and 𝑋, respectively, of the study variable 𝑦 and the auxiliary variable π‘₯. Let 𝐢𝑦(=𝑆𝑦/π‘Œ)), 𝐢π‘₯(=𝑆π‘₯/𝑋), 𝜌(=𝑆𝑦π‘₯/𝑆π‘₯𝑆𝑦) (correlation coefficient between the variables 𝑦 and π‘₯) and π‘˜=(πœŒπΆπ‘¦/𝐢π‘₯), where𝑆2𝑦=(π‘βˆ’1)π‘βˆ’1𝑖=1ξ‚€π‘¦π‘–βˆ’π‘Œξ‚2,𝑆2π‘₯=(π‘βˆ’1)π‘βˆ’1𝑖=1ξ‚€π‘₯π‘–βˆ’π‘‹ξ‚2,𝑆π‘₯𝑦=(π‘βˆ’1)π‘βˆ’1𝑖=1ξ‚€π‘₯π‘–βˆ’π‘‹π‘¦ξ‚ξ‚€π‘–βˆ’π‘Œξ‚.(1.1)

The remaining part of the paper is organized as follows. Section 2 gives the brief review of some estimators for the population mean of study variable 𝑦 with its properties. In Section 3, a new class of estimators for the population mean is described, and the expressions for the asymptotic bias and variance are obtained. Asymptotic optimum estimator (AOE) in the suggested class is obtained with its variance formula. Section 4 addresses the problem of efficiency comparisons, while in Section 5 an empirical study is carried out to evaluate the performance of different estimators.

2. Reviewing Estimators

It is very well known that the sample mean 𝑦 is an unbiased estimator of population mean π‘Œ, and under (SRSWOR), its variance is given byξ€·Var𝑦=(1βˆ’π‘“)𝑛𝑆2𝑦=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦,(2.1) where 𝑓=(𝑛/𝑁).

The usual ratio and product estimators of population mean π‘Œ of the study variable 𝑦 are, respectively, defined as𝑦𝑅=𝑦𝑋π‘₯,𝑦𝑃=𝑦π‘₯𝑋.(2.2)

Bahl and Tuteja [25] suggested exponential ratio-type and product-type estimators for population mean π‘Œ, respectively, as𝑦Re=𝑦expπ‘‹βˆ’π‘₯𝑋+π‘₯ξ‚Ά,𝑦Pe=𝑦expπ‘₯βˆ’π‘‹π‘₯+𝑋.(2.3)

Kadilar and Cingi [31] suggested a chain ratio-type estimator for population mean π‘Œ as𝑦CR=𝑦𝑋2π‘₯2ξƒͺ.(2.4)

Following the procedure adopted by Kadilar and Cingi [31], one may define a chain product-type estimator for population mean π‘Œ as𝑦CP=𝑦π‘₯2𝑋2ξƒͺ.(2.5) To the first degree of approximation, the biases and variances of estimators 𝑦𝑅, 𝑦𝑝, 𝑦Re, 𝑦Pe, 𝑦CR, and 𝑦CP are, respectively, given by𝐡𝑦𝑅=(1βˆ’π‘“)π‘›π‘ŒπΆ2π‘₯(𝐡1βˆ’π‘˜),(2.6)𝑦𝑃=(1βˆ’π‘“)π‘›π‘ŒπΆ2π‘₯π΅ξ€·π‘˜,(2.7)𝑦Reξ€Έ=(1βˆ’π‘“)𝑛𝐢2π‘₯ξ‚΅π‘Œ8ξ‚Ά(𝐡3βˆ’4π‘˜),(2.8)𝑦Peξ€Έ=(1βˆ’π‘“)𝑛𝐢2π‘₯ξ‚΅π‘Œ8𝐡(4π‘˜βˆ’1),(2.9)𝑦CRξ€Έ=(1βˆ’π‘“)π‘›π‘ŒπΆ2π‘₯𝐡(3βˆ’2π‘˜),(2.10)𝑦CPξ€Έ=(1βˆ’π‘“)π‘›π‘ŒπΆ2π‘₯ξ€·(1+2π‘˜),(2.11)Var𝑦𝑅=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+𝐢2π‘₯(ξ€»ξ€·1βˆ’2π‘˜),(2.12)Var𝑦𝑃=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+𝐢2π‘₯ξ€»ξ€·(1+2π‘˜),(2.13)Var𝑦Reξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+𝐢2π‘₯4ξ‚Ήξ€·(1βˆ’4π‘˜),(2.14)Var𝑦Peξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+𝐢2π‘₯4(ξ‚Ήξ€·1+4π‘˜),(2.15)Var𝑦CRξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+4𝐢2π‘₯(ξ€»ξ€·1βˆ’π‘˜),(2.16)Var𝑦CPξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+4𝐢2π‘₯ξ€».(1+π‘˜)(2.17) From (2.1) and (2.12)–(2.17), we have made some efficiency comparisons between the estimators 𝑦, 𝑦𝑅, 𝑦𝑝, 𝑦Re, 𝑦Pe, 𝑦CR, and 𝑦CP, as shown in Table 1.

tab1
Table 1: Efficiency comparisons between different estimators.

3. Suggested Class of Estimators

We define the following class of estimators for the population mean π‘Œ as𝑑(𝛼,𝛿)=π‘¦βŽ‘βŽ’βŽ’βŽ£ξ‚΅2βˆ’π‘₯π‘‹ξ‚Άπ›ΌβŽ§βŽͺ⎨βŽͺβŽ©π›Ώξ‚€expπ‘₯βˆ’π‘‹ξ‚ξ‚€π‘₯+π‘‹ξ‚βŽ«βŽͺ⎬βŽͺ⎭⎀βŽ₯βŽ₯⎦,(3.1) where (𝛼,𝛿) are suitable chosen scalars.

It is to be mentioned that for 𝛿=0, the class of estimators 𝑑(𝛼,𝛿) reduces to the following class of estimators𝑑(𝛼,0)=𝑦2βˆ’π‘₯𝑋𝛼,(3.2) which is due to Sahai and Ray [13].

While for 𝛼=0, it reduces to the new transformed class of estimators defined as𝑑(0,𝛿)=π‘¦βŽ‘βŽ’βŽ’βŽ£βŽ§βŽͺ⎨βŽͺβŽ©π›Ώξ‚€2βˆ’expπ‘₯βˆ’π‘‹ξ‚ξ‚€π‘₯+π‘‹ξ‚βŽ«βŽͺ⎬βŽͺ⎭⎀βŽ₯βŽ₯⎦.(3.3)

To obtain the bias and variance of suggested class of estimators 𝑑(𝛼,𝛿), we write𝑦=π‘Œξ€·1+𝑒0ξ€Έ,π‘₯=𝑋1+𝑒1ξ€Έ,(3.4)

such that𝐸𝑒0𝑒=𝐸1𝐸𝑒=0,20ξ€Έ=(1βˆ’π‘“)𝑛𝐢2𝑦,𝐸𝑒21ξ€Έ=(1βˆ’π‘“)𝑛𝐢2π‘₯,𝐸𝑒0𝑒1ξ€Έ=(1βˆ’π‘“)π‘›πœŒπΆπ‘¦πΆπ‘₯=(1βˆ’π‘“)π‘›π‘˜πΆ2π‘₯.(3.5) Expressing (3.1) in terms of 𝑒’s, we have𝑑(𝛼,𝛿)=π‘Œξ€·1+𝑒0ξ€Έξ‚Έξ€·2βˆ’1+𝑒1𝛼exp𝛿𝑒12+𝑒1=ξ‚Όξ‚Ήπ‘Œξ€·1+𝑒0ξ€Έξ‚Έξ€·2βˆ’1+𝑒1𝛼exp𝛿𝑒12𝑒1+12ξ‚βˆ’1.ξ‚Όξ‚Ή(3.6) We assume that |𝑒1|<1, so that (1+𝑒1)𝛼, exp{(𝛿𝑒1/2)(1+𝑒1/2)βˆ’1}, and (1+𝑒1/2)βˆ’1 are expandable. Now expanding the right-hand side of (3.6), we have𝑑(𝛼,𝛿)=π‘Œξ€·1+𝑒0ξ€Έξ‚Έξ‚»2βˆ’1+𝛼𝑒1+𝛼(π›Όβˆ’1)2𝑒21ξ‚ΌΓ—ξƒ―+β‹―1+𝛿𝑒12𝑒1+12ξ‚βˆ’1+𝛿2𝑒218𝑒1+12ξ‚βˆ’2=+β‹―ξƒ°ξƒ­π‘Œξ€·1+𝑒0ξ€Έξ‚Έξ‚»2βˆ’1+𝛼𝑒1+𝛼(π›Όβˆ’1)2𝑒21ξ‚ΌΓ—ξƒ―+β‹―1+𝛿𝑒12𝑒1βˆ’12+𝑒214π›Ώβˆ’β‹―+2𝑒218ξ€·1βˆ’π‘’1ξ€Έ=+β‹―ξƒͺξƒ°ξƒ­π‘Œξ€·1+𝑒0ξ€Έξ‚Έξ‚»2βˆ’1+(2𝛼+𝛿)2𝑒1+(2𝛼+𝛿)(2𝛿+π›Ώβˆ’2)8𝑒21=+β‹―ξ‚Όξ‚Ήπ‘Œξ‚Έξ‚»1+𝑒0βˆ’(2𝛼+𝛿)2𝑒1+𝑒0𝑒1ξ€Έβˆ’(2𝛼+𝛿)(2𝛿+π›Ώβˆ’2)8𝑒21+𝑒0𝑒21ξ€Έ.βˆ’β‹―ξ‚Όξ‚Ή(3.7) Neglecting the terms of 𝑒’s having greater than two in (3.7), we have𝑑(𝛼,𝛿)β‰…π‘Œξ‚Έ1+𝑒0βˆ’(2𝛼+𝛿)2𝑒1+𝑒0𝑒1βˆ’(2𝛿+π›Ώβˆ’2)4𝑒21ξ‚Όξ‚Ή(3.8)

or𝑑(𝛼,𝛿)βˆ’π‘Œξ‚β‰…π‘Œξ‚Έπ‘’0βˆ’(2𝛼+𝛿)2𝑒1+𝑒0𝑒1βˆ’(2𝛿+π›Ώβˆ’2)4𝑒21ξ‚Όξ‚Ή.(3.9) Taking expectation of both sides of (3.9), we get the bias of the class of estimators 𝑑(𝛼,𝛿) to the first degree of approximation as𝐡𝑑(𝛼,𝛿)ξ€Έ=(1βˆ’π‘“)π‘›π‘ŒπΆ2π‘₯ξ‚Έβˆ’(2𝛼+𝛿)2ξ‚»(2𝛿+π›Ώβˆ’2)4+π‘˜ξ‚Όξ‚Ή.(3.10) Squaring both sides of (3.9) and neglecting terms 𝑒’s having power greater than two, we have𝑑(𝛼,𝛿)βˆ’π‘Œξ‚2β‰…π‘Œξ‚Έπ‘’20+(2𝛼+𝛿)2ξ‚»(2𝛿+𝛿)4𝑒21βˆ’2𝑒0𝑒1ξ‚Όξ‚Ή.(3.11) Taking expectation of both sides of (3.11), we get the variance of 𝑑(𝛼,𝛿) to the first degree of approximation as𝑑Var(𝛼,𝛿)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+(2𝛼+𝛿)4𝐢2π‘₯ξ‚Ή{(2𝛼+𝛿)βˆ’4π‘˜},(3.12) which is minimum when(2𝛼+𝛿)=2π‘˜.(3.13) Thus, the resulting minimum variance of 𝑑(𝛼,𝛿) is obtained asVarmin𝑑(𝛼,𝛿)ξ€Έ=(1βˆ’π‘“)𝑛𝑆2𝑦1βˆ’πœŒ2ξ€Έ.(3.14) The minimum variance of 𝑑(𝛼,𝛿) equals to the approximate variance of the usual linear regression estimator defined as𝑦lr=̂𝛽𝑦+π‘‹βˆ’π‘₯,(3.15) where ̂𝛽=(𝑠π‘₯𝑦/𝑠2π‘₯), 𝑠π‘₯𝑦=(π‘›βˆ’1)βˆ’1βˆ‘π‘›π‘–=1(π‘₯π‘–βˆ’π‘₯)(π‘¦π‘–βˆ’π‘¦) and 𝑠2π‘₯=(π‘›βˆ’1)βˆ’1βˆ‘π‘›π‘–=1(π‘₯π‘–βˆ’π‘₯)2.

It is interesting to note that if we set (𝛿=0) and (𝛼=0) in (3.12), we get the variances of the classes of estimators 𝑑(𝛼,0) and 𝑑(0,𝛿), respectively, as𝑑Var(𝛼,0)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+𝛼𝐢2π‘₯(ξ€»,ξ€·π‘‘π›Όβˆ’2π‘˜)Var(0,𝛿)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+𝛿4𝐢2π‘₯ξ‚„.(π›Ώβˆ’4π‘˜)(3.16) If we assume the value of 𝛼 is specified by π›Όπ‘œ (say), then the variances of 𝑑(𝛼,0) and 𝑑(𝛼,𝛿) are, respectively, given by𝑑Var(𝛼=π›Όπ‘œ,0)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+π›Όπ‘œπΆ2π‘₯ξ€·π›Όπ‘œ,ξ€·π‘‘βˆ’2π‘˜ξ€Έξ€»Var(𝛼=π›Όπ‘œ,𝛿)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+ξ€·2π›Όπ‘œξ€Έ+𝛿4𝐢2π‘₯ξ€½ξ€·2π›Όπ‘œξ€Έξ€Ύξƒ­.+π›Ώβˆ’4π‘˜(3.17) If the value of 𝛿 is specified π›Ώπ‘œ, then the variances of the estimators 𝑑(0,𝛿) and 𝑑(𝛼,𝛿) are, respectively, given by𝑑Var(0,𝛿=π›Ώπ‘œ)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+ξ‚΅π›Ώπ‘œ4𝐢2π‘₯ξ€·π›Ώπ‘œξ€Έξ‚Ή,ξ€·π‘‘βˆ’4π‘˜Var(𝛼,𝛿=π›Ώπ‘œ)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+ξ€·2π›Όπ‘œξ€Έ+𝛿4𝐢2π‘₯ξ€½ξ€·2𝛼+π›Ώπ‘œξ€Έξ€Ύξƒ­.βˆ’4π‘˜(3.18) To illustrate our general results, we consider a particular case of the proposed class of estimators 𝑑(𝛼,𝛿) with its properties. If we set (𝛼,𝛿)=(1,1) in (3.1), we get an estimator of population mean π‘Œ as𝑑(1,1)=𝑦2βˆ’π‘₯𝑋expπ‘₯βˆ’π‘‹π‘₯+𝑋.ξ‚Άξ‚Ή(3.19) Putting (𝛼,𝛿)=(1,1) in (3.10) and (3.12), we get the bias and variance of 𝑑(1,1), to the first degree of approximation, respectively, as𝐡𝑑(1,1)ξ€Έ=βˆ’3(1βˆ’π‘“)8π‘›π‘ŒπΆ2π‘₯(𝑑1+4𝐾),(3.20)Var(1,1)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2𝑦+3𝐢2π‘₯4(ξ‚Ή.3βˆ’4π‘˜)(3.21) Expression (3.20) clearly indicates that the proposed estimator 𝑑(1,1) is better than conventional unbiased estimator 𝑦 if π‘˜>(3/4), a condition which is usually met in survey situations.

4. Efficiency Comparisons

From (3.17), we have 𝑑Var(𝛼=π›Όπ‘œ,𝛿)ξ€Έξ€·π‘‘βˆ’Var(𝛼=π›Όπ‘œ,0)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2π‘₯2π›Όπ‘œξ€Έ+𝛿4ξ€½ξ€·2π›Όπ‘œξ€Έξ€Ύ+π›Ώβˆ’4π‘˜βˆ’π›Όπ‘œξ€·π›Όπ‘œξ€Έξƒ­ξ‚ƒξ€·βˆ’2π‘˜<0,if2π›Όπ‘œξ€Έ+𝛿2ξ€·βˆ’42π›Όπ‘œξ€Έ+π›Ώπ‘˜βˆ’4𝛼2π‘œ+8π›Όπ‘œπ‘˜ξ‚„ξ€·<0,i.e.,if𝛿𝛿+4π›Όπ‘œξ€Έξ‚»ξ€·βˆ’4π‘˜<0,i.e.,ifeither0<𝛿<4π‘˜βˆ’π›Όπ‘œξ€Έξ€·or4π‘˜βˆ’π›Όπ‘œξ€Έ<𝛿<0,(4.1) or equivalentlyξ€½ξ€·minβ‹…0,4π‘˜βˆ’π›Όπ‘œξ€½ξ€·ξ€Έξ€Ύ<𝛿<maxβ‹…0,4π‘˜βˆ’π›Όπ‘œξ€Έξ€Ύ.(4.2) From (3.18), we have𝑑Var(𝛼,𝛿=π›Ώπ‘œ)ξ€Έξ€·π‘‘βˆ’Var(0,π›Ώπ‘œ)ξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2π‘₯2𝛼+π›Ώπ‘œξ€Έ24βˆ’ξ€·2𝛼+π›Ώπ‘œξ€Έπ›Ώπ‘˜βˆ’2π‘œ4+π›Ώπ‘œπ‘˜ξ€Ίπ›Όξƒ°ξƒ­<0,if2+π›Όπ›Ώπ‘œξ€»ξ‚»ξ€·βˆ’2π›Όπ‘˜<0,i.e.,ifeither0<𝛼<2π‘˜βˆ’π›Ώπ‘œξ€Έξ€·or2π‘˜βˆ’π›Ώπ‘œξ€Έ<𝛼<0,(4.3) or equivalentlyξ€½ξ€·minβ‹…0,2π‘˜βˆ’π›Ώπ‘œξ€½ξ€·ξ€Έξ€Ύ<𝛼<maxβ‹…0,2π‘˜βˆ’π›Ώπ‘œξ€Έξ€Ύ.(4.4)

From (2.1), (2.12)–(2.17), and (3.12), we have(i)𝑑Var(𝛼,𝛿)ξ€Έξ€·βˆ’Var𝑦=(1βˆ’π‘“)4π‘›π‘Œ2𝐢2π‘₯ξ€Ί(2𝛼+𝛿)2ξ€»ξ‚»βˆ’π‘˜(2𝛼+𝛿)<0,ifeither0<(2𝛼+𝛿)<π‘˜orπ‘˜<(2𝛼+𝛿)<0,(4.5) or equivalentlyminβ‹…(0,π‘˜)<(2𝛼+𝛿)<maxβ‹…(0,π‘˜).(4.6)(ii)𝑑Var(𝛼,𝛿)ξ€Έξ€·βˆ’Var𝑦𝑅=(1βˆ’π‘“)π‘›π‘Œ2𝐢2π‘₯ξ‚Έ(2𝛼+𝛿)4ξ‚Ήξ‚»{(2𝛼+𝛿)βˆ’4π‘˜}βˆ’(1+2π‘˜)<0,ifeither2<(2𝛼+𝛿)<2(2π‘˜βˆ’1)or2(2π‘˜βˆ’1)<(2𝛼+𝛿)<2,(4.7) or equivalentlyminβ‹…{2,2(2π‘˜βˆ’1)}<(2𝛼+𝛿)<maxβ‹…{2,2(2π‘˜βˆ’1)}.(4.8)(iii)𝑑Var(𝛼,𝛿)ξ€Έξ€·βˆ’Var𝑦𝑝=(1βˆ’π‘“)π‘›π‘Œ2𝐢2π‘₯ξ‚Έ(2𝛼+𝛿)4ξ‚Ήξ‚»{(2𝛼+𝛿)βˆ’4π‘˜}βˆ’(1+2π‘˜)<0,ifeitherβˆ’2<πœƒ<2(2π‘˜+1)or2(2π‘˜+1)<πœƒ<βˆ’2,(4.9) or equivalentlyminβ‹…{βˆ’2,2(2π‘˜+1)}<(2𝛼+𝛿)<maxβ‹…{βˆ’2,2(2π‘˜+1)}.(4.10)(iv)𝑑Var(𝛼,𝛿)ξ€Έξ€·βˆ’Var𝑦Reξ€Έ=(1βˆ’π‘“)4π‘›π‘Œ2𝐢2π‘₯[]ξ‚»(2𝛼+𝛿){(2𝛼+𝛿)βˆ’4π‘˜}βˆ’(1βˆ’4π‘˜)<0,ifeither1<(2𝛼+𝛿)<(4π‘˜βˆ’1)or(4π‘˜βˆ’1)<(2𝛼+𝛿)<1,(4.11) or equivalentlyminβ‹…{1,(4π‘˜βˆ’1)}<(2𝛼+𝛿)<maxβ‹…{1,(4π‘˜βˆ’1)}.(4.12)(v)𝑑Var(𝛼,𝛿)ξ€Έξ€·βˆ’Var𝑦Peξ€Έ=(1βˆ’π‘“)4π‘›π‘Œ2𝐢2π‘₯[]ξ‚»(2𝛼+𝛿){(2𝛼+𝛿)βˆ’4π‘˜}βˆ’(1+4π‘˜)<0,ifeitherβˆ’1<(2𝛼+𝛿)<(4π‘˜+1)or(4π‘˜+1)<(2𝛼+𝛿)<βˆ’1,(4.13) or equivalentlyminβ‹…{βˆ’1,(4π‘˜+1)}<(2𝛼+𝛿)<maxβ‹…{βˆ’1,(4π‘˜+1)}.(4.14)(vi)𝑑Var(𝛼,𝛿)ξ€Έξ€·βˆ’Var𝑦CRξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2π‘₯ξ‚Έ(2𝛼+𝛿)4ξ‚Ήξ‚»{(2𝛼+𝛿)βˆ’4π‘˜}βˆ’4(1βˆ’π‘˜)<0,ifeither4<(2𝛼+𝛿)<4(π‘˜βˆ’1)or4(π‘˜βˆ’1)<(2𝛼+𝛿)<4,(4.15) or equivalentlyminβ‹…{4,4(π‘˜βˆ’1)}<(2𝛼+𝛿)<maxβ‹…{4,4(π‘˜βˆ’1)}.(4.16)(vii)𝑑Var(𝛼,𝛿)ξ€Έξ€·βˆ’Var𝑦CPξ€Έ=(1βˆ’π‘“)π‘›π‘Œ2𝐢2π‘₯ξ‚Έ(2𝛼+𝛿)4ξ‚Ήξ‚»{(2𝛼+𝛿)βˆ’4π‘˜}βˆ’4(1+π‘˜)<0,ifeitherβˆ’4<(2𝛼+𝛿)<4(π‘˜+1)or4(π‘˜+1)<(2𝛼+𝛿)<βˆ’4,(4.17) or equivalentlyminβ‹…{βˆ’4,4(π‘˜+1)}<(2𝛼+𝛿)<maxβ‹…{βˆ’4,4(π‘˜+1)}.(4.18) From (2.1), (2.12), (2.14), (2.16), and (3.21), we have made some efficiency comparisons between the estimators 𝑑(1,1), 𝑦, 𝑦𝑅, 𝑦Re, and 𝑦CR, as shown in Table 2.

tab2
Table 2: Efficiency comparisons between different estimators.

It is clearly indicated from Table 2 that the estimator 𝑑(1,1) is more efficient than estimators 𝑦, 𝑦𝑅, 𝑦Re, and 𝑦CR if547<π‘˜<3.(4.19)

5. Empirical Study

To judge the merits of the suggested estimator 𝑑(1,1) over usual unbiased estimator 𝑦, usual ratio estimator 𝑦𝑅, Bahl and Tuteja [25] exponential ratio-type estimator 𝑦Re and Kadilar and Cingi [31] chain ratio-type estimator 𝑦CR we have considered three populations. Descriptions of the populations are given below.

Population I (source: Srivastava [7], page 406)
π‘¦βˆΆ fiber per plant in jute fiber crops,π‘₯∢ height, 𝐢2𝑦=0.0568, 𝐢2π‘₯=0.00846, 𝜌=0.7418, π‘˜=1.9221.

Population II (source: Gupta and Shabbir [32])
π‘¦βˆΆ the level of apple production amount (1 unit = 100 tones),π‘₯∢ the number of apple trees (1 unit = 100 trees), 𝐢2𝑦=17.4724, 𝐢2π‘₯=4.0804, 𝜌=0.82, π‘˜=1.6968.

Population III (Source: Kadilar and Cingi [33])
π‘¦βˆΆ the level of apple production amount,π‘₯∢ the number of apple trees, 𝐢2𝑦=34.1056, 𝐢2π‘₯=14.8225, 𝜌=0.92, π‘˜=1.3955.
We have computed the percent relative efficiencies (PREs) of different estimators (β‹…)𝑦, 𝑦𝑅, 𝑦Re,  𝑦CR, and 𝑑(1,1) of the population mean π‘Œ with respect to 𝑦 by using the following formula:ξ€·PREβ‹…,𝑦=ξ€·Var𝑦Var(β‹…)Γ—100,(5.1) and results are summarized in Table 3.
Table 3 exhibits that the proposed estimator 𝑑(1,1) is more efficient than usual unbiased estimator 𝑦, usual ratio estimator 𝑦𝑅, Bahl and Tuteja [25] estimator 𝑦Re, and Kadilar and Cingi [31] estimator 𝑦CR in the sense of having the largest PRE in all three populations. In populations II and III, the proposed estimator 𝑑(1,1) has the largest gain in efficiency over all the estimators, while in population I, there is marginal gain in efficiency as compared to 𝑦CR. It is further noted that the condition 5/4<π‘˜<7/4 has been satisfied in population I (1.25<1.9221<2.33), II (1.25<1.6968<2.33), and III (1.25<1.3955<2.33). Thus, we recommend the proposed estimator 𝑑(1,1) for its use in practice wherever the condition 5/4<π‘˜<7/4 is satisfied.

tab3
Table 3: PREs of different estimators of population mean 𝑦.

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