Some Improved Ratio, Product, and Regression Estimators of Finite Population Mean When Using Minimum and Maximum Values

Abstract

Efficient estimation of finite population mean is carried out by using the auxiliary information meaningfully. In this paper we have suggested some modified ratio, product, and regression type estimators when using minimum and maximum values. Expressions for biases and mean squared errors of the suggested estimators have been derived up to the first order of approximation. The performances of the suggested estimators, relative to their usual counterparts, have been studied, and improved performance has been established. The improvement in efficiency by making use of maximum and minimum values has been verified numerically.

1. Introduction

Supplementary information in form of the auxiliary variable is rigorously used for the estimation of finite population mean for the study variable. Ratio and product estimators due to Cochran [1] and Murthy [2], respectively, are good examples when information on the auxiliary variable is incorporated for improved estimation of finite population mean of the study variable. When correlation between the study variable (𝑦) and the auxiliary variable (𝑥) is positive, ratio method of estimation is effective and when correlation is negative, product method of estimation is used. There are a lot of improvements and advancements in the construction of ratio, product, and regression estimators using the auxiliary information. For recent details, see Haq et al. [3], Haq and Shabbir [4], Yadav and Kadilar [5], Kadilar and Cingi [6], and Koyuncu and Kadilar [7] and the references cited therein.

The ratio method of estimation is at its best when the relationship between 𝑦 and 𝑥 is linear and the line of regression passes through the origin but as the line departs from origin, the efficiency of this method decreases. In practice, the condition that the line of regression passes through the origin is rarely satisfied and regression estimator is used for estimation of population mean.

Let 𝑈=(𝑈₁,𝑈₂,…,𝑈_𝑁) be a population of size 𝑁. Let (𝑦_𝑖,𝑥_𝑖) be the values of the study and the auxiliary variables, respectively, on the 𝑖th unit of a finite population.

Let us assume that a simple random sample of size 𝑛 is drawn without replacement from 𝑈 for estimating the population mean 𝑌=∑^𝑁_𝑖=1𝑦_𝑖/𝑁. It is further assumed that the population mean 𝑋=∑^𝑁_𝑖=1𝑥_𝑖/𝑁 of the auxiliary variable 𝑥 is known. The minimum say (𝑥_min) and maximum say (𝑥_max) values of the auxiliary variables are also assumed to be known.

The variance of mean per unit estimator 𝑦=∑^𝑛_𝑖=1𝑦_𝑖/𝑛 is given by 𝑉𝑦=𝜆𝑆²_𝑦, (1) where 𝜆=((1/𝑛)−(1/𝑁)) and 𝑆²_𝑦=(1/(𝑁−1))∑^𝑁_𝑖=1(𝑦_𝑖−𝑌)².

Some time there exists unusually very large (say 𝑦_max) and very small (say 𝑦_min) units in the population. The mean per unit estimator is very sensitive to these unusual observations and as a result population mean will be either underestimated (in case the sample contains 𝑦_min) or overestimated (in case the sample contains 𝑦_max). To overcome the situation Sarndal [8] suggested the following unbiased estimator: 𝑦_𝑠=𝑦+𝑐ifsamplecontains𝑦_minbutnot𝑦_max,𝑦−𝑐ifsamplecontains𝑦_maxbutnot𝑦_min,𝑦forallothersamples, (2) where 𝑐 is a constant.

The variance of 𝑦_𝑠 is given by 𝑉𝑦_𝑠=𝜆𝑆²_𝑦−2𝜆𝑛𝑐𝑦𝑁−1^max−𝑦_min−𝑛𝑐. (3)

Further, 𝑉(𝑦_𝑠)<𝑉(𝑦) if 0<𝑐<(𝑦_max−𝑦_min)/𝑛.

For, 𝑐_opt=(𝑦_max−𝑦_min)/2𝑛, variance of 𝑦_𝑠 is given by 𝑉𝑦_𝑠opt=𝑉𝑦−𝜆𝑦_max−𝑦_min²2(),𝑁−1 (4) which is always smaller than 𝑉(𝑦).

The usual ratio and product estimators of population mean (𝑌) are given by 𝑦_𝑅=𝑦𝑋𝑥,𝑦_𝑃=𝑦𝑥𝑋, (5) where 𝑦=∑^𝑛_𝑖=1𝑦_𝑖/𝑛 and 𝑥=∑^𝑛_𝑖=1𝑥_𝑖/𝑛 are the sample means of variables 𝑦 and 𝑥, respectively.

The expressions for biases (𝐵(⋅)), and mean square errors (𝑀(⋅)), of the conventional ratio and product estimators, are given by 𝐵𝑦_𝑅≅𝑌𝜆𝐶²_𝑥−𝜌_𝑦𝑥𝐶_𝑦𝐶_𝑥,𝐵𝑦_𝑃=𝑌𝜆𝜌_𝑦𝑥𝐶_𝑦𝐶_𝑥,𝑀𝑦_𝑅≅𝑌²𝜆𝐶²_𝑦+𝐶²_𝑥−2𝜌_𝑦𝑥𝐶_𝑦𝐶_𝑥,𝑀𝑦_𝑝≅𝑌²𝜆𝐶²_𝑦+𝐶²_𝑥+2𝜌_𝑦𝑥𝐶_𝑦𝐶_𝑥, (6)(7)(8)(9) where 𝐶_𝑦=𝑆_𝑦/𝑌 and 𝐶_𝑥=𝑆_𝑥/𝑋 are the coefficients of variation of 𝑦 and 𝑥, respectively, 𝜌_𝑦𝑥=𝑆_𝑦𝑥/𝑆_𝑦𝑆_𝑥 is the correlation coefficient between 𝑦 and 𝑥, 𝑆²_𝑥=∑^𝑁_𝑖=1(𝑥_𝑖−𝑋)²/(𝑁−1), and 𝑆_𝑦𝑥=∑^𝑁_𝑖=1(𝑦_𝑖−𝑌)(𝑥_𝑖−𝑋)/(𝑁−1) are the population variance and population covariance, respectively.

Usual regression estimator is given by 𝑦_𝑙𝑟=𝑦+𝑏𝑋−𝑥, (10) where 𝑏 is the sample regression coefficient.

The variance of the estimator 𝑦_𝑙𝑟 is given by 𝑉𝑦_𝑙𝑟=𝜆𝑆²_𝑦1−𝜌²_𝑦𝑥. (11)

2. Proposed Estimators

Motivated by Sarndal [8], we extend this idea to estimators which make use of the auxiliary information for increased precision. It is well known that ratio and product estimators are used when 𝑦 and 𝑥 are positively and negatively correlated, respectively. We suggest estimator for each case separately as follows.

Case 1 (positive correlation between 𝑦 and 𝑥). When 𝑦 and 𝑥 are positively correlated, then with selection of a larger value of 𝑥, a larger value of 𝑦 is expected to be selected and when smaller value of 𝑥 is selected, selection of a smaller value of 𝑦 is expected. So we define the following estimators: 𝑌_𝑅𝐶=𝑦_𝑐11𝑥_𝑐21𝑋=𝑦+𝑐₁𝑥+𝑐₂𝑋,𝑦−𝑐₁𝑥−𝑐₂𝑋,𝑦𝑥𝑋 (12) and similarly 𝑦_{𝑙𝑟𝐶1}=𝑦_𝑐11+𝑏𝑋−𝑥_𝑐21, (13) where (𝑦_𝑐11=𝑦+𝑐₁,𝑥_𝑐21=𝑥+𝑐₂) if the sample contains 𝑦_min and 𝑥_min; (𝑦_𝑐11=𝑦−𝑐₁,𝑥_𝑐21=𝑥−𝑐₂) if the sample contains 𝑦_max and 𝑥_max, and (𝑦_𝑐11=𝑦,𝑥_𝑐21=𝑥) for all other combinations of samples.

Case 2 (negative correlation between 𝑦 and 𝑥). When 𝑦 and 𝑥 are negatively correlated then with selection of a larger value of 𝑥, a smaller value of 𝑦 is expected to be selected and when smaller value of 𝑥 is selected, a larger value of 𝑦 is expected to be selected. Keeping these points in view, the following estimators are therefore suggested: 𝑌_𝑃𝐶=𝑦_𝑐12𝑥_𝑐22𝑋=𝑦+𝑐₁𝑥−𝑐₂𝑋,𝑦−𝑐₁𝑥+𝑐₂𝑋,𝑦𝑥𝑋 (14) and similarly 𝑦_{𝑙𝑟𝐶2}=𝑦_𝑐11+𝑏𝑋−𝑥_𝑐22, (15) where (𝑦_𝑐12=𝑦+𝑐₁,𝑥_𝑐22=𝑥−𝑐₂) if sample contains 𝑦_min and 𝑥_max; 𝑦_𝑐12=(𝑦−𝑐₁,𝑥_𝑐22=𝑥+𝑐₂), if sample contains 𝑦_max and 𝑥_min, and (𝑦_𝑐12=𝑦,𝑥_𝑐22=𝑥) for all other combinations of samples.

To find the bias and mean square error of these suggested estimators, we first prove two theorems which will be used in subsequent derivations.

Theorem 1. If a sample of size 𝑛 units is drawn from a population of size 𝑁 units, then the covariance between 𝑦_𝑐12 and 𝑥_𝑐12, when they are positively correlated, is given by Cov𝑦_𝑐11,𝑥_𝑐21=𝜆𝑆_𝑦𝑥−𝑛𝜆×𝑐𝑁−1₁𝑥_max−𝑥_min+𝑐₂𝑦_max−𝑦_min−2𝑛𝑐₁𝑐₂. (16)

Proof. Let us assume that 𝑛 units have been drawn without replacement from a population of size 𝑁. Let 𝑆_𝑛 denote a sample space. We partition the whole sample space into three mutually exclusive and collectively exhaustive sets, that is, 𝑆₁, 𝑆₂, and 𝑆₃ such that 𝑆_𝑛=𝑆₁∪𝑆₂∪𝑆₃. Further 𝑆₁ is the set of all possible samples which contains 𝑦_min and 𝑥_min, and 𝑆₂ consists of all samples which contains 𝑦_max and 𝑥_max, and 𝑆₃=𝑆−𝑆₁−𝑆₂. The number of sample points in 𝑆₁, 𝑆₂, and 𝑆₃ is given by 𝑁−2𝑛−1, 𝑁−2𝑛−1, and ^𝑁_𝑛−2^𝑁−2_𝑛−1, respectively.
By definition of covariance, we have Cov𝑦_𝑐11,𝑥_𝑐21=𝑁𝑛^{−1𝑠∈𝑆1}𝑦+𝑐₁−𝑌𝑥+𝑐₂−𝑋+_𝑠∈𝑆2𝑦−𝑐₁−𝑌𝑥−𝑐₂−𝑋+_𝑠∈𝑆3𝑦−𝑌𝑥−𝑋=𝑁𝑛^{−1𝑠∈𝑆1}𝑦−𝑌𝑥−𝑋+_𝑠∈𝑆2𝑦−𝑌𝑥−𝑋+_𝑠∈𝑆3𝑦−𝑌𝑥−𝑋−𝑐_{1𝑠∈𝑆2}𝑥−𝑋−_𝑠∈𝑆1𝑥−𝑋−𝑐_{2𝑠∈𝑆2}𝑦−𝑌−_𝑠∈𝑆1𝑦−𝑌+𝑐₁𝑐_{2𝑠∈𝑆1}+_𝑠∈𝑆2=𝑁𝑛^{−1𝑠∈𝑆}𝑦−𝑌𝑥−𝑋−𝑁𝑛⁻¹×𝑐₁𝑥_max−𝑥_min𝑛𝑁−2𝑛−1−𝑐₂𝑦_max−𝑦_min𝑛𝑁−2𝑛−1+2𝑐₁𝑐₂𝑁−2𝑛−1=Cov𝑦,𝑥−𝑁−𝑛𝑁()×𝑐𝑁−1₁𝑥_max−𝑥_min+𝑐₂𝑦_max−𝑦_min−2𝑛𝑐₁𝑐₂=𝜆𝑆_𝑦𝑥−𝑛𝜆()×𝑐𝑁−1₁𝑥_max−𝑥_min+𝑐₂𝑦_max−𝑦_min−2𝑛𝑐₁𝑐₂. (17)

Theorem 2. If a sample of size 𝑛 units is drawn from a population of size 𝑁 units, then the covariance between 𝑦_𝑐12 and 𝑥_𝑐22, when they are negatively correlated, is given by Cov𝑦_𝑐12,𝑥_𝑐22=𝜆𝑆_𝑦𝑥+𝑛𝜆×𝑐𝑁−1₁𝑥_max−𝑥_min+𝑐₂𝑦_max−𝑦_min−2𝑛𝑐₁𝑐₂. (18)

The above Theorem 2 can be proved similarly as Theorem 1.

We define the following relative error terms.

Let 𝑒₀=(𝑦_𝑐1−𝑌)/𝑌 and 𝑒₁=(𝑥_𝑐2−𝑋)/𝑋, such that 𝐸𝑒₀=𝐸𝑒₁𝐸𝑒=0,₂₀=𝜆𝑌²𝑆²_𝑦−2𝑛𝑐₁𝑦𝑁−1^max−𝑦_min−𝑛𝑐₁,𝐸𝑒²₁=𝜆𝑋²𝑆²_𝑥−2𝑛𝑐₂𝑥𝑁−1^max−𝑥_min−𝑛𝑐₂,𝐸𝑒₀𝑒₁=𝜆𝑌𝑋𝑆_𝑦𝑥−𝑛×𝑐𝑁−1₂𝑦_max−𝑦_min+𝑐₁𝑥_max−𝑥_min−2𝑛𝑐₁𝑐₂. (19)(20)(21)(22)

Expressing 𝑌_𝑅𝐶 in terms of 𝑒’s, we have 𝑌_𝑅𝐶=𝑌1+𝑒₀1+𝑒₁⁻¹. (23)

Expanding and rearranging right-hand side of (23), to first degree of approximation, we have 𝑌_𝑅𝐶−𝑌≅𝑌𝑒₀+𝑒₁−𝑒₀𝑒₁+𝑒²₁. (24)

Using (24), the bias of 𝑌_𝑅𝐶 is given by 𝐵𝑌_𝑅𝐶≅𝜆𝑋𝑅𝑆²_𝑥−2𝑛𝑐₂𝑥𝑁−1^max−𝑥_min−𝑛𝑐₂−𝑆_𝑦𝑥−𝑛×𝑐𝑁−1₂𝑦_max−𝑦_min+𝑐₁𝑥_max−𝑥_min−2𝑛𝑐₁𝑐₂, (25) where 𝑅=𝑌/𝑋.

Using (24), the mean square error of 𝑌_𝑅𝐶, to the first degree of approximation, is given by 𝑀𝑌_𝑅𝐶≅𝜆𝑆²_𝑦−2𝑛𝑐₁𝑦𝑁−1^max−𝑦_min−𝑛𝑐₁+𝑅²𝑆²_𝑥−2𝑛𝑐₂𝑥𝑁−1^max−𝑥_min−𝑛𝑐₂−2𝑅𝑆_𝑦𝑥−𝑛×𝑐𝑁−1₂𝑦_max−𝑦_min+𝑐₁𝑥_max−𝑥_min−2𝑛𝑐₁𝑐₂ (26) or 𝑀𝑌_𝑅𝐶≅𝑀𝑦_𝑅−2𝜆𝑛×𝑐𝑁−1₁−𝑅𝑐₂𝑦_max−𝑦_min−𝑅𝑥_max−𝑥_min−𝑛𝑐₁−𝑅𝑐₂. (27)

To find optimum values of 𝑐₁ and 𝑐₂, we differentiate (27) with respect to 𝑐₁ and 𝑐₂ as 𝜕𝑀𝑌_𝑅𝐶𝜕𝑐₁=0⟹𝑦_max−𝑦_min−𝑅𝑥_max−𝑥_min−2𝑛𝑐₁−𝑅𝑐₂=0,𝜕𝑀𝑌_𝑅𝐶𝜕𝑐₂=0⟹𝑦_max−𝑦_min−𝑅𝑥_max−𝑥_min−2𝑛𝑐₁−𝑅𝑐₂=0 (28)

Here we have one equation with two unknowns so unique, solution is not possible, so we let 𝑐₂=(𝑥_max−𝑥_min)/2𝑛, and then 𝑐₁=(𝑦_max−𝑦_min)/2𝑛.

For optimum values of 𝑐₁ and 𝑐₂, the optimum mean square error of 𝑌_𝑅𝐶 is given by 𝑀𝑌_𝑅𝐶opt≅𝑀𝑦_𝑅−𝜆2()×𝑦𝑁−1_max−𝑦_min−𝑅𝑥_max−𝑥_min². (29)

Similarly the bias and mean square error or optimum mean square error of 𝑌_𝑃𝐶 are, respectively, given by 𝐵𝑌_𝑃𝐶≅𝜆𝑋𝑆_𝑦𝑥−𝑛×𝑐𝑁−1₂𝑦_max−𝑦_min+𝑐₁𝑥_max−𝑥_min−2𝑛𝑐₁𝑐₂,𝑀𝑌_𝑃𝐶≅𝑀𝑦_𝑃−2𝜆𝑛×𝑐𝑁−1₁+𝑅𝑐₂𝑦_max−𝑦_min+𝑅𝑥_max−𝑥_min−𝑛𝑐₁+𝑅𝑐₂. (30)(31)

For optimum values of 𝑐₁ and 𝑐₂, the optimum mean square error of 𝑌_𝑃𝐶 is given by 𝑀𝑌_𝑃𝐶opt≅𝑀𝑦_𝑃−𝜆2()×𝑦𝑁−1_max−𝑦_min−𝑅𝑥_max−𝑥_min². (32)

The variance of regression estimator 𝑦_{𝑙𝑟𝐶1} in case of positive correlation is given by 𝑉𝑦_{𝑙𝑟𝐶1}=𝑉𝑦_𝑙𝑟−2𝜆𝑛×𝑐𝑁−1₁−𝛽𝑐₂𝑦_max−𝑦_min−𝛽𝑥_max−𝑥_min−2𝑛𝑐₁−𝛽𝑐₂, (33) where 𝛽=𝜌_𝑦𝑥(𝑆_𝑦/𝑆_𝑥) is the population regression coefficient of 𝑦 on 𝑥.

For 𝑐₂=(𝑥_max−𝑥_min)/2𝑛 and 𝑐₁=(𝑦_max−𝑦_min)/2𝑛, optimum variance of 𝑦_{𝑙𝑟𝐶1} is given by 𝑉𝑦_{𝑙𝑟𝐶1opt}=𝑉𝑦_𝑙𝑟−𝜆2()×𝑁−1(𝑦_max−𝑦_min)−𝛽(𝑥_max−𝑥_min)². (34)

For negative correlation, variance of the regression estimator 𝑦_{𝑙𝑟𝐶2} is given by 𝑉𝑦_{𝑙𝑟𝐶2}=𝑉𝑦_𝑙𝑟−2𝜆𝑛×𝑐𝑁−1₁+𝛽𝑐₂𝑦_max−𝑦_min+𝛽𝑥_max−𝑥_min−2𝑛𝑐₁+𝛽𝑐₂. (35)

For 𝑐₂=(𝑥_max−𝑥_min)/2𝑛 and 𝑐₁=(𝑦_max−𝑦_min)/2𝑛, optimum variance of 𝑦_{𝑙𝑟𝐶2} is given by 𝑉𝑦_{𝑙𝑟𝐶2opt}=𝑉𝑦_𝑙𝑟−𝜆2()×𝑦𝑁−1_max−𝑦_min+𝛽𝑥_max−𝑥_min². (36)

So in general we can write 𝑉(𝑦_𝑙𝑟𝐶)_opt as 𝑉𝑦_{𝑙𝑟𝐶opt}=𝑉𝑦_𝑙𝑟−𝜆2()×𝑦𝑁−1_max−𝑦_min−𝛽𝑥_max−𝑥_min². (37)

3. Comparison

The conditions under which the suggested estimators 𝑌_𝑅𝐶, 𝑌_𝑃𝐶, and 𝑦_𝑙𝑟𝐶 perform better than the usual mean per unit estimator and their usual counterpart is given below.

(a) Comparison of Proposed Ratio Type Estimator. A proposed estimator 𝑌_𝑅𝐶 will perform better than

   (i) mean per unit estimator (by (1) and (27)) if 𝑉(𝑦)−𝑀(𝑌_𝑅𝐶)>0 or if 𝜌_𝑦𝑥>𝑅𝑆_𝑥2𝑆_𝑦−𝑛𝑅𝑆_𝑦𝑆_𝑥()×𝑐𝑁−1₁−𝑅𝑐₂𝑦_max−𝑦_min−𝑅𝑥_max−𝑥_min−2𝑛𝑐₁−𝑅𝑐₂; (38)

    (ii) usual ratio estimator (by (8) and (27)) if 𝑀(𝑦_𝑅)−𝑀(𝑌_𝑅𝐶)>0 or if min𝑅𝑐₂,𝑅𝑐₂−𝑅𝑥_max−𝑥_min−𝑦_max−𝑦_min𝑛<𝑐₁<max𝑅𝑐₂,𝑅𝑐₂−𝑅𝑥_max−𝑥_min−𝑦_max−𝑦_min𝑛. (39)

(b) Comparison of Proposed Product Type Estimator. A proposed product type estimator will perform better than

    (iii) mean per unit estimator if 𝑉(𝑦)−𝑀(𝑌_𝑃𝐶)>0 (by (1) and (31)) or if 𝜌_𝑦𝑥<−𝑅𝑆_𝑥2𝑆_𝑦+𝑛𝑅𝑆_𝑦𝑆_𝑥()×𝑐𝑁−1₁+𝑅𝑐₂𝑦_max−𝑦_min+𝑅𝑥_max−𝑥_min−2𝑛𝑐₁+𝑅𝑐₂; (40)

    (iv) usual product estimator if 𝑀(𝑦_𝑃)−𝑀(𝑌_𝑃𝐶)>0 (by (9) and (31)) or if min−𝑅𝑐₂,−𝑅𝑐₂+𝑅𝑥_max−𝑥_min+𝑦_max−𝑦_min𝑛<𝑐₁<max−𝑅𝑐₂,−𝑅𝑐₂+𝑅𝑥_max−𝑥_min+𝑦_max−𝑦_min𝑛. (41)

(c) Comparison of Proposed Regression Type Estimator. A proposed regression type estimator (positive correlation) will perform better than

    (v) mean per unit estimator if 𝑉(𝑦)−𝑉(𝑦_{𝑙𝑟𝐶1})>0 (by (1) and (33)) or if 𝜌²_𝑦𝑥>−2𝑛𝑆²_𝑦()×𝑐𝑁−1₁−𝛽𝑐₂𝑦_max−𝑦_min−𝛽𝑥_max−𝑥_min−2𝑛𝑐₁−𝛽𝑐₂; (42)

    (vi) usual regression estimator if 𝑉(𝑦_𝑙𝑟)−𝑉(𝑦_{𝑙𝑟𝐶1})>0 (by (11) and (33)) or if min𝛽𝑐₂,𝛽𝑐₂−𝛽𝑥_max−𝑥_min−𝑦_max−𝑦_min𝑛<𝑐₁<max𝛽𝑐₂,𝛽𝑐₂−𝛽𝑥_max−𝑥_min−𝑦_max−𝑦_min𝑛. (43)

A proposed regression type estimator (negative correlation) will perform better than

    (vii) mean per unit estimator if 𝑉(𝑦)−𝑉(𝑦_{𝑙𝑟𝐶2})>0 (by (1) and (35)) or if 𝜌²_𝑦𝑥>−2𝑛𝑆²_𝑦()×𝑐𝑁−1₁+𝛽𝑐₂𝑦_max−𝑦_min+𝛽𝑥_max−𝑥_min−2𝑛𝑐₁−𝛽𝑐₂; (44)

    (viii) usual regression estimator if 𝑉(𝑦_𝑙𝑟)−𝑉(𝑦_{𝑙𝑟𝐶1})>0 (by (11) and (33)) or if min−𝛽𝑐₂,−𝛽𝑐₂+𝛽𝑥_max−𝑥_min−𝑦_max−𝑦_min𝑛<𝑐₁<max−𝛽𝑐₂,−𝛽𝑐₂+𝛽𝑥_max−𝑥_min−𝑦_max−𝑦_min𝑛. (45)

(d) Comparison of Suggested Estimators for Optimum Values of 𝑐₁ and 𝑐₂ with Usual Estimators. For optimum values of 𝑐₁ and 𝑐₂, the proposed estimator will always perform better than usual mean per unit estimator and their usual counterparts (ratio, product and regression estimators).

4. Empirical Study

We consider the following datasets for numerical comparison.

Population 1 (Singh and Mangat [9, page 193]). Let 𝑦 be the milk yield in kg after new food and let 𝑥 be the yield in kg before new yield. 𝑁=27, 𝑛=12, 𝑋=10.41111, 𝑌=11.25185, 𝑦_max=14.8, 𝑦_min=7.9, 𝑥_max=14.5, 𝑥_min=6.5, 𝑆²_𝑦=4.103, 𝑆²_𝑥=4.931, 𝑆_𝑥𝑦=4.454, and 𝜌_𝑦𝑥=0.990.

Population 2 (Singh and Mangat [9, page 195]). Let 𝑦 be the weekly time (hours) spent in nonacademic activities and let 𝑥 be the overall grade point average (4.0 bases). 𝑁=36, 𝑛=12, 𝑋=2.798333, 𝑌=14.77778, 𝑦_max=33, 𝑦_min=6, 𝑥_max=3.82, 𝑥_min=1.81, 𝑆²_𝑦=38.178, 𝑆²_𝑥=0.3504, 𝑆_𝑥𝑦=−2.477, and 𝜌_𝑦𝑥=−0.6772.

Population 3 (Murthy [10, page 399]). Let 𝑦 be the area under wheat crop in 1964 and let 𝑥 be the area under wheat crop in 1963. 𝑁=34, 𝑛=12, 𝑋=208.882, 𝑌=199.441, 𝑦_max=634, 𝑦_min=6, 𝑥_max=564, 𝑥_min=5, 𝑆²_𝑦=22564.56,𝑆²_𝑥=22652.05, 𝑆_𝑥𝑦=22158.05, and 𝜌_𝑦𝑥=0.980.

Population 4 (Cochran [11, page 152]). Let 𝑦 be population size in 1930 (in 1000) and 𝑥 be the population size in 1920 (in 1000). 𝑁=49, 𝑛=12, 𝑋=103.1429, 𝑌=127.7959, 𝑦_max=634, 𝑦_min=46, 𝑥_max=507, 𝑥_min=2, 𝑆²_𝑦=15158.83, 𝑆²_𝑥=10900.42, 𝑆_𝑥𝑦=12619.78 and 𝜌_𝑦𝑥=0.98.

The conditional values and results are given in Tables 1 and 2, respectively.

Table 1 

Numerical values of conditions (38)–(45).

Table 2 

PRE of different estimators with respect to 𝑦.

For percentage relative efficiency (PRE), we use the following expression: PRE𝑦_𝑖,𝑦=𝑉𝑦𝑉𝑦_𝑖or𝑀𝑦_𝑖×100for𝑖=𝑅,𝑃,𝑅𝐶,𝑃𝐶,𝑙𝑟,𝑙𝑟𝐶. (46)

5. Conclusion

From Table 2, it is observed that the ratio estimator 𝑌_𝑅𝐶 is performing better than 𝑦_𝑅 in Populations 1, 3, and 4 because of positive correlation. The product estimator 𝑌_𝑃𝐶 is better than 𝑦_𝑃 just in Population 2 because of negative correlation. The regression estimator 𝑦_𝑙𝑟𝐶 outperforms than all other considered estimators and is preferable.

Acknowledgments

The authors are thankful to the learned referees for their valuable suggestions and helpful comments in revising the manuscript.