#### 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 () and the auxiliary variable () 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 and 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 be a population of size . Let () be the values of the study and the auxiliary variables, respectively, on the th unit of a finite population.

Let us assume that a simple random sample of size is drawn without replacement from for estimating the population mean . It is further assumed that the population mean of the auxiliary variable is known. The minimum say and maximum say values of the auxiliary variables are also assumed to be known.

The variance of mean per unit estimator is given by where and .

Some time there exists unusually very large (say ) and very small (say ) 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 ) or overestimated (in case the sample contains ). To overcome the situation Sarndal [8] suggested the following unbiased estimator: where is a constant.

The variance of is given by

Further, if .

For, , variance of is given by which is always smaller than .

The usual ratio and product estimators of population mean are given by where and are the sample means of variables and , respectively.

The expressions for biases (), and mean square errors (), of the conventional ratio and product estimators, are given by where and are the coefficients of variation of and , respectively, is the correlation coefficient between and , , and are the population variance and population covariance, respectively.

Usual regression estimator is given by where is the sample regression coefficient.

The variance of the estimator is given by

#### 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 and are positively and negatively correlated, respectively. We suggest estimator for each case separately as follows.

*Case 1 (positive correlation between and ). *When and are positively correlated, then with selection of a larger value of , a larger value of is expected to be selected and when smaller value of is selected, selection of a smaller value of is expected. So we define the following estimators:
and similarly
where if the sample contains and ; if the sample contains and , and for all other combinations of samples.

*Case 2 (negative correlation between and ). *When and are negatively correlated then with selection of a larger value of , a smaller value of is expected to be selected and when smaller value of is selected, a larger value of is expected to be selected. Keeping these points in view, the following estimators are therefore suggested:
and similarly
where if sample contains and ; , if sample contains and , and 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 units is drawn from a population of size units, then the covariance between and , when they are positively correlated, is given by
*

*Proof. *Let us assume that units have been drawn without replacement from a population of size . Let denote a sample space. We partition the whole sample space into three mutually exclusive and collectively exhaustive sets, that is, , , and such that . Further is the set of all possible samples which contains and , and consists of all samples which contains and , and . The number of sample points in , , and is given by , , and , respectively.

By definition of covariance, we have

Theorem 2. *If a sample of size units is drawn from a population of size units, then the covariance between and , when they are negatively correlated, is given by
*

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

We define the following relative error terms.

Let and , such that

Expressing in terms of ’s, we have

Expanding and rearranging right-hand side of (23), to first degree of approximation, we have

Using (24), the bias of is given by where .

Using (24), the mean square error of , to the first degree of approximation, is given by or

To find optimum values of and , we differentiate (27) with respect to and as

Here we have one equation with two unknowns so unique, solution is not possible, so we let , and then .

For optimum values of and , the optimum mean square error of is given by

Similarly the bias and mean square error or optimum mean square error of are, respectively, given by

For optimum values of and , the optimum mean square error of is given by

The variance of regression estimator in case of positive correlation is given by where is the population regression coefficient of on .

For and , optimum variance of is given by

For negative correlation, variance of the regression estimator is given by

For and , optimum variance of is given by

So in general we can write as

#### 3. Comparison

The conditions under which the suggested estimators , , and 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 will perform better than

(i) mean per unit estimator (by (1) and (27)) if or if

(ii) usual ratio estimator (by (8) and (27)) if or if

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

(iii) mean per unit estimator if (by (1) and (31)) or if

(iv) usual product estimator if (by (9) and (31)) or if

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

(v) mean per unit estimator if (by (1) and (33)) or if

(vi) usual regression estimator if (by (11) and (33)) or if

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

(vii) mean per unit estimator if (by (1) and (35)) or if

(viii) usual regression estimator if (by (11) and (33)) or if

*(d) Comparison of Suggested Estimators for Optimum Values of ** and ** with Usual Estimators.* For optimum values of and , 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 be the milk yield in kg after new food and let be the yield in kg before new yield. , , , , , , , , , , , and .

*Population 2 (Singh and Mangat [9, page 195]). *Let be the weekly time (hours) spent in nonacademic activities and let be the overall grade point average (4.0 bases). , , , , , , , , , , , and .

*Population 3 (Murthy [10, page 399]). *Let be the area under wheat crop in 1964 and let be the area under wheat crop in 1963. , , , , , , , , , , and .

*Population 4 (Cochran [11, page 152]). *Let be population size in 1930 (in 1000) and be the population size in 1920 (in 1000). , , , , , , , , , , and .

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

For percentage relative efficiency (PRE), we use the following expression:

#### 5. Conclusion

From Table 2, it is observed that the ratio estimator is performing better than in Populations 1, 3, and 4 because of positive correlation. The product estimator is better than just in Population 2 because of negative correlation. The regression estimator 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.