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

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.


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 = ( 1 , 2 , . . . , ) 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 = ∑ =1 / . It is further assumed that the population mean = ∑ =1 / of the auxiliary variable is known. The minimum say ( min ) and maximum say ( max ) values of the auxiliary variables are also assumed to be known.
Some time there exists unusually very large (say max ) and very small (say 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 min ) or overestimated 2 The Scientific World Journal (in case the sample contains max ). To overcome the situation Sarndal [8] suggested the following unbiased estimator: if sample contains max but not min , for all other samples, where is a constant. The variance of is given by For, opt = ( max − min )/2 , variance of is given by which is always smaller than ( ). The usual ratio and product estimators of population mean ( ) are given by where = ∑ =1 / and = ∑ =1 / are the sample means of variables and , respectively.
Usual regression estimator is given by where is the sample regression coefficient. The variance of the estimator is given by

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 ( 11 = + 1 , 21 = + 2 ) if the sample contains min and min ; ( if the sample contains max and max , and ( 11 = , 21 = ) 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 To find the bias and mean square error of these suggested estimators, we first prove two theorems which will be used in subsequent derivations. and 12 , 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, 1 , 2 , and 3 such that = 1 ∪ 2 ∪ 3 . Further 1 is the set of all possible samples which contains min and min , and 2 consists of all samples which contains max and max , and 3 = − 1 − 2 . The number of sample points in 1 , 2 , and 3 is given by By definition of covariance, we have The above Theorem 2 can be proved similarly as Theorem 1.
We define the following relative error terms. 4 The Scientific World Journal Expressinĝin terms of 's, we havê Expanding and rearranging right-hand side of (23), to first degree of approximation, we have Using (24), the bias of̂is given by where = / . Using (24), the mean square error of̂, to the first degree of approximation, is given by To find optimum values of 1 and 2 , we differentiate (27) with respect to 1 and 2 as Here we have one equation with two unknowns so unique, solution is not possible, so we let 2 = ( max − min )/2 , and then 1 = ( max − min )/2 .
For optimum values of 1 and 2 , the optimum mean square error of̂is given by Similarly the bias and mean square error or optimum mean square error of̂are, respectively, given by For optimum values of 1 and 2 , the optimum mean square error of̂is given by The variance of regression estimator 1 in case of positive correlation is given by where = ( / ) is the population regression coefficient of on .
For 2 = ( max − min )/2 and 1 = ( max − min )/2 , optimum variance of 1 is given by The Scientific World Journal 5 For negative correlation, variance of the regression estimator 2 is given by For 2 = ( max − min )/2 and 1 = ( max − min )/2 , optimum variance of 2 is given by So in general we can write ( ) opt as (37)

Comparison
The conditions under which the suggested estimatorŝ, , and perform better than the usual mean per unit estimator and their usual counterpart is given below.

(d) Comparison of Suggested Estimators for Optimum Values of
1 and 2 with Usual Estimators. For optimum values of 1 and 2 , the proposed estimator will always perform better than usual mean per unit estimator and their usual counterparts (ratio, product and regression estimators).

Empirical Study
We consider the following datasets for numerical comparison.

Conclusion
From Table 2, it is observed that the ratio estimator̂is performing better than in Populations 1, 3, and 4 because The Scientific World Journal 7 of positive correlation. The product estimator̂is better than just in Population 2 because of negative correlation. The regression estimator outperforms than all other considered estimators and is preferable.