1. Introduction

To increase the efficiency of the proposed estimator of population mean of the study variable, we use the information of auxiliary variable which is highly correlated with. Many authors viz. Sisodia and Dwivedi [1], Mohanty and Sahoo [2], Singh and Tailor [3], Khoshnevisan et al. [4], Gupta and Shabbir [5], Iqbal and Maqbool [6], Mehta and Sharma [7] etc. have adapted transformation method on auxiliary variable by using several known parameters such as to reduce the bias and mean square error.

In literature, the classical ratio estimator for population mean is defined as,

(1.1)

The bias and MSE of , up to first order of approximation, is

(1.2)

(1.3)

Sisodia and Dwivedi [1] using the information of and , defined ratio-type estimator for population mean as

(1.4)

where is the known population coefficient of variation of the auxiliary variable . The bias and MSE of , up to first order of approximation, is

(1.5)

(1.6)

Where is a constant.

Use of higher order parameters in the proposed estimators restricts their applicability. To overcome such type of restrictions, Mohanty and Sahoo [2] using known minimum and maximum values besides population mean of auxiliary variable and then taking the transformation on auxiliary variable as,

defined two estimators of as

(1.7)

The expressions for bias and MSE of these estimators, up to first order of approximation, are

(1.8)

(1.9)

(1.10)

(1.11)

where and are constants respectively.

It is to be noted that the transformations in and depend upon both and but estimators generated through these transformations depend only upon and respectively. To utilize the information on both and , in this paper, we propose a modified ratio-type estimator of population mean in simple random sampling using transformation on auxiliary variable in different forms. The expressions for bias and mean square error have been obtained. The comparison has been made with Classical ratio estimator, Mohanty and Sahoo [2] and Sisodia and Dwivedi [1]. The results have also been illustrated numerically and graphically.

2. Sampling Design and Terminology

A simple random sample of size is drawn from a finite population of size and both auxiliary variable and study variable are measured on it.

Let and denote the respective values of variables and on the unit of the population. Corresponding small letters denote the values in the sample.

Taking,

Defining,

So we have,

3. Proposed Estimator

Using known minimum and maximum values besides population mean of auxiliary variable and then taking the transformation on auxiliary variable as , we propose an estimator of as

(3.1)

When is also known, we propose an another estimator of population mean as,

(3.2)

To find the bias and the MSE of the estimator , we expand in terms of , up to the second degree of approximation, as

(3.3)

where

Taking the expectation in (3.3), we obtain

(3.4)

Using the expression (3.3), the mean square error of the proposed estimator , up to the first order of approximation, is

(3.5)

(3.6)

and

(3.7)

where

We have proved the following theorems:

Theorem 3.1. Up to the first order of approximation, the bias of estimator is

and its MSE is

Theorem 3.2. Up to the first order of approximation, the bias of estimator is

and its MSE is

4. Comparison

First, we compare the bias of the proposed estimators and one by one with Classical ratio estimator , Mohanty and Sahoo [2] estimators , and Sisodia and Dwivedi [1] estimator .

Up to first order of approximation, we have

Second, we compare MSE of the proposed estimators and one by one with Classical ratio estimator , Mohanty and Sahoo [2] estimators , and Sisodia and Dwivedi [1] estimator .

5. Numerical

To have the rough idea about the performance of the proposed estimators over the existing one, we consider a real finite population presented in Cochran [8] on page 34. This data set concerns food cost (as study variable) and weekly income (as auxiliary variable). Note that we omitted the data of 1 family because it was an outlier. The values of the population parameters are given in Table 1. The Bias, MSE and relative efficiency of proposed estimators with respect to Classical ratio estimator , Mohanty and Sahoo [2] estimators , and Sisodia and Dwivedi [1] estimator are given in Table 2 and Table 3 respectively. The results are also illustrated graphically.

Table 1. Parameters and Constants of the Population

Table 2. Bias and Relative Efficiencies of Existing and Proposed Estimators

Table 3. MSE and Relative Efficiencies of Existing and Proposed Estimators

Figure 1. Bias and Relative Efficiencies of Existing and Proposed Estimators

Figure 2. MSE and Relative Efficiencies of Existing and Proposed Estimators

6. Conclusions

From the Table 2 and Table 3, we observe that although there is small increase in efficiency of the proposed estimators w. r. t. the MSE yet there is a significant increase in efficiency w. r. t. the bias. So we conclude that where we have to give the more weightage to precision as compared to the accuracy, then the proposed estimators with new linear transformation are recommended to use.