1. Introduction

[1] initiated the use of auxiliary information to estimate population mean in sample survey. He proposed the ratio estimator to estimate population mean which can also be used to estimate population total where and denote the sample means of study and auxiliary variables respectively and denotes the population mean of auxiliary variable. This ratio estimator is more efficient than mean per unit estimator if where is the correlation coefficient between the auxiliary and the study variables; and are coefficients of variation for auxiliary and study variables respectively.

Further, on the same pattern, [2] proposed a product estimator for The product estimator is more efficient than mean per unit estimator if Using the power transformation, [3] proposed a new estimator

to estimate and he found the optimum value of for which the MSE of is minimum. After this, many statisticians proposed different kinds of estimators using auxiliary information. But [4] proposed a modified ratio estimator for using the additional known information of coefficient of variation of auxiliary variable as:

Thereafter, it became a trend to propose modified ratio estimators for by using additional known information of auxiliary variable. Some related examples are as follows:

[5]

[6]

where is coefficient of kurtosis of auxiliary variable.

[7]

[8]

[9]

where and are real numbers or some known parameters of auxiliary variable.

[10]

where is some real constant.

[11]

where is some real constant.

[12]

[13]

where is the coefficient of skewness of auxiliary variable.

All the above estimators are more efficient than the classical ratio estimator but in these estimators, researchers have added a pure number in the quantity which possesses a unit which is mathematically not possible. For example, the mean of auxiliary variable is measured in centimeters and the coefficient of variation is always a pure number, still [4] and [5] added these two quantities. The main question which arises here is "Can we add these two quantities?" This question was first raised by [14]. If the answer is in the negative, then the question arises about the authenticity of the estimators proposed by different researchers using above assumption.

The aim of the present paper is to discuss these issues and find out the situation where one can use the classical ratio estimator. In Section-2, the basic conditions for the ratio estimator of population mean have been discussed. The effects of change of origin of variables on ratio estimator of population mean have been presented in Section-3. A detailed discussion about the clarification given by [15] on the transformation of auxiliary variable has been presented in Section-4 and the idea for further study has been proposed in Section-5. In Section-6, a simulation study has been carried out to verify theoretical results. In the last section, the conclusion of the present paper has been drawn.

2. Ratio Estimator

In all the modified ratio estimators, researchers had changed either origin only or both the origin and scale of auxiliary variable. Due to this change, there is some reduction in bias and MSE of modified ratio estimators as compared to those of the classical ratio estimator. It means that there is some effect of change of origin and scale of auxiliary variable. To understand this reduction of bias and MSE, there is a need to understand the assumptions of classical ratio estimator which are as follows:-

1. To obtain the approximate results for ratio estimator, the conditions are

(1)

From the Chebychev's inequality, we have

where and

The above inequality shows that is the upper bound of with probability more than Using this upper bound in (1), we get

or

After ignoring the finite population correction, we get

(2)

Similarly, one can obtain

(3)

Now one can easily verify the conditions given in equation (1) with the help of inequalities (2) and (3).

If the inequalities (2) and (3) are not satisfied then one can use new variables using the transformation and Since the left hand side of these inequalities is independent of change of origin, these transformations affect only the right side. These new variables can further be used in ratio estimation of population mean. In the present paper, we assume that these inequalities are satisfied for variables and

2. Classical ratio estimator of population mean is more efficient than mean per unit estimator provided

(4)

In above inequality, left hand side is independent of change of origin of variables whereas right hand side is dependent on change of origin of variables. In the following section, we will first discuss some results of ratio estimator.

3. Effect of Change of Origin of Variables

After assuming the conditions given in (1), one can easily obtain the approximate bias and MSE for classical ratio estimator as

(5)

(6)

It is known that

(7)

where is the minimum value of if bias of the estimator become zero. may be positive or negative. Both the cases have been discussed as follows:-

We have from equation (5),

or

(8)

If we use the transformation where there is reduction in that is

(9)

One can choose the optimum value of as

such that

(10)

Now we propose a modified ratio estimator of population mean as

(11)

Approximate expressions for bias and MSE of are as

and

(12)

Clearly, the modified ratio estimator is an approximately unbiased estimator of population mean and the estimator is always more efficient than mean per unit estimator.

In practical situations, the value of is generally unknown. So one can use the estimated value of as

From equation (5), we get

or

(13)

If we use the transformation where there is increase in that is

(14)

But due this transformation, the new variable may violate the condition given in (1). So here, we shall use a new transformation where If we compare the coefficients of variation of these variables, we get

By using the optimum value of as

we get

(15)

We define the new ratio estimator for as

(16)

The ratio estimator defined above is approximately unbiased for and the approximate variance is same as the variance of as given in (12).

Modified ratio estimator for may be defined as

(17)

Clearly, remains unbiased for Also the variance of remains same as the variance of given in (12).

The optimum value of contains some unknown parameters, so one can use the estimated value of in practical situations as

From the above discussion, one concludes that if there is a problem of over estimation during ratio estimation of population mean, then it can be solved by shifting the origin of the auxiliary variable. Similarly, if there is a problem of under estimation, then it can be reduced by shifting the origin of the study variable. Further, one more question arises- "What will happen if we change the origin and scale of the auxiliary variable?" To answer this, we consider the following transformation on auxiliary variable.

which implies that

In this case, the modified ratio estimator may be defined as

It shows that in ratio estimation, the transformation is equivalent to the transformation

One can modify the ratio estimator by shifting the origin of auxiliary variable or study variable and this modified estimator is always more beneficial than the existing one. Different modified ratio estimators proposed by the researchers by using the additional known information of auxiliary variable are mathematically not correct. But these estimators perform better than classical ratio estimator due to the change of origin of auxiliary variable; not due to additional known information.

4. Clarification by Singh and Kumar

[14] and [16] raised a question regarding transformation of auxiliary variables by adding a unit free constant. But [15] try to clarify the same with the help of following three examples:

1. “Let

(18)

The area under the function will be which is in fact the distance between two points and on a real line, say Now, to make as area, we have to multiply it by The required area will be:

2. “If we integrate speed of an object with respect to time, then we get distance in Now distance is not area, as expected after integrating a function, but it is distance in centimeter.”

3. is constant and an irrational number, but has radian as units of measurements.”

On the basis of these three examples, they asserted that these kinds of adjustments of units of measurements are common in practice.

Our Clarification:

First of all, in the above three examples we need to understand whether an adjustment has really been done or not. Let us recall the definition of integration as follows:

Let be a bounded real-valued function on It means that is also bounded on each sub-interval corresponding to each partition Let be the least upper bound and greatest lower bound of in Then there are two sums,

and

respectively called the upper and lower sums of corresponding to the partition

Further, infimum of the set of upper sums is called the upper integral and supremum of the set of lower sums is called the lower integral over That is

and

If two integrals are equal, i.e.

then is said to be Riemann integrable or simply integrable over and the common value of these integrals is called the integral of over For more details of the concept of integration, see [17].

From the above definition, it is clear that

The units of integral of

In equation (18), if both and have units in then from the definition, we have

Similarly, when we integrate speed(cm/sec) of an object with respect to time(sec), then we get units of the integral in cm. Further, we know that each individual real number is a constant and is one of them. The number does not possess any units like radian. For example, if height of a person is 72 inches, it does not mean that 72 has inches as units. The units are possessed by variable height but not by 72.

From above discussion, it is clear that in all the three examples given by [15], there is no adjustment of units.

5. Further Study

In the last two decades, a number of modified estimators have been developed under various sampling strategies using transformations of auxiliary variables by shifting origin by adding units free constants or quantities with different units. This type of modifications are mathematically not justifiable. So there is a need to find out the actual reasons about what happens when we change the origin under those sampling strategies. Some of the work which has been done using this pattern by different researchers is mentioned in [18-36] and many more.

6. Simulation Study

A simulation study has been carried out to verify the theoretical results by using software R. Firstly, results have been obtained on the basis of 1,00,000 samples each of size n=3, 5 and 7 from each of normal populations , where For these populations, auxiliary variable does not satisfy the condition

If we use the transformation then the new auxiliary variable satisfies the above condition. Actual and approximate values for the biases and MSEs of the usual ratio estimator and the modified ratio estimator after using transformation are given in Table-1.

Table 1. When The Conditions are not Satisfying

Secondly, the problem of over estimation has been considered. For this purpose, 1,00,000 samples each of size have been drawn from each of normal population where The values of biases and MSEs of and have been presented in Table-2.

Table 2. Problem of Over Estimation

Results Based on Table-1 and Table-2:

1. Table-1 shows that if the study and auxiliary variables satisfy the conditions given in (2) and (3), then the approximate values for biases and MSE's of usual ratio estimator are very close to the actual values.

2. Table-2 shows that if we use the optimum transformation of auxiliary variable, then we can get an almost unbiased ratio estimator for population mean.

7. Conclusions

A number of modified ratio estimators have been proposed by different researchers using the additional known information of auxiliary variable. Basically, they have applied a transformation on auxiliary variable by adding some unit-less quantity or a quantity with different units in auxiliary variable. Such type of transformations are mathematically not correct. But from these modified ratio estimators, one can form an idea that the change of origin of auxiliary variable affect the efficiency of the ratio estimator. In the present paper, it has been shown that the efficiency of ratio estimator also depends upon the origin of the auxiliary and study variables. Also, the efficiencies of the modified estimators have improved due to the change of origin and not due to additional known information of auxiliary variable. Further, the conditions for validity of the approximate results for ratio estimators have been developed in (2) and (3), which can easily be checked in practical situations before using the ratio estimators.