1. Introduction

In the theory of survey sampling, the auxiliary information plays paramount role in developing and searching improved estimators of population parameters of the study variable. The auxiliary information is used at both the stages of designing and estimation. Here we have used this information at estimation stage only. The auxiliary variable (X) and the main variable (Y) under study are highly closely related with each other. When there is a close positive association between the study variable and the auxiliary variable and the line of regression of the study variable Y on the auxiliary variable X passes through origin, then the ratio type estimator is used for improvement over the parameters of the population under consideration. On the other hand the product type estimators are used for improved estimation of parameters when the auxiliary variable x and the study variable y have negative correlation between them. While the regression type estimators are used for the improved estimation of population parameters, when the line of regression does not pass through the origin.

Let the population under investigation is finite and it consists of N distinct and identifiable units. Let be a random sample of size n from above bivariate population (X, Y) of size N using a SRSWOR scheme. Let respectively are the population means of the auxiliary and the study variables, and let and are the corresponding sample means which are unbiased estimators of population means respectively. Let denote the correlation coefficient between the variables X and Y and is the inter-quartile range of the auxiliary variable X. In this manuscript, we have proposed an improved ratio type estimator of population variance of study variable by suitably using the correlation coefficient between the two variables and , inter-quartile range of the auxiliary variable X. Further we assume that a reliable estimate of the correlation coefficient is available in advance from pilot surveys etc.

2. Variance Estimators in Literature

The sample variance is the most appropriate estimator of population variance and is given by:

(2.1)

This estimator of population variance is unbiased, and it has the variance up to the first degree of approximation as:

(2.2)

Isaki (1983) proposed the following ratio estimator of population variance using auxiliary information as:

(2.3)

where

The first order of approximations for the Bias and Mean Square Error (MSE) respectively are given by

(2.4)

(2.5)

where and .

Several authors proposed different estimators by utilizing auxiliary information in different forms. They used it in the form of different parameters of auxiliary variable for estimating the population variance of the main variable under study. Some of them from the literature are as follows,

Upadhyaya and Singh (1999) utilized coefficient of kurtosis of auxiliary variable and proposed the following estimator of population variance as,

(2.6)

The bias and Mean Squared Error of above estimator up to the first order of approximations respectively are,

(2.7)

Where,

Kadilar and Cingi (2006) proposed the following estimators using different parameters of auxiliary information as,

The bias and Mean Squared Error of above estimators up to the first order of approximations respectively are,

(2.8)

Where,

Subramani and Kumarpandiyan (2012), utilizing various population parameters of auxiliary variable proposed the following estimators of population variance as,

The expressions for the bias and Mean Squared Error of above estimators up to the first order of approximations respectively are,

(2.9)

Where,

Where are the quartiles, the three points dividing the whole distribution into four equal parts. Further the functions of quartiles are, the inter quartile range, , the semi-quartile range and the quartile average .

Khan and Shabbir (2013) proposed the following estimator using correlation coefficient and the third quartile of the auxiliary variable as,

The expressions for the bias and Mean Squared Error of the estimator up to the first order of approximations respectively are,

(2.10)

Where,

Yadav et al. (2014), utilizing the correlation coefficient of the inter-quartile range of auxiliary variable proposed the following estimator as,

The bias and Mean Squared Error of the above estimator up to the first order of approximations respectively are,

(2.11)

Where, .

3. Proposed Estimator

Here, an improved ratio estimator of population variance is being suggested in the light of estimators proposed by Yadav et al. (2014) and Prasad (1989) as,

(3.1)

where is a characterizing scalar to be determined such that the MSE of the proposed estimator t is minimized.

To obtain the bias and Mean squared error of the proposed estimator, we wish to define

and such that for and , , .

The proposed estimator can be written in terms of , as

Expanding the right hand side of above equation and considering the terms in up to the first degree of approximation, we get:

After subtracting the population variance of study variable on both the sides of above equation, we have,

(3.2)

The bias of proposed estimator is obtained by taking expectations on both sides of (3.2) and putting the values of different expectations, as:

(3.3)

where .

The mean squared error of the proposed estimator is obtained by squaring both sides of (3.2), simplifying and taking expectation on both sides, up to the first order of approximation as,

(3.4)

is minimum for,

(3.5)

where,

and

The minimum MSE of the estimator, , for this optimum value of , is:

(3.6)

4. Efficiency Comparison

The proposed estimator performs better than the estimator in the sense having lesser mean squared error under the condition:

(4.1)

The proposed estimator in (3.1) will perform better than the estimator (2.3), under the condition if:

(4.2)

The proposed estimator has more efficiency as compared to all other estimators mentioned in this manuscript under the condition if:

(4.3)

5. Numerical Illustration

Following populations have been considered to examine the performances of different estimators of population variance,

Population I: Italian bureau for the environment protection-APAT Waste 2004

Y: Total amount (tons) of recyclable-waste collection in Italy in 2003.

X: Total amount (tons) of recyclable-waste collection in Italy in 2002.

Population II: Italian bureau for the environment protection-APAT Waste 2004

Y: Total amount (tons) of recyclable-waste collection in Italy in 2003.

X: Number of inhabitants in 2003.

Population III: Murthy (1967)

Y: Output for 80 factories in a region.

X: Fixed capital.

Population IV: Singh and Cahudhary

The population consists of 70 wheat farms in 70 villages in certain region of India and the variables under considerations are defined as:

Y = area under wheat crop (in acres) during 1974,

X = area under wheat crop (in acres) during 1973,

Table 1. Comparison of Bias and Mean square error of different estimators

6. Results and Conclusions

This paper deals with the estimation of population variance using improved ratio type estimator. An efficient estimator of population variance using coefficient of correlation and the inter quartile range of the auxiliary variable has been proposed. Up to the first degree of approximation, the expressions for the bias and mean square error of the proposed estimator have been obtained. The optimum value of the characterizing scalar kappa, which minimizes the mean squared error of the proposed estimator, is also obtained. Further the minimum value of the mean square error for this optimum value of kappa has also been obtained. It has been proved theoretically as well as empirically that the proposed estimator performs much better than all of the other mentioned estimators of population variance in the sense of having lesser Bias and MSE. It is of worth to be mentioned that the knowledge regarding the correlation coefficient should be available in advance. This knowledge of correlation coefficient is either available in advance (generally) or it is obtained from prior studies like pilot surveys etc. In case if we do not have prior knowledge of correlation coefficient, then it in the expression of estimator is replaced by its estimate and there is no effect on the mean squared error the estimator. Therefore it is strongly recommended that the proposed estimator should be preferred over the estimators mentioned in this manuscript for the estimation of population variance under simple random sampling scheme.

ACKNOWLEDGEMENTS

The authors are very much thankful to the editor in chief of AJOR and the anonymous learned referees for critically examining the manuscript and giving the valuable suggestions for further improvement in the earlier draft.