1. Introduction

In survey sampling, auxiliary information is used for enhancing the efficiency of the estimate of the parameters of the population for the characteristic under study. Auxiliary information is supplied by the auxiliary variable which is highly positively or negatively correlated with the main variable under study. Ratio type estimators are used when the variables x and y are positively correlated and the line of regression of y on x passes through origin, while the product type estimators are used when x and y are negatively correlated to each other otherwise regression estimators are used. In the present study, we are dealing with only the positive correlation.

Let be the n pair of observations for the auxiliary and study variables, respectively from the population of size N using simple random sampling without replacement. Let and be the population means of auxiliary and study variables respectively and and be the respective sample means.

Following are the notations which have been used in this manuscript and are already being discussed by Subramani and Kumarpandiyan (2015) as:

Size of the population

Size of the sample

Study variable

Auxiliary variable

Correlation coefficient between and

Population means

Sample means

Population variances

Sample variances

Coefficient of variations

Skewness of the auxiliary variable

Kurtosis of the auxiliary variable

First quartile of the auxiliary variable

Third quartile of the auxiliary variable

Inter quartile range of the auxiliary variable

Semi quartile range of the auxiliary variable

Semi quartile average of the auxiliary variable

Bias of the estimator

Variance of the estimator

Mean squared error of the estimator

Percentage relative efficiency of the estimator over

The appropriate estimator of the population variance is the sample variance defined as

(1.1)

which is an unbiased estimator of the population variance and its variance up to the first order of approximation is given as

(1.2)

where and

Isaki (1983) utilizing the auxiliary information, proposed the following classical ratio estimator for the population variance as

(1.3)

where

,

The expressions for the Bias and Mean Square Error (MSE) of the estimator in (1.3) up to the first order of approximation respectively are as

(1.4)

(1.5)

(Vide Kendall and Stuart (1977))

In the series of improvement, many authors have proposed different estimators of population variance utilizing the parameters of the auxiliary variable. Following table-1 which was also given by Subramani and Kumarpandiyan (2012) represents different estimators along with their bias, mean square error and constant.

Table 1. Bias, MSE and Constants of different estimators

Where symbols have their usual meanings. is the coefficient of variation of auxiliary variable, is the coefficient of kurtosis, are the quartiles of the auxiliary variable. and are the functions of quartiles defined by,

Thus, in general the mean square errors of the estimators given in above table may be written as,

(1.6)

The latest references in the series of improvement can be made of Shukla et al. (2015) and Yadav et al. (2015a, 2015b, 2016).

2. Proposed Estimator

Motivated by Prasad (1989) and Khan and Shabbir (2013), we propose an efficient ratio estimator of population variance as

(2.1)

where is a suitable constant to be determined such that the mean square error of is minimum.

In order to study the large sample properties of the proposed estimator, we define and with for

In case of simple random sampling without replacement, ignoring finite population correction term, the following expectations could be obtained either directly or by the method due to Kendall and Stuart (1977) as

Expressing in terms of we have

After simplifying and retaining terms up to the first order of approximation, we have:

Subtracting on both the sides, we obtain,

(2.2)

Taking expectation on both sides of (2.2), we have the bias of proposed estimator as:

(2.3)

where

The mean squared error of the proposed estimator is obtained by squaring both sides of (2.2), simplifying and taking expectation on both sides, up to the first order of approximation as,

(2.4)

is minimum for,

(2.5)

where,

and

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

(2.6)

Here, we pass an important remark that the value of the unknown parameters involved in and can be obtained from the previous surveys or the experience gathered in due course of time, for instance, vide Murthy (1967), Reddy (1973, 1974), Srivenkataramana & Tracy (1980), Singh & Vishwakarma (2008), Singh & Kumar (2008) and Singh & Karpe (2010). If the unknown parameters in and are replaced by their estimates, then we obtain the same minimum mean squared error as in (2.6).

3. Efficiency Comparison

From (2.6) and (1.2), we have:

(3.1)

From (2.6) and (1.5), we have:

(3.2)

From (2.5) and (1.6), we have:

(3.3)

4. Numerical Illustration

To justify the theoretical findings of different estimators, we have considered the following real populations.

Population-1: 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-2: 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-3: Murthy (1967)

Y: Output for 80 factories in a region.

X: Fixed capital.

Population-4: Singh and Chaudhary (1986)

5. Observation and Conclusions

Table-1 shows the Bias, MSE and Constants for previously existing estimators given by different researchers. Proposed estimator which has been subjected to comparisons with previous estimators nothing but is a ratio estimator. In this paper, we have been able to develop more efficient estimator whose bias is significantly far much less as compared to previous estimators as evidently presented in table-2. Similarly, MSE is also far much less as compared to previous estimators given in the table-2. Moreover, table-3 shows that relative efficiency is comparatively much higher. Thus, we can finally conclude with passing remarks that from theoretical discussions in section-3 and the results in table-2 and 3, we infer that the proposed estimator is much better than the previously existing estimators of population mean in simple random sampling scheme, therefore proposed estimator should be preferred for the estimation of population variance.

Table 2. Bias and Mean square error of different estimators

Table 3. Percentage Relative Efficiency (PRE) of different estimators with respect to

ACKNOWLEDGMENTS

The authors are very much thankful to the editor of American Journal of Operational Research and the anonymous referees for critically examining the manuscript and giving the valuable suggestions to improve the manuscript in present form.