A Family of Difference-cum- Exponential Type Estimators for Estimating the Population Variance Using Auxiliary Information in Sample Surveys

Using auxiliary information, a family of difference-cum-exponential type estimators for estimating the population variance of variable under study have been proposed under double sampling design. Expressions for bias, mean squared error and its minimum values have been obtained. The comparisons have been made with the regression-type estimator by using simple random sampling at both occasions in double sampling design. It has also been shown that better estimators can be obtained from the proposed family of estimators which are more efficient than the linear regression type estimator. Results have also been illustrated numerically as well as graphically


Introduction
In survey sampling, taking the advantage of the correlation between the auxiliary variable x and study variable y, the construction of efficient ratio and product type estimators for the population variance has widely been made by many statisticians to assess the variations in population. Because it is well known that the use of auxiliary information in the construction of estimators can increase the efficiency of the estimators of parameter of interest. When the population variance of auxiliary variable x, which is highly correlated with the study variable y, is known in advance, several estimators have been defined by different authors such as Das and Tripathi (1978), Isaki (1983), Ahmed et al.(2003), Jhajj et al.(2005Jhajj et al.( ,2011, Kadilar and Cingi (2006), Pradhan et al.(2010) in the literature for estimating the population variance of study variable y. Sometimes information on population variance of auxiliary variable x is not known in advance then we generally use two phase (double) sampling design. In the two-phase sampling design, a large preliminary random sample (called first phase sample) is drawn and auxiliary information are taken on sample units which are used to estimate the value of unknown population variance of auxiliary variable x. Then second phase sample is drawn either from the first phase sample or independently from the population and observations are taken on study as well as auxiliary variables.
In present paper we have proposed a family of Difference-Cum-Exponential type estimators for estimating the population variance of study variable using auxiliary information under double sampling design. The efforts have been made to compare the proposed family of estimators under simple random sampling design as a special case.

Notations and Results
A preliminary large random sample (first phase sample) of size n' is drawn from a finite population of size N and both auxiliary variable x and study variable y are measured on it. Then second phase random sample of size n(<n') is drawn from the first phase sample.

The Proposed Family of Estimators and Its Results
When we don't have an information about any parameter of the auxiliary variable, then we propose a family of estimators of the population variance of the study variable y under the sampling design defined in section 2 as (3.1) Where are any constants.
To obtain the bias and mean square error of estimator , upto first order of approximation, we expand in terms of and retaining terms upto second degree of approximation.
(3.2) Noting that expectation exists and finite, we take expectation in (3.2) and using the results of section 2, we have which implies that From (3.11), we see that will vary with the change in variation of θ so range of variation of θ can be obtained at which proposed family of estimator is better than the existing ones.

Comparison
For comparing the proposed family of estimators with the linear regression type estimator considered by we first obtain expressions of its bias and mean square error, under double sampling design using simple random sampling at both phases, upto first order of approximation, (4.1) Using (3.11) and (4.1) and after some algebra, we obtain

Numerical Illustration
For comparing the efficiency of the proposed estimator with Regression-type estimate under double sampling design, we take the empirical population considered in literature (source: Institute of Statistics, Republic of Turkey). This empirical data concerns the level of apple production (1 unit=100 tonnes) as the variate of interest and number of apple trees (1 unit=100 trees) as the auxiliary variate in 106 villages in the Marmarian Region respectively in 1999. The values of population parameters obtained are given in Table 1. The biases, mean squared error and relative efficiency of proposed estimator w.r.t regression-type estimator are given for some different values of θ in table 2. representation also predicts that there is gain in efficiency of proposed estimator over the Regression-type estimator for 0 < θ < 2. Hence we conclude that the proposed estimator will always be better than existing Regression-type estimator under double sampling design for 0 < θ < 2