A Ratio-cum-Dual to Ratio Estimator of Population Variance Using Qualitative Auxiliary Information Under Simple Random Sampling

In this paper we have proposed a class of ratio-cum-dual to ratio estimators for estimating population variance of the variable under study, using known values of some population parameters of auxiliary variable, which is available in the form of an attribute. The expressions for the bias and mean squared error of the proposed estimators have been derived upto the first order of approximation. A comparison has been made with some well known estimators of population variance available in the literature when auxiliary information is in qualitative form. It has been shown that the proposed estimator is better than the existing estimators under the optimum condition. For illustration an empirical study has been carried out.


INTRODUCTION
T he use of auxiliary information increases the precision of an estimator when study variable Y is highly correlated with auxiliary variable X.When the variable under study y is highly positively correlated with the auxiliary variable x, then the ratio type estimators are used to estimate the population parameter and product estimators are used when the variable under study y is highly negatively correlated with the auxiliary variable x for improved estimation of parameters of variable under study.But there are situations when information on auxiliary variable is not available in quantitative form but in practice, the information regarding the population proportion possessing certain attribute ψ is easily available (see Jhajj et.al.), which is highly correlated with the study variable Y.For example (i) Y may be the use of drugs and ψ may be the gender (ii) Y may be the production of a crop and ψ may be the particular variety.(iii) Y may be the Amount of milk produced and ψ a particular breed of cow.(iv) Y may be the yield of wheat crop and ψ a particular variety of wheat etc. (see Shabbir and Gupta).
Let there be N units in the population.Let ( , ) y i i ψ , i = 1,2,----,N be the corresponding observation values of the i th unit of the population of the study variable Y and the auxiliary variable ψ respectively.Further we assume that ψ i = 1 and ψ i = 0, i = 1,2,----,N if it possesses a particular characteristic or does not possess it.Let Ratio type estimator for population variance of the study variable based on ISAKI (1983) estimator is as follows where The mean squared error (MSE) of the estimator t R , up to the first order of approximation is given as A dual to ratio type estimator for the population variance of the study variable y is defined as

PROPOSED ESTIMSTOR
Motivated by Sharma & Tailor, we propose the following ratio-cum-dual to ratio type estimator for the population variance of the study variable as Where α is a suitably chosen constant to be determined such that MSE of the estimator t R d is minimum.For α = 1, the estimator t R d reduces to the estimator t R and for α = 0 it is the estimator t d ( ) .
Thus the above estimators are the particular case of the proposed estimator t R d .
In order to study the large sample properties of the proposed family of estimators, we define s S y y Expanding terms on right hand side of (2.2), we get Taking expectation, after subtracting S y 2 on both sides, upto the first order of approximation, we get where α α From equation (2.3), we have Taking expectation on both sides of (2.5), we get the MSE of t R d , upto the first order of approximation as The MSE of the estimator t R d is minimized for the optimum value of α as

EFFICIENCY COMPARISON
It is well known that the variance of estimator t s y 0 2 = is given as proportion of units in the population and sample respectively possessing the attribute ψ.Let a simple random sample of size n from this population is taken without replacement having sample values ( , ) to the variance of the linear regression estimator of population variance, To analyze the performance of various estimators of population variance 2 y S of study variable y, we have considered the following two data: Data [Source: Mukhopadhyay, page 44] Y : Household size.ψ : Whether households have agricultural loan.