Ratio Type Exponential Estimator for the Estimation of Finite Population Variance under Two-stage Sampling

In this study we propose an improved exponential ratio type estimator in estimating the finite population variance in two stage sampling under two cases: (i) when sum of the weights cannot equal to one and (ii) when sum of the weights are equal to one. The bias and Mean Squared Error (MSE) are derived up to first order of approximation. The efficiency conditions, under which proposed estimator is more efficient than the usual sample variance estimator and regression estimator are obtained. Numerical and simulated studies are conducted to support the superiority of the estimators. Real data set is used to observe the performances of estimator.


INTRODUCTION
In many situations, information on the auxiliary variable is required either at the designing stage or estimation stage or both stages, to increase precision of the estimators.Ratio, product and regression estimators are often used when advance knowledge of population mean of the auxiliary variable is readily available.
In application when sampling frame is not available, then it is expense or not feasible to obtain the sampling units directly from the population.Instead one can use the two stage sampling which is generally preferable for large scale surveys and at each stage frame is easily accessible.Mahalanobis (1967) was the first one, who introduced the procedure of two-stage sampling which was further extended by Godambe (1951).A better approach for multi-stage design was introduced by Saxena et al. (1984).Mostly research papers appeared in two-stage sampling for estimation of population mean or total including Yunusa (2010), Saini (2013) and Singh et al. (2013).Many researchers like Wolter (1985), Das and Tripathi (1978), Isaki (1983), Upadhyaya et al. (2004), Shabbir and Gupta (2007), Singh and Vishwakarma (2008) and Shabbir and Gupta (2010) worked in estimating the finite population variance by using the simple random sampling.To best of our knowledge very few research papers exist in estimating the population variance, so in this study an attempt has been made for estimation of finite population variance by using the auxiliary information in two stage sampling.
Assume that a sample of n psus is drawn from U and then a sample of ݉ from ‫ܯ‬ ssus units i.e., from the i-th selected psu using simple random sampling without replacement at both stages.We define the following relative error terms. where, The usual variance estimator for population variance in two-stage sampling is given by: The MSE of ܵ መ ሺଶ௦ሻ ଶ is given by: The traditional regression estimator for population variance under two-stage sampling is given by: where, b is the sample regression coefficient in two stage sampling.

PROPOSED ESTIMATOR
On the lines of Gupta and Shabbir (2008), we propose the following an improved exponential ratio type estimator for population variance (ܵ ௬ ଶ ) in two-stage sampling, given by: where, ݇ ଵ and ݇ ଶ are suitably chosen constants.We discuss the two cases: To find the properties of our proposed estimator (ܵ መ ሺଶ௦ሻ ଶ ), we consider the following two cases.
Using notations from above section, we have: To first order of approximation, we have: Using ( 6), the bias of ܵ መ ሺଶ௦ሻ ଶ to first order of approximation, is given by: Using ( 6), the MSE of ܵ መ ሺଶ௦ሻ ଶ to first order of approximation, is given by: where, Now differentiating (8) with respect to ݇ ଵ and ݇ ଶ , we get: Putting ݇ ଶ = 1 − ݇ ଵ in ( 7) and ( 8), we get the bias and MSE of ܵ መ ሺଶ௦ሻ * ଶ respectively as given by: The proposed estimator will be more efficient than the usual variance and regression estimators in two-stage sampling under two cases, when above Conditions (1)-( 4) are satisfied.

Numerical illustration:
We use the following real data sets to observe the performances of estimators.
Population: (Sarndal et al., 1992) y : Revenues from the 1985 municipal taxation x : 1975 population for M = 284 municipalities (ssus) divided into N = 50 clusters (psus) We use the following expression to obtain the Percent Relative Efficiency (PRE): where, (.For simulation study, we selected 10,000 independent first-stage samples of different sizes from a population.From every selected psus, a second-stage sample of different sizes, ssus was again selected.Thus, we had 10,000 independent samples each of different sizes.For each sample from 1 to 10,000, values of the estimators were computed and then on the basis of these values simulated MSE of different estimators were calculated.Percentage Relative Efficiency (PRE) of different estimators with respect to ܵ ଶ௦ሺሻ ଶ for different sample sizes are given in Table 1.

CONCLUSION
We proposed an improved exponential ratio type estimator for population variance in two stage sample under two cases: • When sum of the weights cannot equal to one • When sum of the weights are equal to one Percentage Relative Efficiency (PRE) of different estimators for different sample sizes are given in Table 1.Both numerical and simulation studies show the same behavior of results.
From Table 1, we observed that the proposed estimator (ܵ መ ሺଶ௦ሻ

Table 1 :
Percent relative efficiency of different estimators with respect to ܵ መ ሺଶ௦ሻ