Generalized Modified Ratio Type Estimator for Estimation of Population Variance

In this paper a generalized modified ratio type estimator for estimation of population variance of the study variable using the known parameters of the auxiliary variable has been proposed. The bias and mean squared error of the proposed estimators are derived. It has been shown that the ratio type variance estimator and existing modified ratio type variance estimators are the particular cases of the proposed estimators. Further the proposed estimators have been compared with that of the existing (competing) estimators for simulated data and two natural populations


Introduction to the Research Problem
When there is no auxiliary information available, the simplest estimator of population variance is the sample variance obtained by using simple random sampling without replacement (SRSWOR). Sometimes in sample surveys, along with the study variable , information on auxiliary variable , which is positively correlated with , is also available. This information on auxiliary variable may be utilized to obtain a more efficient estimator of the population variance. Ratio method of estimation is an attempt in this direction. This method of estimation may be used when (i) represents the same character as , but measured at some previous date when a complete count of the population was made and (ii) Any other character which is closely related to the study variable and it is cheaply, quickly and easily available (see page 77 in Gupta and Kabe (2011)).

Statement of Problem
Consider a finite population * + of distinct and identifiable units. Let be a study variable with value measured on giving a vector of values * +.
The problem is to estimate the population variance ( ) ∑ ( ̅ ) on the basis of a random sample of size , selected from the population with some desirable properties like:  Unbiasedness / Minimum Bias  Minimum Variance / Mean squared error

Notations
The notations to be used in this article are described below: In the case of simple random sampling without replacement (SRSWOR), the sample variance is used to estimate the population variance which is an unbiased estimator and its variance is given below: ( ) ( ( ) ) (1)

Ratio type estimator for estimation of population variance
Isaki (1983) suggested a ratio type variance estimator for the population variance when the population variance of the auxiliary variable is known. The estimator together with its bias and mean squared error are given below: where ( ) ( )

Existing modified ratio type estimators for estimation of population variance
The ratio type variance estimator given in (2)

Motivations and Investigations
Moving along this direction we intend in this paper to show the problem of estimating the population variance of a study variable can be treated in a cohesive framework by defining a class of estimators which may or may not be biased and covers many that are present in the literature. The bias and mean squared error of the class are obtained. The aim is to avoid the large number of estimators that appear different from each other but, as a matter of fact, can be included in the class and therefore, their efficiency is known in advance. In this paper an attempt has been made to suggest a generalized modified ratio type estimator for estimating population variance using known parameters of the auxiliary variable and its linear combination. The materials of the present work are arranged as given below. The proposed estimators using known parameters of the auxiliary variable are presented in section 2 whereas the proposed estimators are compared theoretically with that of the SRSWOR sample variance, ratio estimator and existing modified estimators in section 3. The performance of the proposed estimators with that of the ratio and existing modified ratio estimators are assessed for certain natural populations in section 4 and the conclusion is presented in section 5

Generalized Modified Ratio Type Estimator
In this section, a generalized modified ratio type estimator using the known parameters of the auxiliary variable for estimating the population variance of the study variable has been suggested. The proposed modified ratio type estimator ̂ for estimating the population variance is given below: The bias and mean squared error of the proposed estimators ̂ have been derived (see Appendix B) and are given below: where Remark 2.1: When the study variable and auxiliary variable are negatively correlated and the population parameters of the auxiliary variable are known, the following generalized modified product type variance estimator can be proposed: Remark 2.2: When in (5), the proposed estimator ̂ reduces to ratio type estimator ̂ suggested by Isaki (1983). Remark 2.3: When the proposed estimator ̂ reduces respectively to the existing estimators ̂ listed in Table 1.

Efficiency of the Proposed Estimators
The mean squared error of the modified ratio type estimators ̂ given in Table 1 (Appendix A) are represented in single class as given below: Comparing (1) and (7) we have derived (see Appendix C) the condition for which the proposed estimator ̂ is more efficient than the SRSWOR sample variance and it is given below: Comparing (4) and (7) we have derived (see Appendix D) the conditions for which the proposed estimator is more efficient than the ratio type estimator and it is given below: Comparing (7) and (9) we have derived (see Appendix E) the conditions for which the proposed estimator is more efficient than the modified ratio type variance estimator ̂ respectively and it is given below: Let us consider the lower limit point as and upper limit point as in (12). At the average of limit points, ( ), the proposed estimator always performs better than the existing estimators. That is,

Numerical Study
The performance of the proposed modified ratio type estimators for variance are assessed with that of SRSWOR sample variance, ratio type estimator and existing modified ratio type variance estimators for two natural populations. The population 1 is taken from Singh and Chaudhary (1986, page141) and the population 2 is taken from Murthy (1967, page 228). The population parameters of the above populations are given below: Variance of SRSWOR sample variance and Mean Squared Error of the ratio type estimator for the two populations are given below: Further to show the efficiency of the proposed estimators (p), the Percent Relative Efficiencies (PREs) of the proposed estimators with respect to the existing estimators (e) given in Table 3 are computed by using the formula given below:  Table 3, it is observed that the proposed estimators are performed better than the existing estimators. In fact PRE of the proposed estimators varies from 130.48 to 131.10 for population 1 and from 100.13 to 130.24 for population 2. Hence one may conclude from the numerical comparison that the proposed estimators are more efficient than the existing estimators.

Simulation study
However to assess more about the efficiency of the proposed estimators, we have undertaken a simulation study as given below: We generate values ( ) from a Bi-variate normal distribution with means (50, 50) and standard deviation (10, 10). The correlation coefficient is fixed at values 0.90 and 0.95. Simple random sampling without replacement has been considered for sample size . Since the PREs are independent to the sample size we have restricted the simulation study for sample size 20 only. Variance of SRSWOR sample variance and Mean Squared Error of the ratio type estimator for simulated data are given below:  shows that the proposed estimators are more efficient than the existing estimators.
In order to show the performances of the proposed estimators graphically, we have simulated 200 samples 1000 times and repeated the same procedures 10 times and shown the average MSE values of the estimators in the below tables and graphs. We have considered only three estimators namely ̂ ̂ and ̂ for comparison with the proposed estimators.

Conclusion
In this paper a generalized modified ratio type estimator for estimating population variance using the known parameters of the auxiliary variable has been proposed. The bias and mean squared error of the proposed modified ratio type estimators are derived. Further it has been shown that ratio and existing modified ratio type estimators are the particular cases of the proposed estimators. We have also assessed the performances of the proposed estimators with that of the existing estimators for simulated data and two natural populations. It is observed from the numerical comparison that the mean squared error of the proposed estimators is less than the mean squared error/variance of the existing (competing) estimators. Hence we strongly recommend that the proposed modified ratio type estimators for the use of practical applications for estimation of population variance.  The proposed estimator ̂ is given below: Expanding and neglecting the terms more than 3 rd order, we get ̂ ̂ (A) By taking expectation on both sides of (A), we get Squaring both sides of (A), neglecting the terms more than 2 nd order and taking expectation, we get: