Variance Estimation Using Linear Combination of Tri-mean and Quartile Average

In this paper, we have proposed a class of modified ratio type variance estimator for estimation of population variance of the study variable, when Tri Mean and Quartile average of the auxiliary variable are known. The bias and mean square error (MSE) of the proposed estimator are obtained. From the numerical study it is observed that the proposed estimator performs better than the existing estimators in the literature.


Introduction
The use of an auxiliary variable x in the estimation of the finite population total or mean of a characteristic y is a common occurrence in practice. The theory of basic sample survey as available in standard text books on sampling deals with the case which comprises linear estimators such as mean, total, proportion and simple sample design using simple random sampling with replacement. It continues to supply new and improved procedures for estimation of variances assuming independence of observations. Ratio, difference and regression estimators utilize an auxiliary variable for more efficient estimation of the parameter in question. Such estimators take advantage of the correlation between the x variable and the characteristic y. In a similar manner, then, it seems reasonable that under suitable conditions efficient estimation of the variance of the estimator of the finite population total or mean of the characteristic y is also possible using such estimation techniques. With the increasing growth in the number and diverse uses of sample surveys worldwide, it is often desired to analyze and interpret the resulting voluminous data by swifter methods (Cochran (1970) on the basis of random sample selected from the population U. Sometimes in sample surveys along with the study variable Y, information on auxiliary variable X, which is positively correlated with Y, is also available. The information on auxiliary variable X, may be utilized to obtain a more efficient estimator of the population. In this paper, our aim is to estimate the population variance on the basis of a random sample of size n selected from the population U. Ogus and Clark (1971) proposed the use of ratio or difference estimators of the variance under a passion sampling design (a design in which each sampling unit is given an independent chance of being selected into the sample without replacement) for the purpose of reducing the effect of the random sample size on the variance estimator. Under a sample design in which one sample unit is selected in each stratum with probability proportional to size (PPS). Shapiro and Bateman (1978) considered reducing the bias of the estimator of the variance in a one-per stratum design by using a variance estimator as the Yates Grundy variance estimator for a two sample units per stratum design with joint inclusion probabilities. The problem of constructing efficient estimators for the population variance has been widely discussed by various authors such as Isaki (1983), who proposed ratio and regression estimators. Prasad and Singh (1990) considered a ratio type estimator for estimating population variance by improving Isaki's (1983) estimator. Gupta and Shabir (2008) proposed a new hybrid class of estimators and showed that in some cases their efficiency is better than the traditional ratio estimators, where as Kadilar & Cingi (2006,2007  proposed a modified ratio estimator using non-conventional location parameters because these parameters take care of outliers in the data. Outliers can be generated by from a simple operational mistake to including small sample from a different population, and they make serious effects of statistical inference. Even one outlying observation can destroy least squares estimation, resulting in parameter estimates that do not provide useful information for the majority of the data.

Ratio Type Variance Estimator Proposed by Isaki (1983)
Isaki (1983) suggested a ratio type variance estimator for the population variance 2 Y S when the population variance 2 x S of the auxiliary variable X is known. Its bias and mean square error are given by Here γ is the sampling fraction and λ 22 is covariance between the study and auxiliary variable. Kadilar and Cingi (2006) suggested four ratio type variance estimators using known values of C.V. and coefficient of kurtosis of an auxiliary variable

Recent Developments
Subramani and Kumarapandiyan (2015) proposed a generalized modified ratio type estimator for estimating population variance using the known parameters of the auxiliary variable and their estimator is given as The bias and MSE of the estimator in (6) as suggested by the Subramani and kumarapandiyan (2015) is given as When the study variable Y and the auxiliary variable X are negatively correlated and the population parameters of the auxiliary variable are known, Subramani & Kumarapandiyan (2015) proposed the following generalized modified product type variance estimator as  Isaki (1983) estimator.

Proposed Estimator
The accuracy of the variance estimator is an important criterion for the choice of a variance estimator and it can be assessed by mean square error (MSE) of the variance estimator that has the best statistical properties for the proposed analysis and interpretation of data. Compromises may have to be made because different analysis of the same data may suggest different variance estimator. In short, it is, therefore seen that the process of evaluating alternative variance estimator and selecting a specific estimator for use in particular complex survey in a difficult and knotty problem, involving both subjective and objective elements. Keen judgment in making intelligent compromises between the consideration of accuracy, cost, timeliness and simplicity is required. The performance of the estimator of the study variable can be improved by using known population parameters of an auxiliary variable, which are positively correlated with a study variable. We have proposed a new modified ratio type variance estimator of the auxiliary variable by using linear combination of Tri-mean and population semi inter quartile average of the auxiliary variable. It is highly sensitive to outliers as its design structure is based on only extreme values of the data (for more details see Ferrell (1953)).
We have derived here the bias and mean square error of the proposed estimator 2 SJ S to first order of approximation as given below. The Taylor series linearization approach derives a linear approximation to the survey estimator and then obtains variance estimate for this linear combination taking account of the sampling design.
Squaring both sides of (8) and (9), neglecting the terms more than 2 nd order and taking expectation, we get 2  The bias and mean square error of the above estimator in (7) after simplification is given as

Numerical Illustration
The performance of the proposed estimator is assessed with that of simple random sampling without replacement (SRSWOR) sample variance and existing estimators. We use the data of Murthy (1967) page 228 in which fixed capital is denoted by X(auxiliary variable) and output of 80 factories are denoted by Y(study variable). We apply the proposed and existing estimators to this data set and the data statistics are given below: The results obtained are shown in Table 1.

Conclusion
The paper proposes a ratio type variance estimator using known values of an auxiliary variable. From Table 1 we see that the bias and MSE of the existing estimators ranges from 6.12 to10.87 and 3180 to 3925 respectively, while as the proposed estimator suggested by Maqbool & Shakeel has a bias of 3.02 and MSE 2820.06 which show that the bias and mean square error are less than the already existing estimators in the literature and the percentage relative efficiency comes out be 139.18%. Hence the proposed estimator may be preferred over existing estimators for use in practical applications. In future work, we hope to adopt the estimator presented here to stratified random sampling and hope to develop a variance estimator using two auxiliary variables.