Introduction

In our everyday life variations are available all over the place. It is the idea of Estimating of these restricted population variance (variation) has enormous significance in various fields such as manufacturing, cultivation, health and natural sciences where we come across the populations which are expected to be skewed.law that people or no two things are precisely same. For example, an agriculturist needs a sufficient comprehension of the varieties in climatic factors particularly from place to place (or time to time) to have the capacity to anticipate when, how and where to plant his yield. For consistent information of the level of variations in individuals' response a maker need to lessen or increment cost of his item, or make strides the nature of his item. A doctor needs a full comprehension of variations in the body temperature, level of human circulatory strain and heartbeat rate for full medicine.

Variation is at hand everywhere in our day to day life. It is law of natural world that no two things or individuals are closely alike. For instance, a medical doctor needs a full understanding of dissimilarity in the degree of human blood stress, body temperature and beat rate for sufficient prescription (Singh 2005).

Simple Random Sampling With Out Replacement    Sample Vriance

In the case of simple random sampling without replacement sample variance sy2 is used to estimate the population variance Sy2  which is an unbiased estimator and variance is given below:

Ratio Type Estimation For Estimation Of Population Variance

Isaki (1983) planned the ratio type variance estimator for the population variance Sy2 when the population variance Sx2 of the auxiliary variable X is known the estimator together with its bias, mean square error given below:

The ratio type variance estimator used to improve the precision of the estimate of the population variance compared to SRSWOR sample variance. Further improvements are also achieved on the ratio estimator by introducing a number of modified ratio estimators with the use of known parameters like Median, Quartiles and Coefficient of correlation. The problem of constructing efficient estimators for the population variance had been widely discussed. For the purpose of this study we reviewed the estimators developed by  Subramani and Kumarapandiyan, (2012a, 2012b, 2012c)  in Table 1. Further, interested readers see; Shahzad (2016) and Shahzad et al. (2017).

Table 1:  Bias, Mean Squared error of the Existing Estimators Click here to view table

Material and Methods

Assume a sample with size n from a population with size N, selected by a precise sampling design. Let Y be the variable which is the entity of study and X, the available auxiliary variable. For a condition in which the population means, X is available, some estimators of the population variance Y had been planned. We have considered variance ratio method, for estimating a population variance. Selecting sample according to simple random sampling, and we have proposed a general class of estimators. The presentation properties of the planned estimators are analyzed with respect to the bias, mean squared error criteria using asymptotic theory, and we find the most favorable values in each planned class. The planned estimators are legitimated, advanced on the usual estimators reducing the errors obtained.

Notations

The following notation are used for numerical illustrations

proposed class. For estimating the population variance the first degree of approximation is used, the proposed estimators, bias, constant and mean squared error are given below:

Proposed Estimators

Taking motivation from Subramani and Kumarapandiyan, (2012a, 2012b, 2012c), we propose the following estimators

1^st proposed estimator

proposed estimator we used ^st1Md,Q1and ρ.linear combination of

2^nd proposed estimator

proposed estimator we used ^nd2Md,Q2and ρ.linear combination of

3^rd  proposed estimator

3^rd proposed estimator we used linear combination of  Md,Q2and ρ

5^th  proposed estimator

Empirical study

Population

In the first population, the mean of the auxiliary variable X =11.2646 and the standard deviation Sx=8.4563 respectively.

Auxiliary variable and study variable are highly correlated with ρ=0.9413. Both the variable contains 80 units. We calculated X elements on the auxiliary characteristic and Y elements on the study characteristics. Another fact of interest is that in our population the efficiency gain achieved by the proposed estimators. The Population is taken from the Murthy (1967, Page 228). Descriptive statistics, constants, bias, mean squared error are given below:

Table 3.1: Population Characteristics and Values of Data 1 Click here to view table

Table 3.2: Constant of the Existing Estimators and Proposed Estimators Click here to view table

Table 3.3: Biases of the Existing Estimators and Proposed Estimators Click here to view table

Table 3.4: Mean Squared Error of the existing and proposed estimators Click here to view table

The characteristics, constants, bias and mean squared error of the proposed and existing estimators are given in table 4.1, 4.2, 4.3, 4.4 respectively. We use the linear combination of measures of location for numerical illustrations. However these all existing ratio variance estimators are biased but have minimum values of bias and mean squared error as compared to classical ratio estimators.

New modified variance ratio estimators introduced by using linear combination of measures of location shows better results than the existing modified variance ratio estimators.

Population

In the 2^nd population, the mean of the auxiliary variable X =46.37 and the standard deviation Sx  =25.4 respectively.

Auxiliary variable and study variable are highly correlated with ρ=0.9773. Both the variable contains 33 units. We calculated X elements on the auxiliary characteristic and Y elements on the study characteristics. Another fact of interest is that in our population the efficiency gain achieved by the proposed estimators. The Population Data 2 is taken from Government of Pakistan Statistics Division Federal Bureau of Statistics (Economic Wing) Islamabad (AREA & PRODUCTION OF FOOD CROPS IN PUNJAB, Area in "000" Hectares, Production in "000" Tones, Barley) from 1981-2014.Descriptive statistics, constants, bias, mean squared error are given below:

Table 3.5: Population Characteristics and Values of Data 2 Click here to view table

Table 3.6: Constant of the Existing Estimators and Proposed Estimators Click here to view table

Table 3.7: Biases of the Existing Estimators and Proposed Estimators Click here to view table

Table 3.8: Mean Squared Error of the existing and proposed estimators Click here to view table

The characteristics, constants, bias and mean squared error of the proposed and existing estimators are given in table 4.5, 4.6, 4.7, 4.8 respectively. We use the linear combination of measures of location for numerical illustrations. However these all existing ratio variance estimators are biased but have minimum values of bias and mean squared error as compared to classical ratio estimators till date no attempt has been made to utilized the linear combination of measures of location.

New modified variance ratio estimators introduced by using linear combination of measures of location shows better results than the existing modified variance ratio estimators.

Note that, on replacing the unknown population quantities in the optimum values of constants of an estimator of interest with their respective consistent estimators based on the same sample, the efficiency of the estimator of interest remains the same, up to first order of approximation.

Summary and Conclusion

This Research proposed ratio type variance estimator by using a known linear combination of median quartile and correlation coefficient of an auxiliary variable. The bias, mean squared error of the planned estimator were obtained and compared with the typical ratio kind and obtainable modified ratio kind variance estimators. Further the conditions for which the planned estimator is more capable than the conventional and accessible estimators were derived. The performance of the planned estimator was experienced using five known populations. Results explain that the bias, mean squared error of the planned estimator are lesser than the biased, mean squared errors of the conventional and existing estimators for the known populations measured. Based on results, the planned modified ratio type variance estimator may be favored over conventional ratio kind and obtainable modified ratio type variance estimators for the use in realistic applications.

References

1. Cochran, W. G. 1977. Sampling Techniques, Third Edition, Wiley Eastern Limited. Murthy, M. N. 1967. Sampling theory and methods, Statistical Publishing Society, Calcutta, India.
2. Isaki, C. T. 1983. Variance estimation using auxiliary information, Journal of American Statistical Association 78, 117–123.
3. Subramani, J. and G. Kumarapandiyan, 2012a. Estimation of population mean using coefficient of variation and median of an auxiliary variable. International Journal of Probability and Statistics, 1(4): 111-118.
4. Subramani, J. and G. Kumarapandiyan, 2012b. Variance estimation using median of the auxiliary variable. International Journal of Probability and Statistics, 1(3): 36-40.
5. Subramani, J. and G. Kumarapandiyan. 2012c. Variance estimation using quartiles and their functions of the auxiliary variable. International Journal of Statistics and Applications, 2(5):67-72.
6. Singh, H. P. 2005. Estimation of finite population mean using known correlation coefficient between auxiliary characters. Statistica, 65(4):407–418.            
7. Shahzad, U., 2016, On the Estimation Of Population Mean Under Systematic Sampling Using Auxiliary Attributes, Oriental Journal Physical Sciences, 1 (1 & 2):17-22.
8. Shahzad, U., Hanif, M., Koyuncu, N., Garcia Luengo, A.V., 2017, Estimation of population  variance using quartiles in absence and presence of non-response, Gazi University Journal of Science, 30 (2):205-218.