Improved Estimator of Population Variance using Measure of Dispersion of Auxiliary Variable

This research study is designed to obtain a more precise class of estimators of a population variance by taking advantage of relation between auxiliary variable and study variable. Here a class of new modified ratio type estimators of population variance by using coefficient of variation (CV), standard deviation, mean and median of auxiliary variable. Further empirical study is made to compare bias and mean square error (MSE) of proposed estimators with the existing estimators. Expressions for bias and MSE are obtained. Few secondary data sets are used to check the efficiency of proposed estimators of population variance. Article history Received: 29 March 2018 Accepted: 14 June 2018


Introduction
In our everyday life variations are available all over the place.It is the idea of 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 learning of the level of varieties in individuals' response a maker need to decrease or increase cost of his item, or make strides the nature of his item.A doctor needs a full comprehension of varieties in the body temperature, level of human circulatory strain and heartbeat rate for full remedy.In this article we estimate one of the measure of variation which is known as variance.
The following Notations are used throughout this paper.
The improved ratio type variance estimators discussed above are biased but have minimum mean square errors compared to the old ratio type variance estimator.The list of estimators given in table 1 uses the known values of the parameters like S 2 χ , C χ , b 2 , median and their linear combinations.The materials of the present study are arranged as given below.The proposed estimators with known population C χ , S χ , mean, median are presented in section 2 and the condition in which the proposed estimator performs better than the existing estimators are derived in section 3. The performances of the proposed and the existing estimators are measured for certain natural populations in section 4 and conclusion is presented in section 5.

Proposed Estimators
The accurateness of the estimator may be increased by using the supporting evidence in ratio type variance estimator.Estimators depend on the population characteristics in applied field.We use ratio estimator to estimate the values of the population variance by using different characteristics like kurtosis, skewness, mean, median, quartiles.deciles , percentiles and coefficient of variation etc. Supporting evidences are utilizedin sampling survey to improve the estimate of S 2 y .In current section, we have proposed a class of ratio type variance estimator utilizing different parameters of auxiliary variate.

The new and improved class of ratio type variance estimators for population variance S 2
y is defined as  The theoretical properties of after some simple algebra are derived as given below:  3, shows that the B(.) of the new class of variance estimators is less as compare to the B(.) of the reviewed estimators.Similarly,numerical calculations of Table 4, shows that the MSE(.) of the new class of variance estimators is less as compare to the B(.) of the reviewed estimators.

Conclusions
In this Study we have defined improved some new variance estimators by utilizing coefficient of variation, standard deviation, mean and M d of supplementary variate.The B(.) and MSE(.) of the new variance estimators are obtained and compared with that of existing improved ratio type variance estimators.Further we have derived some theoretical conditions for which 2 ˆNKi S estimators are performing much better as compare to the reviewed estimators.So on behalf of the results of population 1, 2, 3, 4 and 5 we claim that the proposed-ratio-type variance estimators are competent as compare to existing ratio type variance estimators.

y.
"N" : Size of Population "n" : Size of Sample γ : 1/n Y : Variate of interest X : Supplementary variate X : A.M of Supplementary variate Y : A.M of Variate of interest ȳ : Sample A.M of Variate of interest x : Sample A.M of Supplementary variate ρ : Correlation coefficient 2 y S : Variance of Variate of interest 2 x S : Variance of Supplementary variate 2 y s : Sample Variance of Study variable 2 x s : Sample Variance of Auxiliary variable y C : CV of Study variate x C : CV of Supplementary variate Md : Population Median of Auxiliary variable B(.) : Bias of the estimator MSE(.) : Mean squared error of the estimato b 1 = Coefficient of skewness b 2 = Coefficient of Kurtosis 2 ˆKCi S : Existing improved ratio type variance estimators of S 2 y by Kadilar &Cingi, (2006a) and SK (2012) proposed a class ofratio type variance estimators for the population variance S 2 It is assumed that variance of supplementary variable (S 2 x )is known.Family members of KC(2006) and SK (2012) utilizing supplementary information with their theoretical properties such as bias and mean squared error are given below.

.
Estimators As mentioned the B(.) and MSE(.) of KC (2006) variance estimators are given below: ....(5) ....(6) As mentioned the B(.) and MSE(.) of SK (2012) variance estimator are given below: ....(7) ....(8) The B(.) and MSE(.) of proposed class ( 2 ˆNKi S ) are derived as given below.....(9) ....(10) From the expression given in (6) and (8) we have derived the condition for which the Subramani and Kumarapandiyan (2012) estimator 2 ˆNKi S is more efficient than Kadilar and Cingi (2006) estimators and it is given below if From the (8) and (10) we have obtained the theoretical condition for which the proposed estimators 2 ˆNKi S are more efficient than SK (2012) estimator 2 ˆNKi S and it is given below If numerical study The performance of the proposed ratio type variance estimators are evaluated with that of existing modified ratio type estimators listed in Table 1 for certain natural population.The population 1 has taken from Singh and Chaudhary (1986, page 141), population 2 has taken from Cochran (1977, page 151), population 3 is real data set taken from Bureau of statistics.The data is about area and production of sugarcane in the districts of Punjab.Where Y= Production of sugar cane in Punjab in 2008-2009 and X= Area of sugar cane in Punjab in 2007-2008, Population 4 and 5 are real data set taken from the Report on Waste 2004 drew up by the Italian bureau for the environmental-protection.2003X=Number of inhabitants in 2003.The population parameters and the constants computed from the above populations are given below The B(.) and MSE(.) of the reviewed and proposed improved ratio type variance estimators are given in the following Tables: Numerical results of Table