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

Muhammad Khalil , Muhammad Ali , Usman Shahzad* , Muhammad Hanif and Nasir Jamal

1Pir Mehr Ali Shah Arid Agriculture University, Rawalpindi, Pakistan .

DOI: http://dx.doi.org/10.13005/OJPS03.01.06

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.


Bias; Mean Squared Error; Natural Populations; Simple Random Sampling

Copy the following to cite this article:

Khalil M, Ali M, Shahzad U, Hanif M, Jamal N. Improved Estimator of Population Variance Using Measure of Dispersion of Auxiliary Variable. Orient J Phys Sciences 2018;3(1).

DOI:http://dx.doi.org/10.13005/OJPS03.01.06

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Khalil M, Ali M, Shahzad U, Hanif M, Jamal N. Improved Estimator of Population Variance Using Measure of Dispersion of Auxiliary Variable. Orient J Phys Sciences 2018;3(1). Available from: https://bit.ly/3rswZ17

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Article Publishing History

Received: 29/3/2018
Accepted: 14/6/2018

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.

KC (2006) and SK (2012) proposed a class of ratio type  variance estimators for the population variance S2y. It is assumed that variance of supplementary variable (S2x ) 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.

Table 1: Reviewed variance estimators with their theoretical properties

 

Click here to view table

 

 

The purpose of above mentioned ratio estimators is to reduce the variance of estimates when there exist a positive relation between X and Y. The problem of estimating  has been extensively discussed by many authors namely Isaki5,  Singh et al.3, Cebrian and Garcia.3, Ahmad et al.1, Singh & Singh.11 , KC.7, Singh & Singh.11,Shabbir & Gupta.10,   KC.6, Gupta and Shabbir4, Singh and Solanki.12, S.K17,18 ,Khan and Shabbir8 and Yadav et al.20,21,22 etc.

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 Sx2, Cx, B2 median and their linear combinations. The materials of the present study are arranged as given below. The proposed estimators with known population Cx, Sx 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 utilized in sampling survey to improve the estimate of S2y. 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 S2y is defined as  

Theoretical Comparisons of Proposed

Estimators

As mentioned the B(.) and MSE(.) of KC (2006) variance estimators are given below:

 As mentioned the B(.) and MSE(.) of SK (2012) variance estimator are given below:

The B(.) and MSE(.) of proposed class (S2NKi ) are derived as given below 

From the expression given in (6) and (8) we have derived the condition for which the   Subramani and Kumarapandiyan (2012) estimator 

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.

APAT Where Y=Total amount (tons) of recyclable-waste collections in Italy in 2003 X= amount (tons) of recyclable-waste collection in Italy in 2002 and Y=Total amount (tons) of recyclable-waste collections in Italy in 2003 X=Number of inhabitants in 2003. The population parameters and the constants computed from the above populations are given below

Table 2: Decriptive of the considered Populations

Click here to view table

 

The B(.) and MSE(.) of the reviewed and proposed improved ratio type variance estimators are given in the following Tables:

Table 3: B(.) of the Reviewed and New estimators
 

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Table 4: MSE(.) of the Reviewed and New estimators

 

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Numerical results of Table  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 Md 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 (S2NKi ) 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.

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