Estimation of Finite Population Mean of Median Based Using Power Transformation

This paper deals with the assessment of finite population mean. An estimator is suggested for estimation of finite population mean of study variable. The purpose of this study is to evolve a ratio-type estimator to enhance the proficiency of the existing estimators considered in the study in sample random sampling without replacement using information of auxiliary variable. Expressions of the bias and mean square error (MSE) of the proposed estimator was derived by Taylor series method. The efficiency conditions under which the proposed ratio-type estimator is better than sample mean, ratio estimator, and other estimators considered in this study have been established. Theoretical and empirical findings are incentive and brace the robustness of the proposed estimator for mean estimation. The empirical results shown that the suggested estimator is more efficient than the sample mean, ratio estimator and other estimators. CONTACT J. O. Muili jamiunice@yahoo.com Department of Mathematics, Kebbi State University of Science and Technology Aliero, Nigeria. © 2021 The Author(s). Published by Oriental Scientific Publishing Company This is an Open Access article licensed under a Creative Commons license: Attribution 4.0 International (CC-BY). Doi: http://dx.doi.org/10.13005/OJPS06.01-02.05 Oriental Journal of Physical Sciences www.orientaljphysicalsciences.org ISSN: 2456-799X, Vol.06, No.(1-2) 2021, Pg. 26-31 Article History Received: 6 October 2021 Accepted: 3 January 2022


Introduction
Median is one of the auxiliary variables aid in improving the precision of estimates of the finite population mean. Auxiliary variables associated with the study variables have been identified to be helpful in improving the efficiency of ratio, product and regression estimators both at planning and estimation stages. Cochran 1 invented the use of auxiliary information and developed a ratio estimator for population mean. Ratio type estimator provides effective estimate (minimum value of MSE) in comparison to simple mean estimator provided the variable of interest and auxiliary variable are positively associated. If the association between the study and auxiliary variables is positive, then ratio type estimator is applicable. Product estimator is useful where the association between the study variable and auxiliary variable is negative, and more efficient than sample mean. This concept has been utilized by several researchers in order to increase the precisions of ratio and product type estimators in estimating population mean of study variable using auxiliary information for assessments to maximize precisions. Bahl and Tuteja 2 initiated exponential estimators with the used of exponential function in simple random sampling. Singh et al. 3 developed exponential ratio estimator with the used of known values of coefficient of variation, correlation coefficient and coefficient of kurtosis. Sanaullah et al. 4 , Riaz et al. 5 , Yadav and Adeware 6 , Rashid et al. 7 , and Kadilar 8 have developed different exponential estimators purposely to solve the problem of estimation of finite population mean. Few researchers have made used of median as the only auxiliary information in their works such as Subramani 9 and Kumar et al. 10 Other researchers have also used linear combinations of median and other auxiliary parameters for the estimation of population mean such as Subramani and Kumarapandiyan, 11,12,13 Subramani and Kumarapandiyan, 14 Yadav et al. 15 , Subzar et al. 16 , Muili et al. 17 , and Muili et al. 18 The purpose of this research is to evolve a ratio-type estimator to improve the precision of estimation of finite population mean in sample random sampling without replacement with the use of available known information of auxiliary variable.
Let a finite population Ψ={Ψ 1 , Ψ 2 ,...,Ψ N } having N units where each Ψ i = (X i , Y i ), i=1,2,3,4,...,N has a pair of values. X is the auxiliary variable which Y is the study variable and is correlated with X, where y ={y 1 , y 2 ,...,y n } and x ={x 1 , x 2 ,...,x n } are the n sample values. ȳ is the sample mean of the study variable and x ̅ is the sample mean auxiliary variable. Let and be the population mean squares of Y and X respectively and be sample mean square of study variable and be sample mean squares based on the random sample of size n drawn without replacement. N: Population size, Y : Study variable, ȳ Sample mean of study variable and x ̅ sample mean of auxiliary variables, f : Sampling fraction,

Literature Review
Sample mean (ȳ) of simple random sampling is given as: And variance of ȳ is given by: Watson 19 developed what is known as linear regression estimator of finite population mean using information of auxiliary variable that are highly correlation with variable of interest as where is the regression coefficient of study variable on auxiliary variable.

Proposed Estimator
Motivated by Subramani, 9 of finite population mean based on the information on auxiliary variable for estimation of population mean of study variable is proposed by: ...(14)

VERIFIABLE STUDY
To assess the performance of the suggested estimator, a natural population is used as: Source: Subramani 9  Table 1 shows the values of the populations' parameters  Table 2 shows MSE of the estimators using the three set of populations. The result revealed that the suggested estimator has minimum mean square error compared to the conventional estimators.
This implies that the proposed estimator is better and can produce better estimates of population mean than the conventional estimators, Bahl and Tuteja 2 and Subramani. 9  Table 3 shows PRE of the proposed and some existing estimators using the three set of populations.
The result revealed that the suggested estimator has the highest value of PRE compared to the conventional estimators, Bahl and Tuteja 2 and Subramani. 9 This implies that the suggested estimator is more efficient and can produce better estimates of population mean than the conventional estimators, Bahl and Tuteja 2 and Subramani. 9

Results and Discussion
Proficient ratio estimator of finite population mean is suggested. The properties of the suggested estimator were obtained. Table 2 shows MSE of the suggested and some existing estimators using the three set of populations. The result revealed that the suggested estimator has minimum MSE compared to the conventional estimators, Bahl and Tuteja 2 and Subramani. 9 Estimators. Table 3 shows PRE of the estimators using the three set of populations. The result revealed that the suggested estimator has highest PRE compared to the conventional estimators. This implies that the suggested estimator is more efficient and can produce better estimates of population mean than the conventional estimators, Bahl and Tuteja 2 and Subramani 9 estimators.

Conclusion
Based on the empirical study conducted on the efficiency comparison of the suggested estimator with related estimators, it is obtained that the suggested estimator is highly efficient and can produce better estimates of finite population mean than the conventional and existing estimators considered in the study. Future scope of this study can be study under different sampling schemes like stratified sampling, successive sampling or cluster sampling.

Funding
The authors received no financial support for the research, authorship and publication of this article.