On Modification of Some Ratio Estimators using Parameters of Auxiliary Variable for the Estimation of the Population Mean

Some existing estimators based on auxiliary attribute have been proposed by many authors. In this paper, we use the concept of power transformation to modify some existing estimators in order to obtain estimators that are applicable when there is positive or negative correlation between the study and auxiliary variable. The properties (Biases and MSEs) of the proposed estimators were derived up to the first order of approximation using Taylor series approach. The efficiency comparison of the proposed estimators over some existing estimators considered in the study were established. The empirical studies were conducted using existing population parameters to investigate the proficiency of the proposed estimators over some existing estimators. The results revealed that the proposed estimators have minimum Mean Square Errors and higher Percentage Relative Efficiencies than the conventional and other competing estimators in the study. These implies that the proposed estimators are more efficient and can produce better estimates of the population mean compared to the existing estimators considered in the study.


Introduction
In sample surveys, auxiliary attribute is always used to increase the precision of estimated of population parameters.This can be done at either estimation or selection stage or both stages.The commonly used estimators, which make use of auxiliary variables, include ratio estimator, product estimator, regression and difference estimator.The classical ratio estimator is preferred when there is a high positive correlation between the variable of interest, Y and the auxiliary variable, X with the regression line passing through the origin.The classical product estimator, on the other hand is most preferred when there is a high negative correlation between Y and ONWUKA et al., Orient.J. Phys.Sciences, Vol. 8 (1) 27-35 (2023) X while the linear regression estimator is most preferred when there is a high positive correlation between the two variables and the regression line of the study variable on the auxiliary has intercept on Y axis.The classical ratio and product estimators even though considered to be more useful in many practical situations have efficiencies which does not exceed that of the linear regression.
The use of auxiliary information has become indispensable for improving the exact of the estimators of population parameters like the mean and variance of the variable under study.A great variety of the techniques such as the ratio, product and regression methods of estimation are commonly known in this esteem.Keeping this fact in view, large number of estimators have been suggested in sampling literature.Some noteworthy contribution in this direction have been made by 1,2,4,5,6,7,8,9,11,12,13,1  5,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34 and many others.
The weaknesses discovered in this research is that 21  estimators are ratio-based, therefore they are only efficient when the correlation between study and auxiliary variables is positive.The efficiency of the estimator by 33 reduces as approaches zero in the presence of negative correlation between study and auxiliary variables.
To address the weaknesses in the 21,33 estimators, the estimators were modified using power transformation technique so as to obtain estimators that are applicable when the correlation between the study and auxiliary variables is either positive or negative.This study focuses on the modification of some ratiobased estimators using power transformation under simple random sampling in the presence of auxiliary variables and limited to the work of. 21,33  Methodology Let U denotes a finite population consisting of N units {U 1 , U 2 ,....., U N }.Also, let (Y, X) denote the study variable and auxiliary variable taking values (y i , x i ), (i = 1,2,.........., N), respectively, on the i th unit U i of the population U. On the assumption that the population mean ( ) of X is known, the estimate of the population mean ( ) of Y is obtained by selecting a sample of size n (n<N) from the population U using Simple Random Sampling without Replacement (SRSWOR) scheme.
N: Population size, n: Samplesize being selected from the entire population, f= n/N: is the sampling fraction, : The population mean of the study variable Y, : The population coefficient of skewness of auxiliary variable X.
: The population coefficient of kurtosis of auxiliary variable X.
: The population quartile deviation of auxiliary variable X.

Review of existing estimators
The conventional unbiased sample mean estimator is given by .
The Bias and MSE of the proposed estimator are given by: ... (32) where, the optimum value of k is

The Proposed Estimators
Having studied the estimators of 21,33 and identified some weaknesses, the following proposed exponential-type estimators for estimating population mean under Simple Random Sampling without Replacement (SRSWOR) were suggested based on the motivation from the works of. 4,34The proposed estimators are as given in ( 33) and (34).

Properties of the Proposed Estimators
In this section, the bias and MSE of the estimator proposed in this paper are derived and discussed.where Take expectation of ( 46), (47) and apply the results of (35), theorem 1.2 is proved.

Efficiency Comparison
In this section, conditions for the efficiency of the new estimators over some existing related estimators established were established.43) from ( 29) and ( 30), theorem 1.5 is proved.

Empirical Study
In this section, real life data was conducted to examine the superiority of the proposed estimators over the existing estimators considered in the study.

Population 2
The data is defined as follows:

Population 3
The data is defined as follows:

Population 4
The data is defined as follows: Table 1 above show the numerical results of the Mean Square Errors (MSEs) of the estimators and using four natural data sets of all the subjects examined, the two proposal have a minimum MSE for all data sets.This implies that the proposed methods have shown a high level of efficiency on others considered in the study, and can produce better estimate of the population parameters than the existing estimators.
Table 2 above show the numerical results of the Percentage Relative Efficiencies (PREs) of the estimators and using four natural data sets of all the subjects examined, the two proposal have the highest PREs for all data sets.This implies that the proposed methods have shown a high level of efficiency on others considered in the study, and can produce better estimate of the population parameters than the existing estimators.
where is the variance of sample mean, is the mean square error values of the proposed estimator in section 3 and is the mean square error values of the existing estimators mentioned in section 2.

Conclusion
By considering the results obtained from the empirical study on the efficiency of the suggested estimators over some exists related estimators considered in the study.From the empirical study, the results revealed that the suggested estimators n 1 (*) , and n 2 (*) have minimum mean square error and higher percentage relative efficiency compared to other estimators considered in the numerical computations carried out in the study.In the other words, the suggested estimators n 1 (*) , and n 2 (*) have higher chance of producing estimate that is closer to the true value of the population mean than other estimators considered in the literature of this study.