A Sine Type Median Based Estimator for the Estimation of Population Mean

In the literature, there are numerous estimators for estimating population means when auxiliary information is provided. Subramani suggested ratio median based estimator when the median of the study variable is available and the regression estimator was shown to be significantly less efficient than the estimator. In this research, we suggested an estimator for the population mean of the studied variable based on a sine type median. Using Taylor series expansion, the bias and mean square error of the estimator were obtained up to the first order of approximation. The condition under which the proposed estimator is more efficient than the existing estimators was established. An empirical investigation was done to compare the suggested estimator's efficiency to that of the existing estimators


Introduction
One of the measures of central tendency is the median. It is employed to describe the distribution's center, when there is presence of outliers or extreme values in a dataset. Despite the fact that the mean is the most widely used indicator of central tendency but it has a weakness, that is, it sensitive to outliers or extreme values. The mean is susceptible to the influence of outliers in a dataset, hence, it is not a center-resisting measure. Contrarily, regardless of how significant these changes are, the median is unaffected or only marginally affected by changes in the numerical value of a tiny subset of the observations. A reliable indicator of the center is the median.
In sampling theory, auxiliary information is used in practice increasing the efficiency of estimators for the estimation of population mean of the study variable using the estimation strategies such as ratio, product and regression. When there is a positive correlation between the study and the auxiliary variable, the ratio technique of estimation is applied. Cochran 3 was the first to propose the ratio method of estimation under this assumption. Bahl and Tuteja 2 developed the estimators of a finite population's exponential ratio and product type. The main importance of the exponential estimators is that they give more precise estimate than ratio method of estimation when there is little relationship between the study and the auxiliary variable. To address the issue of population mean estimation, authors including Singh et al., 9 Sanaullah, 8 Riaz et al., 7 Yadav and Adewara, 14 Audu and Singh 1 and Yunusa et al., 16 Hussain et al, 17 Rather et al. 18 have developed various exponential estimators. Because of the resistant nature of median to outliers, some authors have used median as the auxiliary information in their works such as Subramani, 10 Srija et al., 12 Subramani and Kumaranpandiyan, 11 Yadav et al., 15 Muili et al., 4 Muili et al., 5 Muili and Audu, 6 Rather and Yousuf 19 and Zakari et al. 20 In order to increase the accuracy of estimating the mean of a finite population, this study proposes a sine type estimator that can yield estimates that are more closely related to the true population mean of the study variable.
Consider a finite population V=(V 1 , V 2 ,...,V N ). Utilizing the simple random sampling without replacement (SRSWOR) strategy, we select a sample of size n from the population. Let y and x respectively be the study and the auxiliary variables and y 1 and x 1 , respectively be the observations on the ith unit. and are the sample means. and be the corresponding population means of the study and auxiliary variables respectively. and are the sample variances and and , are the corresponding population variances. p is the correlation coefficient between y and x. Finally let C y =S y Y ̅ -1 and C x =S x X ̅ -1 respectively be the coefficients of variation for y and x.

Some Literature-Based Estimators
In this part, we look at a few existing finite population mean estimators in use. It is said that the typical sample mean is: The estimator's variance is provided by ... (2) Cochran 3 initiated the ratio method of estimation and it is defined as: ... (3) As a first order approximation, the MSE of y ̅ R is given by First-order approximation MSE of y ̅ s is provided by ... (11) Where,

Proposed Estimator
Motivated by the work of Subramani. 10 and Watson, 13 a sine type median-based estimator of the study variable's population mean is suggested by ... (12) where, b is the unknown constant to be obtained by means of differentiating the MSE of (t d )

Efficiency Comparisons
The conditions under which the proposed estimator outperforms various other estimators were developed in this section.
Estimator t d is more efficient than the estimator y ̅ if Var(y ̅ )>MSE min (t d ) Estimator t d is more efficient than the estimator y ̅ R if MSE(y

Results and Discussion
In this section, numerical analyses to assess the performance of the estimators are illustrated. The MSEs and PREs of the proposed estimator and the existing estimators are displayed in Table 1. The results showed that the proposed estimator has minimum MSE and higher PRE among other estimators for the three populations considered in the study. These findings suggest that the proposed estimator outperforms the existing ones taken into account in this investigation.

Conclusion
In this study, a sine type median based estimator is proposed for the estimation of finite population mean. The empirical results revealed that the new proposed estimator outperformed estimators in literature considered in this study. Therefore, it is advised to apply the proposed estimator in actual cases.