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. Cochran³ was the first to propose the ratio method of estimation under this assumption. Bahl and Tuteja² 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, Sanaullah⁸, Riaz et al.⁷, Yadav and Adewara¹⁴, Audu and Singh¹ and Yunusa et al.¹⁶, Hussain et al¹⁷, 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 Subramani¹⁰, Srija et al.¹², Subramani and Kumaranpandiyan¹¹, Yadav et al.¹⁵, Muili et al.⁴, Muili et al.⁵, Muili and Audu⁶, Rather and Yousuf¹⁹ and Zakari et al²⁰.

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 = ( V₁,V₂,…..,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_i and x_i, respectively be the observations on the ith unit.


are the sample means


be the corresponding population means of the study and auxiliary variables respectively.


are the sample variances and


are the corresponding population variances.p is the correlation coefficient between y and x.

n : the sample size, M: the median, ^NC_n : number of size n samples that can be drawn from a population of size N, M^- : Average of Sample Median, m: Sample median,

Finally let


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 (y) variance is provided by

Cochran³ initiated the ratio method of estimation and it is defined as:

As a first order approximation, the MSE of y_R is given by

Watson¹³ suggested linear regression estimators of Y, and it is given as:

is the regression slope

The variance of y_lr to first order of approximation is given by:

Yunusa et al¹⁶ proposed an efficient exponential type estimators for Y as

The MSE of the estimator is given by:

is the value of the unknown in the estimator (T_M)

The estimator's minimum MSE is provided by:

Subramani¹⁰ developed an efficient estimator of population mean as

First-order approximation MSE of y_s is provided by:

Proposed Estimator

Motivated by the work of Subramani.¹⁰ and Watson¹³, a sine type median-based estimator of the study variable's population mean is suggested by

where, b is the unknown constant to be obtained by means of differentiating the MSE of (t_d)

Bias and Mean Square Error (MSE) of the proposed estimator's (t_d) Derivation  

To derive the proposed estimator's bias and mean square error (t_d) in (12), we write

such that



Expressing estimator t_d  in terms of error terms in (13), we have   

By expanding (1+e₂)^-1 and (­^_e₂) to first order of approximation, (15) becomes

By simplifying and expanding (16) to first order approximation, we have,

Subtract Y from both sides of  (17), we have,


Taking expectation of both sides of (18) and apply the results in (13) to obtain the bias of t_d , we have, to first order of approximation,

Squaring and taking the expectation from both sides of equation (18) will gives the estimator's (t_d) MSE as follows:

By differentiating (20) partially with respect to b, equate to zero and solve for b, we obtain

By substituting (21) into (20), we obtain the minimum MSE of t_d as

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:

Estimator t_d is more efficient than the estimator y_R  if:

Estimator t_d is more efficient than the estimator y_lr if:

Estimator t_d is more efficient than the estimator T_m if:

Estimator t_d is more efficient than the estimator y_s if:

Results and Discussion

In this section, numerical analyses to assess the performance of the estimators are illustrated.

Population 1: [Source: Subramani¹⁰]

Population 2: [Source:  Subramani¹⁰]

Population 3: [Source:  Subramani¹⁰]

Table 1: Proposed and Existing Estimators' MSE and PRE

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.

Acknowledgements

The authors thank the Oriental Journal of Physical Sciences Editorial Board and anonymous reviewers for their insightful work and suggestions.

 Funding

The authors received no any form of funding this research, authorship and publication.

Conflict of interest

Authors have declared that no competing interest exist.

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