Modified difference type estimators for estimation of finite population mean

Introduction

In survey sampling, the accurate estimation of population parameters, particularly the population mean, has received considerable attention in statistical research. The use of auxiliary information is widely recognized as an effective technique for improving the precision and efficiency of estimators. When an auxiliary variable correlated with the study variable is available, it can be incorporated into the estimation process to reduce the variance of the estimator and improve its reliability. The theoretical foundations and applications of survey sampling methods have been extensively discussed in classical literature on sampling theory and survey design. Among the various estimators that utilize auxiliary information, the ratio and product estimators are commonly used for estimating the population mean. The ratio estimator is generally applied when the study variable and auxiliary variable are positively correlated, whereas the product estimator is preferred when the correlation is negative. Several modifications and improvements of these classical estimators have been proposed in the literature. Over time, several improved estimators have been proposed by incorporating auxiliary information in different ways. An important contribution in this direction was made by Bahl and Tuteja [1] , who introduced exponential ratio and exponential product type estimators to increase the efficiency of the classical ratio estimator in simple random sampling. Their work inspired many researchers to develop improved estimators by modifying the structure of ratio and product estimators. Further developments include the modified ratio estimator proposed by Sisodia and Dwivedi [9] , which incorporates the coefficient of variation of the auxiliary variable to improve the estimation of the population mean. Tailor and Sharma [15] also suggested a modified ratio-cum-product estimator by using known parameters of the auxiliary variable.

Similarly, the transformation of auxiliary variables has been explored to obtain more efficient estimators, as discussed by Upadhyaya and Singh [16] . The use of known correlation coefficients between the study variable and auxiliary variable has also been proposed to enhance estimator efficiency. Several authors have proposed improved estimators under different sampling frameworks. Koyuncu and Kadilar [3] developed efficient estimators for the population mean in stratified random sampling. Yadav and Kadilar [17] later introduced an efficient family of exponential estimators based on auxiliary information. Zaman and Kadilar [18] further discussed exponential ratio-type estimators and demonstrated their improved performance in survey sampling applications. Additionally, Singh and Solanki [11] proposed a new procedure for variance estimation in exponential type estimators. The theoretical background of sampling techniques and the role of auxiliary information in survey sampling have been widely discussed in standard textbooks on sample survey methods. Singh et al. [13] also highlighted the importance of auxiliary information and prior knowledge of population parameters in constructing improved estimators. In recent years, several new estimators based on auxiliary information have been proposed to further improve the estimation of the population mean. Malik and Tailor [8] introduced ratio and product type exponential estimators using known parameters of the auxiliary variable, while Malik and Singh [7] proposed an improved exponential ratio type estimator for estimating the population mean. Furthermore, Malik and Kusum [4] proposed a new logarithmic type estimator using auxiliary information. Malik and Nikita [5] also introduced a regressioncum exponential type estimator to improve estimation efficiency. Similarly, Malik and Poonam Devi [6] proposed an improved exponential type estimator under simple random sampling. These contributions demonstrate the continuing interest in developing efficient estimators using auxiliary information. Motivated by these developments, the present study proposes new modified RA-type estimators for estimating the population mean under simple random sampling without replacement using auxiliary information. The proposed estimators combine the properties of ratio and product type estimators. Theoretical expressions for the mean squared error (MSE) of the proposed estimators are derived and compared with those of existing estimators. The efficiency of the proposed estimators is also examined to demonstrate their superiority over conventional estimators under certain conditions. This study focuses on improving the estimation of logarithmic-type estimators by utilizing data from an auxiliary variable under simple random sampling. The paper is organized as follows: Section 1 reviews existing estimators along with relevant methodology and notes from the literature. Section 2 introduces a new modified logarithmictype estimator for simple random sampling. Section 3 presents a comparative analysis of the efficiency of the proposed estimators, 𝑃 𝑟 and Pp, against the existing estimators 𝑃 1 , 𝑃 2 , and 𝑃 3 . Section 4 examines the optimization of the variance of the proposed estimators, and Section 5 provides a numerical comparison to illustrate their performance.

Notations

Let a finite population U = (1, ...i, ...N) are created from a simple random sample without replacement (SRSWOR).

Existing Estimators

The standard mean, ratio and product type estimators in simple random sampling are as follows:

Simple Estimator

Ratio Estimator

The ratio estimator P2 is defined by

Product Estimator

The product estimator P3 is defined by

Proposed Estimators

By the event of simple random sampling (SRS), we propose the enhanced combined ratio and Product with exponential type estimators with constants k as

Expressing 𝑃 𝑟 in terms of 𝑒 0 and 𝑒 1 , we obtain the desired result:

After simplification, we get……

By computing the expected value of, it is to determine the bias of proposed estimator 𝑃 𝑟 to the first order of approximation:

To first degree of approximation MSE can be expressed as:

Expressing 𝑃 𝑝 in terms of 𝑒 0 and 𝑒 1 , we get outcome as:

After simplification, we get…….

By computing the expected value of, it is to determine the bias of proposed estimator 𝑃 𝑝 to the first order of approximation:

To first degree of approximation MSE can be expressed as:

Relative Efficiency

1) It can be observed that proposed estimator 𝑃 𝑟 is better in efficiency than Ratio 𝑃 2,

2) Suggested estimator 𝑃 𝑟 is more proficient than original product 𝑃 3,

3) It can be observed that proposed estimator 𝑃 𝑝 is better in efficiency than Ratio 𝑃

Conclusion

In this study, exponential type ratio and product estimators incorporating a scalar constant k were considered for estimating the finite population mean under simple random sampling without replacement (SRSWOR). The mean square errors (MSEs) of both estimators were derived up to the first order of approximation. In this study, new modified RA-type estimators are proposed under simple random sampling without replacement by utilizing an auxiliary variable. The proposed estimators are formulated as a combination of well-known estimators, particularly the ratio and product estimators. The mean squared error (MSE) of the proposed estimators is theoretically compared with the MSEs of the standard estimator P1, the ratio estimator P2, and the product estimator P3. The conditions for obtaining the MSEs are also derived. The findings show that, under certain conditions, the proposed estimators perform more efficiently than the existing estimators.

References

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