Utilizing Auxiliary Information to Develop an Exponential Ratio-type Estimator under the Problem of Non-Response in Sample Surveys

Background and Objectives: In survey sampling, the estimation of population parameters, such as the mean (Mean), proportion (Proportion), variance (Variance), or total (Total), based on sampling methods often faces limitations in the precision of estimators. This is especially true when the sample size is limited due to various factors, such as budget, time, or the resources required for data collection. These factors directly affect the resulting estimators, leading to high variance and an inability to accurately reflect the true values of the population parameters. To mitigate these limitations, most researchers have developed approaches using auxiliary variables to construct or extend the estimators of interest. An auxiliary variable is information or a characteristic of a variable that researchers believe is correlated with the study variable and can be accessed easily or at a low cost. The integration of auxiliary variables into parameter estimation not only reduces bias and variance of the estimators but also increases the efficiency of the estimators under study compared to traditional estimation methods. Therefore, the study of the role of auxiliary variables in the development of estimators is important both theoretically and practically, including applications, as researchers can use the benefits of auxiliary variables in constructing estimators across various fields, including demography, economics, agriculture, industry, and social sciences, all of which may face limitations in resources and data completeness. Moreover, considering the relationship between auxiliary variables and the study variable helps to understand the characteristics of the population in greater depth and can be used to improve survey planning and resource allocation efficiently. This aligns with research that aims to use information from auxiliary variables to develop exponential ratio-type estimators under the presence of non-response in sample surveys, which is important for increasing the accuracy and reliability of survey estimators. Additionally, an in-depth study of auxiliary variables helps to ensure that survey design is appropriate and can reduce measurement errors throughout all stages of the survey process.

Methodology: This research studies certain key properties of the newly proposed estimators, which were developed from the estimators introduced by Rao (1986) and Singh et al. (2009), such as mean squared error (MSE) and minimum mean squared error (MMSE). The efficiency of the proposed estimators will be compared with other related estimators theoretically, through applications to real data, and via simulation studies. The study comprises three components: 1) Theoretical comparison: The conditions under which the proposed estimators perform better than other related estimators will be presented. 2) Comparison using real data: Real data collected by Khare & Sinha (2010) and Yadav et al. (2019) will be used to compare the efficiency of the proposed estimators with other estimators, using percentage relative efficiency (PRE) as the criterion. 3) Comparison using simulation: A bivariate normal population (,) will be generated using the R program, where the study variable  has a mean of 150 and variance of 5, and the auxiliary variable  has a mean of 180 and variance of 10 (when ) while 25% of the total observations are considered as non-respondents. The population size is set to 1,000, and correlation coefficients () are set at 0.2, 0.5, and 0.8, respectively. Samples of the study variable  and auxiliary variable  will be drawn from the simulated population using a two-phase sampling design, with sample sizes of 20 and 120, respectively. The efficiency of the studied estimators will be compared using percentage relative efficiency (PRE). This simulation helps researchers to understand the behavior of the estimators under different scenarios in detail and enhances the reliability of the study findings.

Main Results: From the comparison of the efficiency of the proposed estimators with other related estimators, in theory, applications to real data, and simulation studies, it was found that the proposed estimators perform better than other estimators when the conditions in Table 1 are satisfied. Moreover, the application to real data, as shown in Table 2, indicates that the proposed estimators have lower MSE than other estimators. When considering the efficiency of the estimators from PRE values, the proposed estimators have higher PRE than other studied estimators under the same conditions. In-depth analysis shows that the difference in PRE between the proposed estimators and other estimators tends to increase when the correlation between the study variable and the auxiliary variable is higher. For the comparison of the efficiency of estimators from the simulation study, it was found that when the correlation coefficient and sample size are the same, an increase in the relevant parameter leads to a decrease in PRE for all estimators. However, the proposed estimators remain the estimators with the highest PRE, confirming that the proposed estimators continue to perform better than other related estimators in all scenarios studied.

Conclusions: The study proposed an exponential ratio-type estimator for the population mean using information from auxiliary variables under two-phase sampling with non-response, developed from the estimators introduced by Rao (1986) and Singh et al. (2009). Results from the efficiency comparison between the proposed estimators and other related estimators, using theoretical analysis, real data applications, and simulation studies, consistently show that the newly proposed exponential ratio-type estimator for the population mean outperforms all other related estimators under the same datasets and scenarios. In conclusion, the estimators proposed in this study provide a useful and reliable tool for researchers and statisticians in quantitative surveys. They can be applied across various fields, particularly in situations involving non-response, as they enhance the accuracy of population parameter estimation and have been validated using both real and simulated data.