Modified Exponential Ratio Estimator Using Two Auxiliary Variables

Abstract

This study contributes to the growing body of literature on survey sampling by enhancing 
the efficiency of population mean estimation through the incorporation of auxiliary 
information. In sample survey methodology, the integration of auxiliary variables at the 
estimation stage has become a fundamental strategy for improving estimator precision 
relative to conventional unbiased estimators. Building on this premise, previous research 
has demonstrated that the use of a single auxiliary variable can significantly reduce 
estimation variance; more recent advancements have extended this framework to 
accommodate multiple auxiliary variables for further efficiency gains. In this research, 
two auxiliary variables are simultaneously utilized to develop a class of modified ratio
type estimators under a double sampling (two-phase sampling) scheme. The proposed 
estimator incorporates a system of weights—arbitrarily assigned to each auxiliary 
variable—to optimize their relative contributions during estimation. Furthermore, an 
exponential functional form is embedded within the estimator to attenuate the influence 
of extreme observations and mitigate the effect of outliers, thereby enhancing 
robustness. The theoretical properties of the proposed estimator are rigorously derived 
using first-order approximation techniques. Specifically, expressions for the Mean 
Square Error (MSE) are obtained and compared analytically with those of existing 
estimators in the literature. Conditions under which the proposed estimator outperforms 
conventional ratio and regression-type estimators are formally established. 
To validate the theoretical findings, an empirical investigation based on simulated 
datasets was conducted. The simulation results demonstrate a consistent reduction in 
MSE for the proposed estimator relative to competing estimators, confirming its 
superior efficiency and stability. Overall, the study provides a robust and flexible 
estimation framework that leverages multiple auxiliary variables and exponential 
adjustment to achieve improved performance in practical survey applications.