Abstract

Small area estimation (SAE) methods commonly utilize auxiliary information through linear linking models between the study and auxiliary variables. However, in many practical situations the relationship between these variables is inherently nonlinear, which may lead to inefficiency or bias when linear small area estimators are applied. Motivated by this issue, this paper proposes a new nonlinear small area estimator that incorporates auxiliary information through a logarithmic adjustment to the direct estimator.

The proposed estimator is designed to accommodate nonlinear relationships between the study and auxiliary variables while preserving the simplicity of conventional ratio-based estimators. Approximate expressions for the bias and mean squared error of the proposed estimator are derived under a superpopulation model, and its theoretical properties are examined. The performance of the estimator is evaluated through comparisons with the classical direct, ratio, and regression-type small area estimators.

An extensive simulation study is conducted under various nonlinear population structures and sampling scenarios, including areas with small and zero sample sizes. The results demonstrate that the proposed estimator achieves improved efficiency and reduced mean squared error in the presence of nonlinear auxiliary effects, while maintaining competitive performance under near-linear conditions. An application to a real data set further illustrates the practical usefulness of the proposed approach for small area estimation.