An Unbiased Predictor for Skewed Response Variable with Measurement Error in Covariate

TL;DR

This paper proposes an unbiased small area predictor for skewed response variables with measurement error in covariates, outperforming previous methods and providing more accurate prediction intervals through bootstrap techniques.

Contribution

The authors develop a new unbiased predictor for skewed responses with measurement error, improving upon existing methods and introducing a bootstrap-based prediction interval.

Findings

The proposed predictor is unbiased and performs uniformly better than previous methods.

The bootstrap prediction interval offers improved coverage accuracy.

Simulation studies and real data demonstrate the method's effectiveness.

Abstract

We introduce a new small area predictor when the Fay-Herriot normal error model is fitted to a logarithmically transformed response variable, and the covariate is measured with error. This framework has been previously studied by Mosaferi et al. (2023). The empirical predictor given in their manuscript cannot perform uniformly better than the direct estimator. Our proposed predictor in this manuscript is unbiased and can perform uniformly better than the one proposed in Mosaferi et al. (2023). We derive an approximation of the mean squared error (MSE) for the predictor. The prediction intervals based on the MSE suffer from coverage problems. Thus, we propose a non-parametric bootstrap prediction interval which is more accurate. This problem is of great interest in small area applications since statistical agencies and agricultural surveys are often asked to produce estimates of right skewed variables with covariates measured with errors. With Monte Carlo simulation studies and two Census Bureau's data sets, we demonstrate the superiority of our proposed methodology.