Transformed Fay-Herriot Model with Measurement Error in Covariates

TL;DR

This paper introduces a transformed Fay-Herriot model with measurement error in covariates, aiming to improve small area estimates for skewed variables by stabilizing skewness and normalizing responses.

Contribution

It develops a novel area-level log-measurement error model and derives an empirical Bayes predictor with bias correction for small area estimation.

Findings

The proposed model effectively stabilizes skewness in response variables.

The empirical Bayes predictor shows reduced bias with bias order $O(m^{-1})$.

Simulation studies demonstrate improved estimation accuracy.

Abstract

Statistical agencies are often asked to produce small area estimates (SAEs) for positively skewed variables. When domain sample sizes are too small to support direct estimators, effects of skewness of the response variable can be large. As such, it is important to appropriately account for the distribution of the response variable given available auxiliary information. Motivated by this issue and in order to stabilize the skewness and achieve normality in the response variable, we propose an area-level log-measurement error model on the response variable. Then, under our proposed modeling framework, we derive an empirical Bayes (EB) predictor of positive small area quantities subject to the covariates containing measurement error. We propose a corresponding mean squared prediction error (MSPE) of EB predictor using both a jackknife and a bootstrap method. We show that the order of the bias is $O(m^{-1})$, where $m$ is the number of small areas. Finally, we investigate the performance of our methodology using both design-based and model-based simulation studies.