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.
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