  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.
