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.

翻译：本文针对Fay-Herriot正态误差模型拟合对数变换后的响应变量、且协变量存在测量误差的情形，提出了一种新的小域预测方法。该框架此前由Mosaferi等人（2023）研究过。他们手稿中给出的经验预测变量无法在所有情况下优于直接估计量。本文提出的预测变量具有无偏性，且整体表现优于Mosaferi等人（2023）的方案。我们推导了该预测变量均方误差（MSE）的近似表达式。基于MSE的预测区间存在覆盖缺陷，为此我们提出一种更精确的非参数bootstrap预测区间。该问题在小域应用中具有重要价值，因为统计机构和农业调查常需对右侧偏斜变量进行估计，且协变量存在测量误差。通过蒙特卡洛仿真研究及两个美国人口普查局数据集，我们验证了所提方法的优越性。