Empirical best prediction under a nested error model with log transformation
对数变换嵌套误差模型下的经验最佳预测AI 解读 ↓
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对数变换嵌套误差模型下的经验最佳预测
- 研究问题
- 在对数尺度拟合嵌套误差模型后，如何预测原尺度的个体值与小区域均值，避免直接反变换偏差并合理评估参数估计带来的不确定性？
- 方法与思路
- 作者推导模型下的最佳预测量，再以估计参数构造经验最佳预测量，分析精确及二阶近似均方误差，并用模拟和墨西哥生活条件数据说明。
- 材料陈述的贡献
- 摘要称，直接反变换可能产生显著偏差，甚至不如直接小区域估计；论文给出最佳预测量、均方误差展开及二阶正确的误差估计方法。
- 阅读关注点
- 应核对对数尺度嵌套误差模型及最佳预测的损失标准，检查反变换修正、参数估计步骤与二阶误差条件，并比较小样本区域的表现。
- 阅读边界
- 仅依据稿件版本未注明的OpenAlex索引摘要。“最佳”依赖所设模型与预测准则，摘要未说明模型错设下性质，也未提供应用效果数值。
原文摘要
In regression models involving economic variables such as income, log transformation is typically taken to achieve approximate normality and stabilize the variance. However, often the interest is predicting individual values or means of the variable in the original scale. Under a nested error model for the log transformation of the target variable, we show that the usual approach of back transforming the predicted values may introduce a substantial bias. We obtain the optimal (or “best”) predictors of individual values of the original variable and of small area means under that model. Empirical best predictors are defined by estimating the unknown model parameters in the best predictors. When estimation is desired for subpopulations with small sample sizes (small areas), nested error models are widely used to “borrow strength” from the other areas and obtain estimators with greater efficiency than direct estimators based on the scarce area-specific data. We show that naive predictors of small area means obtained by back-transformation under the mentioned model may even underperform direct estimators. Moreover, assessing the uncertainty of the considered predictor is not straightforward. Exact mean squared errors of the best predictors and second-order approximations to the mean squared errors of the empirical best predictors are derived. Estimators of the mean squared errors that are second-order correct are also obtained. Simulation studies and an example with Mexican data on living conditions illustrate the procedures.