The Annals of Statistics
- Ann. Statist.
- Volume 46, Number 5 (2018), 1961-1993.
Empirical best prediction under a nested error model with log transformation
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.
Ann. Statist., Volume 46, Number 5 (2018), 1961-1993.
Received: October 2016
Revised: March 2017
First available in Project Euclid: 17 August 2018
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Molina, Isabel; Martín, Nirian. Empirical best prediction under a nested error model with log transformation. Ann. Statist. 46 (2018), no. 5, 1961--1993. doi:10.1214/17-AOS1608. https://projecteuclid.org/euclid.aos/1534492825
- Supplement to “Empirical best prediction under a nested error model with log transformation”. This document contains results on the bias of the proposed and existing predictors, simulation results for prediction at the individual level, on the performance of the bootstrap MSE estimator compared with the analytical estimator, and additional results on the application with Mexican data.