Open Access
October 2018 Empirical best prediction under a nested error model with log transformation
Isabel Molina, Nirian Martín
Ann. Statist. 46(5): 1961-1993 (October 2018). DOI: 10.1214/17-AOS1608


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.


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Isabel Molina. Nirian Martín. "Empirical best prediction under a nested error model with log transformation." Ann. Statist. 46 (5) 1961 - 1993, October 2018.


Received: 1 October 2016; Revised: 1 March 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964322
MathSciNet: MR3845007
Digital Object Identifier: 10.1214/17-AOS1608

Primary: 62D05
Secondary: 62G09

Keywords: Empirical best estimator , mean squared error , Parametric bootstrap

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 5 • October 2018
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