Parametric transformed Fay-Herriot model for small area estimation

Consider the small area estimation when positive area-level data like income, revenue, harvests or production are available. Although a conventional method is the logtransformed Fay-Herriot model, the log-transformation is not necessarily appropriate. Another popular method is the Box-Cox transformation, but it has drawbacks that the maximum likelihood estimator (ML) of the transformation parameter is not consistent and that the transformed data are truncated. In this paper, we consider parametric transformed Fay-Herriot models, and clarify conditions on transformations under which the ML estimator of the transformation is consistent. It is shown that the dual power transformation satisfies the conditions. Based on asymptotic properties for estimators of parameters, we derive a second-order approximation of the prediction error of the empirical best linear unbiased predictors (EBLUP) and obtain a second-order unbiased estimator of the prediction error. Finally, performances of the proposed procedures are investigated through simulation and empirical studies.