  For estimating area-specific parameters (quantities) in a finite population,
a mixed model prediction approach is attractive. However, this approach
strongly depends on the normality assumption of the response values although we
often encounter a non-normal case in practice. In such a case, transforming
observations to make them suitable for normality assumption is a useful tool,
but the problem of selecting suitable transformation still remains open. To
overcome the difficulty, we here propose a new empirical best predicting method
by using a parametric family of transformations to estimate a suitable
transformation based on the data. We suggest a simple estimating method for
transformation parameters based on the profile likelihood function, which
achieves consistency under some conditions on transformation functions. For
measuring variability of point prediction, we construct an empirical Bayes
confidence interval of the population parameter of interest. Through simulation
studies, we investigate numerical performance of the proposed methods. Finally,
we apply the proposed method to synthetic income data in Spanish provinces in
which the resulting estimates indicate that the commonly used
log-transformation would not be appropriate.
