# Adaptively Transformed Mixed Model Prediction of General Finite Population Parameters

## TL;DR

This paper introduces an adaptive transformation approach for mixed model predictions of finite population parameters, improving estimation accuracy when response data deviate from normality.

## Contribution

It proposes a data-driven method to select suitable transformations using a parametric family and profile likelihood, enhancing prediction robustness.

## Key findings

- The method performs well in simulations, showing improved accuracy over traditional transformations.
- Application to income data demonstrates the method's practical utility and identifies limitations of common transformations.
- Constructs empirical Bayes confidence intervals for population parameters.

## Abstract

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.