Consistent Estimation in Box-Cox Transformed Linear Mixed Models

TL;DR

This paper introduces a simple, consistent moment-based estimator for the Box-Cox transformation parameter in linear mixed models, improving over traditional maximum likelihood methods and ensuring reliable parameter estimation.

Contribution

It proposes a novel, consistent estimator for the Box-Cox transformation parameter in mixed models, addressing inconsistency issues of maximum likelihood estimation.

Findings

The proposed estimator outperforms maximum likelihood in simulations.

The estimator achieves consistency as the number of groups increases.

Application to real data demonstrates practical effectiveness.

Abstract

The Box-Cox transformation is applied to the linear mixed models for analyzing positive and grouped data. The problem in using Box Cox transformation is that the maximum likelihood estimator of the transformation parameter is generally inconsistent. To fix it, we suggest a simple and consistent estimator for the transformation parameter based on the moment method. The consistent estimator is used to construct consistent estimators of the parameters involved in the model and under some conditions, the estimators of model parameters are shown to be consistent under $m\to\infty$, where $m$ is the number of groups. Moreover, in estimation of the expectation of the (future) observations, it is shown that the resulting estimators also hold consistency owing to the consistent estimator of the transformation parameter. The proposed estimating method is compared with the maximum likelihood method via simulation and it is shown that the proposed estimator works better than the maximum likelihood estimator. Finally, the proposed method is applied to a real data set.