Nonparametric Empirical Bayes Prediction in Mixed Models

Abstract

Mixed models are classical tools in statistics for modeling repeated data on subjects, such as data on patients or firms collected over time. They extend conventional linear models to include latent parameters, called random effects, that capture between-subject variation and accommodate dependence within the repeated measurements of a subject. Traditionally, predictions in mixed models are conducted by assuming that the random effects have a zero mean Normal distribution, which leads to the Best Linear Unbiased Predictor (BLUP) of the random effects in these models. However, such a distributional assumption on the random effects is restrictive and may lead to inefficient predictions, especially when the true random effect distribution is far from Normal. In this article, we develop a framework, EBPred, for empirical Bayes prediction in mixed models. The predictions from EBPred rely on the Best Predictor of the random effects, which are constructed without any parametric assumption on the distribution of the random effects and offer a natural extension to the BLUP when the true random effect distribution is not Normal. An extensive simulation study demonstrates the superior prediction performance of EBPred relative to extant approaches across many settings. Extensions to dynamic panel data and cross-classified random effect models are discussed. The method is illustrated on an application involving the prediction of bank stock returns.

Keywords: Kernelized Stein's discrepancy, Mixed models, Nonparametric empirical Bayes, Shrinkage prediction