Asymptotics for EBLUPs: Nested Error Regression Models

Abstract

In this article we derive the asymptotic distribution of estimated best linear unbiased predictors (EBLUPs) of the random effects in a nested error regression model. Under very mild conditions which do not require the assumption of normality, we show that asymptotically the distribution of the EBLUPs as both the number of clusters and the cluster sizes diverge to infinity is the convolution of the true distribution of the random effects and a normal distribution. This result yields very simple asymptotic approximations to and estimators of the prediction mean squared error of EBLUPs, and then asymptotic prediction intervals for the unobserved random effects. We also derive a higher order approximation to the asymptotic mean squared error and provide a detailed theoretical and empirical comparison with the well-known analytical prediction mean squared error approximations and estimators proposed by Kackar and Harville and Prasad and Rao. We show that our simple estimator of the predictor mean squared errors of EBLUPs works very well in practice when both the number of clusters and the cluster sizes are sufficiently large. Finally, we illustrate the use of the asymptotic prediction intervals with data on radon measurements of houses in Massachusetts and Arizona.

KEYWORDS:

• Convolution of distributions
• Mixed model
• Prediction mean squared error
• Random effect

Supplementary Materials

Supplementary material provides second order expansion for multivariate M-estimators, rest of simulation results, and simulation codes.

Additional information

Funding

This research is supported by Australian Research Council Discovery Project DP180100836. The authors are grateful to the Editor, Associate Editor and Referees for helpful comments that improved the article.