Estimating variance of random effects to solve multiple problems simultaneously

Abstract

The two-level normal hierarchical model (NHM) has played a critical role in statistical theory for the last several decades. In this paper, we propose random effects variance estimator that simultaneously (i) improves on the estimation of the related shrinkage factors, (ii) protects empirical best linear unbiased predictors (EBLUP) [same as empirical Bayes (EB)] of the random effects from the common overshrinkage problem, (iii) avoids complex bias correction in generating strictly positive second-order unbiased mean square error (MSE) (same as integrated Bayes risk) estimator either by the Taylor series or single parametric bootstrap method. The idea of achieving multiple desirable properties in an EBLUP or EB method through a suitably devised random effects variance estimator is the first of its kind and holds promise in providing good inferences for random effects under the EBLUP or EB framework. The proposed methodology is also evaluated using a Monte Carlo simulation study and real data analysis.