We develop an empirical Bayes procedure for estimating the cell means in an unbalanced, two-way additive model with fixed effects. We employ a hierarchical model, which reflects exchangeability of the effects within treatment and within block but not necessarily between them, as suggested before by Lindley and Smith [J. R. Stat. Soc., B 34 (1972) 1–41]. The hyperparameters of this hierarchical model, instead of considered fixed, are to be substituted with data-dependent values in such a way that the point risk of the empirical Bayes estimator is small. Our method chooses the hyperparameters by minimizing an unbiased risk estimate and is shown to be asymptotically optimal for the estimation problem defined above, under suitable conditions. The usual empirical Best Linear Unbiased Predictor (BLUP) is shown to be substantially different from the proposed method in the unbalanced case and, therefore, performs suboptimally. Our estimator is implemented through a computationally tractable algorithm that is scalable to work under large designs. The case of missing cell observations is treated as well.
Bibliographical noteFunding Information:
Received May 2016; revised February 2017. 1Supported in part by grant NSF DMS-15-12084. 2Supported in part by the Zumberge individual award from the University of Southern California’s James H. Zumberge faculty research and innovation fund. MSC2010 subject classifications. Primary 62C12; secondary 62C25, 62F10, 62J07. Key words and phrases. Shrinkage estimation, empirical Bayes, two-way ANOVA, oracle optimality, Stein’s unbiased risk estimate (SURE), empirical BLUP.
1Supported in part by grant NSF DMS-15-12084. 2Supported in part by the Zumberge individual award from the University of Southern California’s James H. Zumberge faculty research and innovation fund. We thank Tony Cai, Samuel Kou and Art Owen for helpful discussions. We would also like to thank the Associate Editor and three referees for constructive suggestions to shorten and improve the paper.
© Institute of Mathematical Statistics, 2018
- Empirical BLUP
- Empirical Bayes
- Oracle optimality
- Shrinkage estimation
- Stein’s unbiased risk estimate (SURE)
- Two-way ANOVA