Empirical bayes estimates for a two-way cross-classified model

Lawrence D. Brown, Gourab Mukherjee, Asaf Weinstein

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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.

Original languageAmerican English
Pages (from-to)1693-1720
Number of pages28
JournalAnnals of Statistics
Issue number4
StatePublished - Aug 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© Institute of Mathematical Statistics, 2018


  • Empirical BLUP
  • Empirical Bayes
  • Oracle optimality
  • Shrinkage estimation
  • Stein’s unbiased risk estimate (SURE)
  • Two-way ANOVA


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