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We develop an empirical Bayes procedure for estimating the cell means in anunbalanced, two-way additive model with fixed effects. We employ a hierarchicalmodel, which reflects exchangeability of the effects within treatment andwithin block but not necessarily between them, as suggested before by Lindleyand Smith (1972). The hyperparameters of this hierarchical model, instead ofconsidered fixed, are to be substituted with data-dependent values in such away that the point risk of the empirical Bayes estimator is small. Our methodchooses the hyperparameters by minimizing an unbiased risk estimate and isshown to be asymptotically optimal for the estimation problem defined above.The usual empirical Best Linear Unbiased Predictor (BLUP) is shown to besubstantially different from the proposed method in the unbalanced case andtherefore performs sub-optimally. Our estimator is implemented through acomputationally tractable algorithm that is scalable to work under largedesigns. The case of missing cell observations is treated as well. Wedemonstrate the advantages of our method over the BLUP estimator throughsimulations and in a real data example, where we estimate average nitratelevels in water sources based on their locations and the time of the day.

Empirical Bayes Estimates for a 2-Way Cross-Classified Additive Model