A contrast of EB, modal and EM-algorithm estimates arising in the one-way analysis of variance situation

This paper explores the question of the choice of whether to use a joint posterior or a marginal posterior as basis for analysis with respect to various parameters of interest. It turns out that in the one-way analysis of variance situation there is an optimal choice. Connections with Empirical Bayes estimation is stressed as well as the use of the EM-algorithm. We demonstrate that for the latter, care must be taken with the choice of parameters to be declared "missing", for a wrong choice could lead to inconsistent estimators or estimators with poor mean-square behaviour. The discussion is in the context of one-way analysis of variance.