On Bayes estimates

In this paper the author tries to give general conditions for the existence of Bayes estimates and for the consistency of sequences of Bayes estimates.In Section 3 we prove existence theorems for Bayes estimates, which contain those of DeGroot and Rao 3], as a special case. The proof is based on a theorem of Landers 5].Section 4 gives a characterization of Bayes estimates with convex loss and linear decision space. This theorem is also a generalization of a similar theorem of DeGroot and Rao 3].In Section 5 we generalize the theory of minimum contrast estimates (the foundations of which were laid by Huber 4], cf. Pfanzagl 6]) in such a way that we can apply it to the theory of Bayes estimates.Section 6 tries to give a general theory of consistency for Bayes estimates using the martingale argument of Doob 1] and the theory of minimum contrast estimates. Confer in this connection the results of Schwartz 8].Section 7 contains some auxiliary results.