Summary
A result of Doob regarding consistency of Bayes estimators is extended to a large class of Bayes decision procedures in which the loss functions are not necessarily convex. Rather weak conditions are given under which the Bayes procedures are consistent. One set involves restrictions on the a priori distribution and follows an example in which the choice of a priori distribution determines whether the Bayes estimators are consistent. Another example shows that the maximum likelihood estimators may be consistent when the Bayes estimators are not. However, the conditions given are of an essentially weaker nature than those established for consistency of maximum likelihood estimators.
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This paper was prepared with the partial support of the Office of Ordnance Research, U. S. Army, under Contract DA-04-200-ORD-171. While the paper was in press, its author Dr. Lorraine Schwartz suddenly died. Sit ei terra levis! (L. LC., J. N.)
I am deeply indebted to Professor L. LeCam, under whose guidance I obtained the results established in my thesis and given in this paper.
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Schwartz, L. On Bayes procedures. Z. Wahrscheinlichkeitstheorie verw Gebiete 4, 10–26 (1965). https://doi.org/10.1007/BF00535479
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DOI: https://doi.org/10.1007/BF00535479