On the concepts of admissibility and coherence

Abstract

In the present paper it will be argued that if a parameter value assigns zero probability to an open set containing the actual response, then that parameter value should be excluded from the parameter space. When this is done, the model becomes a restricted model, and sampling theory inferences should be focused on the sampling distribution of an hypothetical independent response from the restricted model.

The above situation may arise when probability densities vanish outside a compact set. This phenomenon arises frequently in the real world, but it is usually ignored for reasons of mathematical convenience. Although many statistical procedures remain substantially the same when we consider this restricted model, admissibility propertics may be drastically changed, and inferences which are known to be inadmissible may turn out to be really admissible. Thus, Stein's phenomenon concerning the inadmissibility of the sample mean as an estimator of the population mean of ap-variate normal distribution whenp≥3may be explained by the fact that the distribution has a compact support but this has been ignored by reasons of mathematical convenience.

The introduction of a restricted model is also important in the study of coherence. Thus, it will be shown that Brunk's theory of countably additive coherence, which admits the use of countably additive improper priors, can be improved with the introduction of a restricted model. Thus, the theory will be unified because it will be proved that the posterior is really coherent if and only if it is a Bayes posterior, and it will be simplified because it will not be required that the prior be minimally compatible with the model.