I have the following questions. They are not homework problems, but they are things that the professor said that I should wonder about. I suspect that I will have to deal with this on an exam in the future. So my questions are:

1. Is a limiting Bayes estimator always admissible?
2. Is a generalized Bayes estimator with constant risk always admissible?

I suspect that the answer to the first question is no, and the answer to the second question is yes, but I am uncertain.

• Maybe it would help to provide the definitions of the terms being used? – Chill2Macht Dec 21 '17 at 14:38

Take $$\delta(x)=x$$, as the estimator of the mean of a Normal vector $$x$$ under squared error loss. This is the limit of Bayes estimators $$\frac{\alpha}{\alpha+1} x$$ under conjugate priors, but it is inadmissible in dimension three and above (Stein effect).
The same example applies: $$\delta(x)=x$$ is a Bayes estimator under the flat prior (which I assume is what you mean by generalised Bayes, as the Bayes risk does not exist). But it is inadmissible in dimension three and above.