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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.

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  • $\begingroup$ Maybe it would help to provide the definitions of the terms being used? $\endgroup$ – Chill2Macht Dec 21 '17 at 14:38
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Is a limiting Bayes estimator always admissible?

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).

Is a generalized Bayes estimator with constant risk always admissible?

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.

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