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Basic example: $X$ has a $p$-variate iid standard Normal distribution; the sample mean is not admissible if $p>2$ and is dominated by the Stein shrinkage estimator.

However, the Stein shrinkage estimator is also not admissible, and is dominated by the positive-part Stein shrinkage estimator. Which is also not admissible.

As far as I can tell, there is no known admissible estimator that dominates the sample mean. Is there a theoretical guarantee that there must be one, or could there just be an infinite sequence of inadmissible estimators, each slightly better than the last?

Thomas Lumley
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  • Wouldn't the limit of such a sequence (where each term dominates the one before it) be an admissible estimator by construction? – whuber Aug 22 '18 at 20:53
  • Replying to the comment by @whuber since I can't do comments: if the limit existed I think it would be an admissible estimator -- though I don't know if something that isn't finitely computable counts as an estimator. – Thomas Lumley Aug 22 '18 at 21:48
  • Please [merge](https://stats.stackexchange.com/help/merging-accounts) your accounts so that you can comment in your own question. – Glen_b Aug 23 '18 at 02:03
  • "Isn't finitely computable" is not a statistical concept and plays no role in standard statistical theories. If that's a concern to you, then please edit your question to explain why. – whuber Aug 23 '18 at 12:56
  • @whuber Does the limit of such a sequence have to be an estimator? I wonder if the limit could fail to be a function. – Dave Jun 08 '21 at 20:34
  • @Dave I don't understand: what kind of limit do you have in mind? – whuber Jun 08 '21 at 20:49
  • @whuber I confess my uncertainty, but I am thinking of something like $\underset{a\rightarrow \infty}{\text{lim}} f(x)$ for $f(x) = ax$ which is a function for every real $a$ but a vertical line in the limit. – Dave Jun 08 '21 at 20:54

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