minimax estimator In the lecture here
https://www.stat.berkeley.edu/~yuekai/201b/lec6.pdf
we have
"minimaxity does not imply admissibility: a minimax estimator has the best worst-case performance, but its performance at other parameters may be suboptimal. However, any estimator that dominates a mini- max estimator is also minimax. Thus unique minimax estimators are admissible".
Can somebody explain this? How can one have several mini-max estimators?
 A: Suppose $X \sim \mathcal N(\mu, I)$ is a multivariate normal random variable with dimension $p \ge 3$ and consider the squared-error loss. Then observe the following:


*

*$X$ is a minimax estimator of $\mu$. This can be seen by observing that it has constant risk and is an appropriate limit of Bayes estimators (in the sense of the Bayes risk converging to the risk of $X$). See Theorem 2 here. 

*$X$ is not admissible because $X$ is dominated by the James-Stein estimator. See here. Because it dominates $X$, the James-Stein estimator is also minimax. 

*This is possible because the $\sup$ of the risk of the James-Stein estimator is indeed the same as the constant risk of $X$, and indeed for most values of $\mu$ (in the sense of Lebesgue measure) the James-Stein estimator and $X$ have virtually identical risk. 

*The James-Stein estimator is itself not admissible, being dominated by the positive-part James-Stein estimator. The positive-part James-Stein estimator is also not admissible because it is not an analytic function of $X$. 

*In sufficiently high dimensions, one can construct proper-Bayes estimators which dominate $X$. This finally gives us admissible, minimax, estimators of $\mu$ since all Bayes rules are admissible. 
