Counterexamples: Minimax but not efficient, efficient but not minimax Are there examples of estimators that are minimax, but not efficient? Perhaps, to be more concrete, any of the following:

*

*(strong) Estimator sequence for each $n$ that is minimax but does not match the Cramer-Rao lower bound for any $n$

*(in-between) Estimator sequence for each $n$ that is minimax but does not match the Cramer-Rao lower bound asymptotically ($n\to\infty$)

*(weak) Estimator sequence that is asymptotically minimax but does not match the Cramer-Rao lower bound asymptotically ($n\to\infty$)

*Converses: Estimator sequence for each $n$ that matches the Cramer-Rao lower bound for each $n$, but is not minimax for any $n$ (or asymptotically)


1 and 4. That Wikipedia link has the example
$$\hat\delta^M= \frac{x+0.5\sqrt{n}}{n+\sqrt{n}}$$
for estimating $\theta$ in $Bin(n,\theta)$. It's not finite-sample efficient because that's one of the models where the MLE, $x/n$, is finite-sample efficient.


*The local asymptotic minimax theorem says that an asymptotically efficient estimator is locally asymptotically minimax (over neighbourhoods of size $n^{-1/2}$) for any bowl-shaped loss function, so if there's a counterexample here I think it would have to be a bit pathological.

