Is an admissible minimax rule unique?

Question

As the title said, for a given state space $\Theta$ and loss function $L(\theta,a)$, if a (randomized) decision rule $\delta$ is a minimax decision rule with respect to $L$, i.e. $$sup_{\theta}E^{X\mid\theta}L(\theta,\delta(X))=inf_{\delta\in\mathcal{D}}sup_{\theta}E^{X\mid\theta}L(\theta,\delta(X))$$ among a class of decision rules $\mathcal{D}=\{\delta:E^{X\mid\theta}L(\theta,\delta(X))<\infty\}$ and it is also admissible, i.e. for all $\delta\neq\delta_1\in\mathcal{D}$ $$E^{X\mid\theta}L(\theta,\delta(X))\leq E^{X\mid\theta}L(\theta,\delta_1(X)),\forall\theta \in \Theta$$ $$E^{X\mid\theta_0}L(\theta_0,\delta(X))\leq E^{X\mid\theta_0}L(\theta_0,\delta_1(X)),\exists\theta_0 \in \Theta$$ Then, must such a decision rule $\delta$ be unique as a function? If so, why; if not, is there any counter-example?

Moreover, what will the answer to the problem change if we assume that $\Theta$ is finite?

Answer 1

It depends on the sampling distribution and on the loss function as well as the dimension of the parameter. For instance, the MLE of a multivariate Gaussian mean is admissible when the dimension is $p=1,2$ under quadratic loss. Since its risk is constant, it is the sole admissible minimax estimator. This does not hold for $p\ge 3$, a phenomenon called the Stein effect. (Brown (1966) shows the wide generality of this phenomenon.)

In Strawderman (1972), the author shows the existence of proper Bayes [hence admissible] minimax estimators of the multivariate Normal mean for dimensions $p\ge 5$. One example of such estimators is associated with the priors$$\theta\sim\text{N}_p(0,\lambda^{-1}\lambda\mathbf{I}_p)\qquad \lambda\sim\lambda^{-a}/(1-a)\quad 0\le a\le 1$$

In Berger and Robert (1990), we extended this result and obtained a family of priors leading to admissible minimax estimators of the multivariate Normal mean. This was further extended by Berger and Strawderman (1996).

In Ghosh and Amin (1981), the authors exhibit a family of proper Bayes [hence admissible] minimax estimators of the p-variate Poisson mean, under the sum of weighted squared error losses, when $p\ge 3$, weights being reciprocals of variances. For instance, for a prior$$\pi(\lambda)\propto\lambda^{m-1}\{1+\lambda\}^{-(n+m)}\qquad 0< m\le p-2,\quad n>0,$$the associated posterior mean $$\mathbb{E}[\lambda|\mathbf{x}]=\dfrac{z+m+p}{z+m+n+p-1}\mathbf{x}\qquad z=\sum_{i=1}^p x_i$$is proper Bayes minimax admissible.