Minimax estimator

Remarks

A minimax estimator need not exist, and it need not be unique. A Bayes estimator whose risk function is constant in $\theta$ is minimax.

Example: let $X$ be binomial with parameters $n$ and $\theta \in [0, 1]$, with squared error loss. Let $\delta^*(X) = (X + \sqrt{n}/2)/(n + \sqrt{n})$. A direct computation of variance plus squared bias gives $R(\theta, \delta^*) = \frac{1}{4(1 + \sqrt{n})^2}$ for every $\theta$. This $\delta^*$ is the Bayes estimator for the beta prior with parameters $\sqrt{n}/2$ and $\sqrt{n}/2$, and its risk is constant, so it is minimax.

Non-example: $\delta(X) = X/n$ in the same problem. Its risk is $\theta(1 - \theta)/n$, with maximum $\frac{1}{4n}$ at $\theta = \frac{1}{2}$. Since $\frac{1}{4n} > \frac{1}{4(1 + \sqrt{n})^2}$, its worst-case risk exceeds $R^*$, so it is not minimax.