Suppose $X_i \sim N(\theta_i, \tau_i^2)$. The goal is to estimate $\theta_i$ under squared loss.
What is the easiest way to prove that the realization of $X_i$, call it $x_i$, is minimax?
One approach I've thought of:
We can think of $x_i$ as the limit of Bayes estimators under the prior $\theta_i \sim N(\mu,\sigma^2)$, with $\sigma^2 \rightarrow \infty$. The risk of these Bayes estimators converges to $\tau_i^2$, the constant risk of the estimator $x_i$. This proves $x_i$ is minimax.
Does this seem correct?
