= Single-observation location minimax bound
{title2=$\inf_T\sup_a\mathbb E(T(a+\epsilon)-a)^2=\operatorname{Var}(\epsilon)$}
For a known mean-zero noise law with finite <variance> $\sigma^2$, observing only $Y=a+\epsilon$ with unrestricted $a\in\mathbb R$ has <minimax risk> $\sigma^2$ under <squared-error loss>. The observation itself attains this risk. For the lower bound, use a flat <prior distribution> on $[-A,A]$ and condition an easier experiment on $|\epsilon|\leq M$. Away from the prior endpoints, the <Bayesian posterior> noise has its original truncated law. Let $A\to\infty$ and then $M\to\infty$ in the resulting <Bayes risk> lower bound. No <normal distribution> or density for the noise is required.
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