Minimax estimator

A minimax estimator minimizes the supremum of its risk over the parameter space.

For $X\sim \mathrm{Bin}(n,\theta )$, the posterior mean under $\mathrm{Beta}(\sqrt{n}/2,\sqrt{n}/2)$ has constant squared-error risk $1/[4(\sqrt{n}+1{)}^2]$. Its unique Bayes property and constant risk make it the unique minimax estimator.

The minimax risk is the infimum over estimators of the supremum of their risk functions over a parameter class. An upper bound supplies an estimator; a lower bound applies to every estimator, often using the Le Cam two-point lemma.

A risk function can decay rapidly for each fixed regression function while its supremum over a function class decays more slowly or fails to decay. Bounds depending on each function's local differentiability remainder need not be uniform. In fixed-design nonparametric regression, a class with no common smoothness bound permits smooth bump functions whose supports shrink between design points. Such sequences obstruct uniform estimation without contradicting fixed-function mean squared error bounds.

For $X\sim N_p(\theta ,I_p)$ under quadratic loss, the estimator ${\delta }_0(X)=X$ has constant risk $p$ and is minimax. Gaussian priors $N_p(0,c^2{I}_p)$ have Bayes risks $p{c}^2/(1+c^2)$, which tend to $p$ and therefore give the matching minimax lower bound.