minimax estimation rates

Suppose nature gets to pick the hardest problem in some class, and you must design the best estimator before knowing which one it picked. The minimax rate is the error you are guaranteed against that worst case: the best — the minimum over estimators — of the worst — the maximum over the class — risk. It answers the fundamental question of a statistical problem, not how well does my method do, but how well can any method possibly do, setting an unbeatable benchmark.

Formally it is the infimum over estimators of the supremum over the parameter class of the expected loss, as a function of sample size. Establishing a rate has two halves that must meet. The upper bound exhibits a concrete estimator achieving the rate. The lower bound shows no estimator can do better, by reducing estimation to testing among a cleverly chosen finite set of hard instances and applying an information inequality — Le Cam's two-point method, Assouad's hypercube, or Fano's method. When the two halves coincide up to constants, the rate is minimax-optimal.

Minimax rates are how statistics certifies optimality: nonparametric regression of smoothness s in dimension d has rate n to the power minus two-s over two-s-plus-d, exposing the curse of dimensionality directly in the exponent. They guide whether an estimator is worth improving or already essentially optimal. The caveat is that minimax is worst-case and global; an estimator optimal in this sense may be far from optimal on the benign instances you actually face, motivating adaptive and instance-dependent refinements.