= Pointwise versus uniform risk distinction
{title2=$\sup_m R_n(m)$}
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.
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