= Two-point lower bound with a triangular bump
Consider <fixed-design nonparametric regression> $Y_i=m(i/n)+\varepsilon_i$, with <independent> errors $\varepsilon_i\sim N(0,\sigma^2)$, and estimation at a fixed interior point $x_0\in(0,1)$. In the class of functions with <Lipschitz continuity> constant at most $L$, compare $m_0=0$ with $m_1(x)=L(h-|x-x_0|)_+$, with $h$ small enough to stay inside the design interval.
The target values differ by $Lh$. At most $2nh+1$ equally spaced design points meet the bump, and its height is at most $Lh$, so the joint <Kullback-Leibler divergence> is bounded by
$$
\frac1{2\sigma^2}\sum_i m_1(i/n)^2\leq\frac{L^2h^2(2nh+1)}{2\sigma^2}.
$$
Taking $h$ a sufficiently small constant times $(\sigma^2/(nL^2))^{1/3}$ keeps this bounded away from a perfectly distinguishable experiment. <Pinsker's inequality> and the <Le Cam two-point lemma> give squared-error risk at least a constant times $L^{2/3}\sigma^{4/3}n^{-2/3}$. The design-spacing assumption is essential to the count of observations inside the bump; it is not valid for arbitrary fixed designs.
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