Two-point lower bound with a triangular bump

Consider fixed-design nonparametric regression $Y_i=m(i/n)+{\epsilon }_i$, with independent errors ${\epsilon }_i\sim N(0,{\sigma }^2)$, 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 byTaking $h$ a sufficiently small constant times $({\sigma }^2/(n{L}^2){)}^{1/3}$ 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.