= Fourth-derivative pilot estimate of density curvature
{title2=$\widehat R_g^{(2)}=n^{-2}\sum_{i,j}K_g^{(4)}(X_i-X_j)$}
Twice integrating by parts gives $R(f'')=\int f f^{(4)}$. Estimate this by averaging a fourth-derivative <kernel density estimator> at the observations. Its expectation contains diagonal term $K^{(4)}(0)/(ng^5)$ and leading smoothing bias $-g^2\mu_2(K)R(f''')/2$. Balancing them yields pilot scale $g\asymp R(f''')^{-1/7}n^{-1/7}$. With suitable derivatives and tails, the estimation error is bounded by $O_p(g^2+(ng^5)^{-1}+n^{-1/2}+n^{-1}g^{-9/2})$, giving an $O_p(n^{-2/7})$ relative-bandwidth bound after substitution. More carefully calibrated bias cancellation can improve that bound.
// Target: statistical-inference.bigb
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