= Biased cross-validation for density bandwidth
{title2=$\operatorname{BCV}(h)=\frac{R(K)}{nh}+\frac{\mu_2(K)^2h^4}{4}[R(\widehat f_h'')-R(K'')/(nh^5)]$}
= Biased density cross-validation
{synonym}
Subtract the diagonal variance contribution from the derivative estimate's squared norm, then insert the result into <asymptotic mean integrated squared error>. The expected curvature estimate equals $R(f'')+O(h^2)$, apart from a negligible finite-n term. This makes the resulting criterion asymptotically correct even though it is not an exactly unbiased estimate of integrated risk. Under standard conditions its selector is ratio-consistent with the optimal bandwidth and has relative fluctuation order $n^{-1/10}$.
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