Biased cross-validation for density bandwidth ($\mathrm{BCV}(h)=\frac{R(K)}{nh}+\frac{{\mu }_2(K{)}^2{h}^4}{4}[R({\hat{f}}_h^{\prime \prime })-R(K^{\prime \prime })/(n{h}^5)]$)

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 $R(f^{\prime \prime })+O(h^2)$, 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 $n^{-1/10}$.