= Pointwise optimal kernel bandwidth
{title2=$h_{\rm AMSE}(x)=\left(\frac{R(K)f(x)}{n\mu_2(K)^2f''(x)^2}\right)^{1/5}$}
For a second-order <kernel density estimator>, if $f(x)>0$ and $f''(x)\ne0$, the leading pointwise <asymptotic mean squared error> is $R(K)f(x)/(nh)+\mu_2(K)^2f''(x)^2h^4/4$. Differentiation gives the displayed interior optimum. At a zero of $f$, the leading variance term vanishes and this two-term approximation has no positive interior minimizer; at a zero of $f''$, a higher-order bias expansion is needed.
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