Pointwise optimal kernel bandwidth ($h_{\mathrm{AMSE}}(x)={\left(\frac{R(K)f(x)}{n{\mu }_2(K{)}^2{f}^{\prime \prime }(x{)}^2}\right)}^{1/5}$)

For a second-order kernel density estimator, if $f(x)>0$ and $f^{\prime \prime }(x)\ne 0$, the leading pointwise asymptotic mean squared error is $R(K)f(x)/(nh)+{\mu }_2(K{)}^2{f}^{\prime \prime }(x{)}^2{h}^4/4$. 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 $f^{\prime \prime }$, a higher-order bias expansion is needed.

For the standard normal density, $f^{\prime \prime }(x)=(x^2-1)f(x)$ and $R(f^{\prime \prime })=3/(8\sqrt{\pi })$. The fifth power of the bandwidth ratio is $3\sqrt{2}{e}^{x^2/2}/[8(x^2-1{)}^2]$. Its global minima occur at $x=\pm \sqrt{5}$; $x=0$ is only the minimum within the interval between the two inflection points. The ratio diverges as $x$ approaches either inflection point.