Normal-density local-to-global bandwidth ratio (${\inf }_x{h}_{\mathrm{AMSE}}(x)/h_{\mathrm{AMISE}}=(9e^5/8192{)}^{1/10}$)

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.