Normal-density local-to-global bandwidth ratio
= Normal-density local-to-global bandwidth ratio
{title2=$\inf_x h_{\rm AMSE}(x)/h_{\rm AMISE}=(9e^5/8192)^{1/10}$}
For the <standard normal density>, $f''(x)=(x^2-1)f(x)$ and $R(f'')=3/(8\sqrt\pi)$. The fifth power of the bandwidth ratio is $3\sqrt2\,e^{x^2/2}/[8(x^2-1)^2]$. Its global minima occur at $x=\pm\sqrt5$; $x=0$ is only the minimum within the interval between the two inflection points. The ratio diverges as $x$ approaches either inflection point.