= Quadratic density derivative functional
{title2=$J_r(f)=\int|f^{(r)}|^2$}
The squared <L2 norm> of a <probability density function> <derivative> measures roughness and appears in optimal smoothing constants. For $r=2$, its <directional derivative> along a smooth compactly supported perturbation $a$ is $2\int f^{\prime\prime}a^{\prime\prime}$. When <integration by parts> is valid twice, this equals $2\int f^{(4)}a$. Thus regular root-$n$ estimation requires substantially more regularity than merely estimating a smooth <probability density function>.
Back to article page
