Quadratic density derivative functional ($J_r(f)=\int |f^{(r)}{|}^2$)

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.

For $A(v)=\int L^{\prime \prime }(t)L^{\prime \prime }(t+v)dt$, this U-statistic has expectation $\Vert L_b\ast f^{\prime \prime }{\Vert }_2^2$. It removes the diagonal term $\Vert L^{\prime \prime }{\Vert }_2^2/(n{b}^5)$ from the integral of a squared second derivative kernel density estimator, but leaves smoothing bias of an estimator.