Derivative kernel density estimator (${\hat{f}}_{n,h,K}^{\prime }$)

Differentiating a smooth kernel estimate givesIts integrated variance is at most $\Vert K^{\prime }{\Vert }_2^2/(n{h}^3)$.

Differentiating a twice differentiable kernel density estimator twice estimates $f^{\prime \prime }$. Its integrated variance of a kernel density estimator analogue is at most $\Vert L^{\prime \prime }{\Vert }_2^2/(n{b}^5)$. This explains why derivative estimation requires more smoothing than ordinary density estimation.

The integrated squared second derivative measures a smooth density's curvature in L2. It supplies the leading integrated squared bias constant of a second-order kernel density estimator. For nonzero second moment ${\mu }_2(K)$, the asymptotic mean integrated squared error has bias term $h^4{\mu }_2(K{)}^{2}R(f^{\prime \prime })/4$, so greater curvature calls for a smaller smoothing bandwidth.

For $f_a(x)=af(ax)$ with a positive, differentiation and change of variables give $R(f_a^{\prime \prime })=a^{5}R(f^{\prime \prime })$, while its standard deviation is $\sigma (f)/a$. Their product D is therefore invariant under scale changes, and translation also leaves it unchanged. Raw integrated squared curvature can be arbitrarily small under dilation even though this dimensionless shape measure is unchanged.

The $L^2$ Nikolsky class $\mathcal{N}(\beta ,L)$ consists of functions with $r=\lfloor \beta \rfloor$ square-integrable derivatives whose highest derivative obeys the translation boundwith the corresponding finite-difference definition at integer smoothness.

If $f^{\prime }\in \mathcal{N}(\beta ,L)$ and the kernel has sufficient order, the mean integrated squared error of ${\hat{f}}^{\prime }$ is bounded byBalancing the terms gives $h\asymp n^{-1/(2\beta +3)}$ and rate $n^{-2\beta /(2\beta +3)}$.