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

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

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 $n^{-2\beta /(2\beta +3)}$.