Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 210 / 2 / Solution

A kernel for density estimation is an integrable function $K$ with $\int K=1$. The kernel density estimator isIt has order $\ell$ when $\int u^{j}K(u)du=0$ for $1\le j<\ell$ and $\int |u{|}^{\ell }|K(u)|du<\infty$.

Differentiation gives the derivative kernel density estimatorDropping the negative square of its mean from the variance and using Tonelli and the substitution $u=(x-X_1)/h$,Thus $\alpha =3$ and $C_1(K)=\Vert K^{\prime }{\Vert }_2^2$.

For $r=\lfloor \beta \rfloor$, the Nikolsky class $\mathcal{N}(\beta ,L)$ consists of functions $g$ with square-integrable derivatives through order $r$ andusing the equivalent finite-difference definition when $\beta$ is an integer. Integration by parts showsTaylor expansion in $L^2$, cancellation of the moments through order $\ell -1$ for $\ell =\lceil \beta \rceil$, and the generalized Minkowski inequality giveConsequentlywhere one admissible choice isHence $\gamma =2\beta$.

The mean integrated squared error isthe sum of integrated variance and squared bias. The preceding bounds giveTaking $h\asymp n^{-1/(2\beta +3)}$ proves the MISE rate for derivative kernel density estimationThus $\delta =2\beta /(2\beta +3)$. Estimating a density of the same smoothness has variance order $(nh{)}^{-1}$ and rate $n^{-2\beta /(2\beta +1)}$; estimating its derivative is harder because differentiation amplifies high-frequency noise, changing $h^{-1}$ to $h^{-3}$.