Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 210 / 2

A kernel for density estimation is an integrable function $K$ with $\int K=1$. With $K_h(u)=h^{-1}K(u/h)$, the kernel density estimator isA kernel of order ell satisfies

Split the integrated $L^p$ risk into stochastic and bias terms:For fixed $x$, put $A_i(x)=K_h(x-X_i)$ and $W_i(x)=A_i(x)-\mathbb{E}{A}_i(x)$. The supplied Rosenthal inequality givesNowAlso $\mathbb{E}{W}_1(x{)}^2\le (K_h^2\ast f)(x)$. The Young convolution inequality with exponent $p/2$ givesAfter integration and multiplication by the outer factor $2^{p-1}$, the stochastic contribution is at most

It remains to control the bias. Here $\ell =\lceil \beta \rceil =m+1$. The Taylor formula with integral remainder, the vanishing kernel moments, and Minkowski integral inequality giveThe defining Nikol'skii smoothness bound is $L|hu{|}^{\beta -m}$, soRaising to the $p$th power and applying the outer factor $2^{p-1}$ proves