 Solution

Put $r=\lceil \beta \rceil -1$ and $\alpha =\beta -r\in (0,1]$. The density Hölder class $\mathcal{F}(\beta ,L)$ consists of nonnegative functions $f$ integrating to one, with derivatives through order $r$, such thatThis is the density version of the Hölder class. A kernel for density estimation is an integrable function $K$ with $\int K=1$. It has order $\ell$ when $\int u^{j}K(u)du=0$ for $1\le j<\ell$ and $\int |u{|}^{\ell }|K(u)|du<\infty$.

Choose a bounded kernel of order at least $\lceil \beta \rceil$ with $\int |u{|}^{\beta }|K(u)|du<\infty$, and use the kernel density estimatorTaylor's theorem at $x$ and the vanishing kernel moments cancel every polynomial term below the remainder. Hence, for a constant depending only on $\beta$ and the fixed kernel,

We also need a uniform density bound. The standard Hölder interpolation argument combines nonnegativity, $\int f=1$, and the Hölder constraint to giveIndeed, near a point where $f$ is close to its maximum $M$, Taylor's theorem and the derivative bounds implied by the Hölder constraint keep $f$ of order $M$ on an interval of length comparable to $(M/L{)}^{1/\beta }$; integrating over that interval gives $M^{1+1/\beta }\le C_{\beta }{L}^{1/\beta }$.

Using this bound and independence,ThusChooseBoth terms then have order $L^{2/(\beta +1)}{n}^{-2\beta /(2\beta +1)}$. Since the infimum over all measurable estimators is no larger than the risk of this particular estimator,This is the pointwise minimax rate for Hölder density estimation upper bound.