= Third-order pointwise kernel error bound
{title2=$\mathbb E|f_n^K(x,h)-f(x)|\leq C_1h^3+C_2(nh)^{-1/2}$}
Suppose a <probability density function> and its third <derivative> are bounded. A square-integrable unit-integral <kernel for density estimation> with first two moments zero and $\mu_3=\int|u|^3|K(u)|du<\infty$ has <bias of a kernel density estimator> at most $\|f^{(3)}\|_\infty\mu_3h^3/6$, by <Taylor theorem with Lagrange remainder> and moment cancellation. Independence gives <variance> at most $\|f\|_\infty\|K\|_2^2/(nh)$. The <Cauchy-Schwarz inequality> bounds the expected centred absolute deviation by the square root of this variance. Adding the bias proves the inequality uniformly in $x$. The choice $h=n^{-1/7}$ gives expected absolute error $O(n^{-3/7})$.
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