Integrated second-order kernel bias bound
= Integrated second-order kernel bias bound
{title2=$\|K_h*f-f\|_2\le h^2\mu_2(K)\|f^{\prime\prime}\|_2/2$}
For a nonnegative symmetric unit-integral <kernel for density estimation> and $f\in H^2(\mathbb R)$, the integral <Taylor remainder> and <Minkowski inequality> give $\|K_h*f-f\|_2\le h^2\mu_2(K)\|f^{\prime\prime}\|_2/2$. Translation preserves the <L2 norm>. For a signed kernel the same proof requires the absolute moment $\int u^2|K(u)|\,du$.