= Second-order pointwise bias bound for kernel density estimation
{title2=$|\mathbb E\widehat f_h(x)-f(x)|\leq h^2\mu_2(K)\|f^{\prime\prime}\|_\infty/2$}
Let a nonnegative <kernel for density estimation> $K$ have integral one, zero first moment and finite second moment $\mu_2(K)$. If a <probability density function> $f$ has bounded <second derivative>, its <kernel density estimator> has pointwise <bias> at most $h^2\mu_2(K)\|f^{\prime\prime}\|_\infty/2$. Apply the <Taylor theorem with Lagrange remainder> to $f(x-hu)$ and integrate: the first-order term vanishes and the remainder is bounded by $h^2u^2\|f^{\prime\prime}\|_\infty/2$. For $K=\mathbb1_{[-1/2,1/2]}$, $\mu_2(K)=1/12$.
Back to article page
