= Canonical kernel for density estimation
{title2=$K_c(u)=c^{-1}K(u/c),\quad c=[R(K)/\mu_2(K)^2]^{1/5}$}
= Canonical kernel
{synonym}
For a nonnegative symmetric unit-integral <kernel for density estimation> with $0<\mu_2(K)=\int u^2K(u)\,du<\infty$ and $R(K)=\int K^2<\infty$, the canonical scaling satisfies $R(K_c)=\mu_2(K_c)^2$. Since $R(K_c)=R(K)/c$ and $\mu_2(K_c)=c^2\mu_2(K)$, the unique scale is $c=[R(K)/\mu_2(K)^2]^{1/5}$. Consequently its <asymptotic mean integrated squared error> is $C(K)[(nh)^{-1}+h^4R(f^{\prime\prime})/4]$, where $C(K)=R(K)^{4/5}\mu_2(K)^{2/5}$ is invariant under kernel rescaling. The leading optimal <smoothing bandwidth> becomes $[nR(f^{\prime\prime})]^{-1/5}$, independent of the kernel shape. Unit second moment is a different normalization and does not have this separation property.
Back to article page
