= Kernel of order ell
{title2=$\ell$}
= Kernels of order ell
{synonym}
In <kernel density estimation>, an <integrable> <kernel for density estimation> $K$ is normalized by $\int K(u)\,du=1$. A common exact-order convention calls it a <kernel of order ell> when, for a positive <integer> $\ell$,
$$
\int u^jK(u)\,du=0\quad(1\leq j<\ell),\qquad
0<\left|\int u^\ell K(u)\,du\right|<\infty,
\qquad\int|u|^\ell|K(u)|\,du<\infty.
$$
Some authors say \i[order at least] $\ell$ without the nonzero final signed <moment> condition. Stating the convention makes clear whether the first surviving <Taylor expansion> term is actually of order $h^\ell$ or merely bounded by that order.
A symmetric <Gaussian density kernel> has order two: its first <moment> is zero and its second <moment> is one. A nonnegative ordinary <probability density function> cannot have zero second <moment> unless it is concentrated at zero, which no <integrable> <probability density function>[density] can do. Hence kernels with the first two <moments> zero must take both signs. These higher-order kernels can reduce smoothing <bias of an estimator>, but the resulting estimate need not itself be nonnegative.
Back to article page
