Kernel of order ell ($\ell$)

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 $\ell$,Some authors say 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 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.

For normalized Legendre polynomials, this kernel for density estimation reproduces evaluation at zero on every polynomial of degree below $\ell$: expand $p$ in the orthonormal basis to obtain $\int K_{\ell }p=p(0)$. Taking monomials proves integral one and zero moments of orders $1,\dots ,\ell -1$; compact support gives finite absolute moments. If exact order requires a nonzero moment of order $\ell$, add $({\varphi }_{\ell }(0)+1){\varphi }_{\ell }{1}_{[-1,1]}$. Lower moments remain unchanged. Writing $a_{\ell }=⟨u^{\ell },{\varphi }_{\ell }⟩\ne 0$ and evaluating the full expansion of $u^{\ell }$ at zero shows that the new moment is $a_{\ell }$. A useful three-derivative example is $K_3=(9-15u^2)1_{[-1,1]}/8$, whose first two moments vanish and whose squared L2 norm is $9/8$.