 Kernel of order 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.