Order of a kernel

In statistics, the order of a kernel is the degree of the first non-zero moment of a kernel.

Definitions

The literature knows two major definitions of the order of a kernel. Namely are:

Definition 1

Let $\ell \geq 1$ be an integer. Then, $K:\mathbb {R} \rightarrow \mathbb {R}$ is a kernel of order $\ell$ if the functions $u\mapsto u^{j}K(u),~j=0,1,...,\ell$ are integrable and satisfy $\int K(u)du=1,~\int u^{j}K(u)du=0,~~j=1,...,\ell .$