Order of a kernel

In statistics, the order of a kernel is the degree of the first non-zero moment of a kernel.<ref>Template:Citation</ref>

Definitions

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

Definition 1

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