Order of a kernel

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

Definitions

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