In statistics, the order of a kernel is the degree of the first non-zero moment of a kernel.
Definitions Let ℓ ≥ 1 {\displaystyle \ell \geq 1} be an integer. Then, K : R → R {\displaystyle K:\mathbb {R} \rightarrow \mathbb {R} } is a kernel of order ℓ {\displaystyle \ell } if the functions u ↦ u j K ( u ) , j = 0 , 1 , . . . , ℓ {\displaystyle u\mapsto u^{j}K(u),~j=0,1,...,\ell } are integrable and satisfy
∫ K ( u ) d u = 1 , ∫ u j K ( u ) d u = 0 , j = 1 , . . . , ℓ . {\displaystyle \int K(u)du=1,~\int u^{j}K(u)du=0,~~j=1,...,\ell .}
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