• Source: 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



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 .}





= Definition 2

=




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