Order of a kernel
From HandWiki
In statistics, the order of a kernel is the degree of the first non-zero moment of a kernel.[1]
Definitions
The literature knows two major definitions of the order of a kernel:
Definition 1
Let [math]\displaystyle{ \ell \geq 1 }[/math] be an integer. Then, [math]\displaystyle{ K: \mathbb{R} \rightarrow \mathbb{R} }[/math] is a kernel of order [math]\displaystyle{ \ell }[/math] if the functions [math]\displaystyle{ u\mapsto u^{j}K(u), ~ j=0,1,...,\ell }[/math] are integrable and satisfy [math]\displaystyle{ \int K(u)du=1, ~ \int u^{j}K(u)du=0,~ ~j=1,...,\ell. }[/math][2]
Definition 2
References
- ↑ Li, Qi; Racine, Jeffrey Scott (2011), "1.11 Higher Order Kernel Functions", Nonparametric Econometrics: Theory and Practice, Princeton University Press, ISBN 9781400841066, https://books.google.com/books?id=Zsa7ofamTIUC&pg=PT63
- ↑ Tsybakov, Alexandre B. (2009). Introduction to Nonparametric Econometrics. New York, NY: Springer. p. 5. ISBN 9780387790510.
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