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 \( {\displaystyle \ell \geq 1} \) be an integer. Then, \( {\displaystyle K:\mathbb {R} \rightarrow \mathbb {R} } \) is a kernel of order \( \ell \) if the functions \( {\displaystyle u\mapsto u^{j}K(u),~j=0,1,...,\ell } \) are integrable and satisty \( {\displaystyle \int K(u)du=1,~\int u^{j}K(u)du=0,~~j=1,...,\ell .} \)[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
Tsybakov, Alexandre B. (2009). Introduction to Nonparametric Econometrics. New York, NY: Springer. p. 5. ISBN 9780387790510.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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