Order of a kernel

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 $\ell \geq 1$ be an integer. Then, $K:\mathbb {R} \rightarrow \mathbb {R}$ is a kernel of order $\ell$ if the functions $u\mapsto u^{j}K(u),~j=0,1,...,\ell$ are integrable and satisty $\int K(u)du=1,~\int u^{j}K(u)du=0,~~j=1,...,\ell .$ [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.

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[1] Li, Qi; Racine, Jeffrey Scott (2011), "1.11 Higher Order Kernel Functions", Nonparametric Econometrics: Theory and Practice, Princeton University Press, ISBN 9781400841066

[2] Tsybakov, Alexandre B. (2009). Introduction to Nonparametric Econometrics. New York, NY: Springer. p. 5. ISBN 9780387790510.