193.204.248.189: /* Definitions */

2024-04-16T08:00:11Z

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In [[statistics]], the '''order of a kernel''' is the degree of the first non-zero [[Moment (mathematics)|moment]] of a [[Kernel (statistics)|kernel]].<ref>{{citation|title=Nonparametric Econometrics: Theory and Practice|first1=Qi|last1=Li|first2=Jeffrey Scott|last2=Racine|publisher=Princeton University Press|year=2011|isbn=9781400841066|contribution=1.11 Higher Order Kernel Functions|contribution-url=https://books.google.com/books?id=Zsa7ofamTIUC&pg=PT63}}</ref>

== Definitions ==
The literature knows two major definitions of the ''order of a kernel''. Namely are:

=== Definition 1 ===
Let <math> \ell \geq 1 </math> be an integer. Then, <math> K: \mathbb{R} \rightarrow \mathbb{R} </math> is a ''kernel of order <math> \ell </math>'' if the functions <math> u\mapsto u^{j}K(u), ~ j=0,1,...,\ell </math> are ''integrable'' and satisfy
<math> \int K(u)du=1, ~ \int u^{j}K(u)du=0,~ ~j=1,...,\ell. </math><ref>{{cite book|last1=Tsybakov|first1=Alexandre B.|title=Introduction to Nonparametric Econometrics|date=2009|publisher=Springer|isbn=9780387790510|location=New York, NY|page=5}}</ref>

=== Definition 2 ===

==References==
{{reflist}}

{{Statistics-stub}}
[[Category:Nonparametric statistics]]

Order of a kernel - Revision history

193.204.248.189: /* Definitions */