Order of a kernel explained

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:

Let

\ell\geq1

be an integer. Then,

K:R → R

is a kernel of order
\ell

if the functions

u\mapstou^jK(u),~j=0,1,...,\ell

are integrable and satisfy

\intK(u)du=1,~\intu^jK(u)du=0,~~j=1,...,\ell.

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Book: Tsybakov. Alexandre B.. Introduction to Nonparametric Econometrics. 2009. Springer. 9780387790510. New York, NY. 5.