Computes the complete or incomplete \(m\)th moment of a
smoothing kernel.

Usage

kernel.moment(m, r, kernel = "gaussian")

Value

A single number, or a numeric vector of the same length as r.

Arguments

m

Exponent (order of moment).
An integer.

r

Upper limit of integration for the incomplete moment.
A numeric value or numeric vector.
Set r=Inf to obtain the complete moment.

kernel

String name of the kernel.
Options are
"gaussian", "rectangular",
"triangular",
"epanechnikov",
"biweight",
"cosine" and "optcosine".
(Partial matching is used).

Kernel estimation of a probability density in one dimension
is performed by density.default
using a kernel function selected from the list above.
For more information about these kernels,
see density.default.

The function kernel.moment computes the partial integral
$$
\int_{-\infty}^r t^m k(t) dt
$$
where \(k(t)\) is the selected kernel, \(r\) is the upper limit of
integration, and \(m\) is the exponent or order.
Here \(k(t)\) is the standard form of the kernel,
which has support \([-1,1]\) and
standard deviation \(sigma = 1/c\) where c = kernel.factor(kernel).