stieltjes.Rd
\name{stieltjes}
\alias{stieltjes}
\title{Compute Integral of Function Against Cumulative Distribution}
\description{
Computes the Stieltjes integral
of a function \eqn{f} with respect to a function \eqn{M}.
}
\usage{
stieltjes(f, M, ...)
}
\arguments{
\item{f}{
The integrand. A function in the \R language.
}
\item{M}{
The cumulative function against which \code{f} will be
integrated. An object of class \code{"fv"}.
}
\item{\dots}{
Additional arguments passed to \code{f}.
}
}
\details{
This command computes the Stieltjes integral
\deqn{I = \int f(x) dM(x)}{I = integral f(x) dM(x)}
of a real-valued function \eqn{f(x)}
with respect to a nondecreasing function \eqn{M(x)}.
One common use of the Stieltjes integral is
to find the mean value of a random variable from its
cumulative distribution function \eqn{F(x)}. The mean value is
the Stieltjes integral of \eqn{f(x)=x} with respect to \eqn{F(x)}.
The argument \code{f} should be a \code{function} in the \R language.
It should accept a numeric vector argument \code{x} and should return
a numeric vector of the same length.
The argument \code{M} should be a function value table
(object of class \code{"fv"}, see \code{\link{fv.object}}).
Such objects are returned
by the commands \code{link{Kest}}, \code{\link{Gest}}, etc.
}
\value{
A list containing the value of the Stieltjes integral
computed using each of the versions of the function \code{M}.
}
\seealso{
\code{\link{fv.object}},
\code{\link{Gest}}
}
\examples{
data(redwood)
# estimate cdf of nearest neighbour distance
G <- Gest(redwood)
# compute estimate of mean nearest neighbour distance
stieltjes(function(x){x}, G)
# estimated probability of a distance in the interval [0.1,0.2]
stieltjes(function(x,a,b){ (x >= a) & (x <= b)}, G, a=0.1, b=0.2)
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{math}