\name{st.cumulants}
\alias{st.cumulants}
\alias{st.cumulants.inversion}
\title{
Cumulants of the skew-t distribution
}
\description{
Cumulants of the skew-t distribution and inverse matching
}
\synopsis{
st.cumulants(location = 0, scale = 1, shape = 0, df = Inf, dp = NULL,  n = 4) 
st.cumulants.inversion(cum, abstol = 1e-08) 
}
\usage{
st.cumulants(location = 0, scale = 1, shape = 0, df = Inf, n = 4)
st.cumulants(dp=, n = 4)
st.cumulants.inversion(cum, abstol = 1e-08)
}
\arguments{
\item{location}{
 location parameter (vector)
 }
\item{scale}{
 scale parameter (vector)
 }
\item{shape}{
 shape parameter (vector)
 }
\item{df}{
 degrees of freedom (scalar); default is \code{df=Inf} which corresponds 
 to the skew-normal distribution.
 }
\item{dp}{
  a vector of four elements, whose elements are \code{(location, scale,
  shape, df)} respectively. If \code{dp} is specified,  then
  the individual parameters must not be.
}
\item{n}{
  a scalar integer of the maximal order or cumulants required;
  it must be from 1 to 4 and smaller than \code{df}
}
\item{cum}{
  a vector of 4 elements which are taken to represent the first
  4 cumulants of a skew-t distribution (hence the second term  must
  be positive)
}
\item{abstol}{
 a scalar which regulates the accuracy of the cumulants matching (default
 value 1e-08)
}}
\value{
\code{st.cumulants} computes the cumulants up to order \code{n} of 
the skew-t distribution with the selected parameters. The returned object
is a vector of length \code{n} if the parameters are all scalar, 
otherwise a matrix with \code{n} columns.

\code{st.cumulants.inversion} returns a vector with the \code{dp} parameters 
of the  matching skew-t distribution
}
\details{
Expressions of the moments and other details on the skew-t distribution are 
given in the reference below. These formulae are used by \code{st.cumulants}
to compute the cumulants.

\code{st.cumulants.inversion} searches the set of \code{shape} and \code{df}
parameters of the skew-t family, attempting to match  the third and fourth 
cumulants with those of the supplied vector \code{cum}. 
This search is done numerically twice,
once using \code{optim} and a second time using \code{nlminb}, 
to the accuracy \code{abstol}; the best matching solution is retained.
If the required accuracy of the matching is not achieved by any of the
two methods, a warning message is issued.
After this step, the other two parameters (\code{location} and
\code{scale}) are computed via simple algebra.
}
\note{
  The joint use \code{st.cumulants.inversion} and
  \code{sample.centralmoments} allows to fit a skew-t distribution by
  the methods of moments; see the example below. Note however, that for
  stability reasons, this is \emph{not} adopted as the standard method
  for producing initial values of MLE search.
}
\references{
Azzalini, A. and Capitanio, A. (2003).
Distributions generated by perturbation of symmetry 
with emphasis on a multivariate skew-\emph{t} distribution.
\emph{J. Roy. Statist. Soc. B }
\bold{65}, 367--389.
}
\seealso{
  \code{\link{sn.cumulants}},\code{\link{dst}},
  \code{\link{sample.centralmoments}}, \code{\link{optim}},
  \code{\link{nlminb}}
}
\examples{
st.cumulants(shape=c(0,3,9), df=5)
cum <- st.cumulants(dp=c(10, 2, -8, 5.2))
st.cumulants.inversion(cum)
#
data(ais, package='sn')
mom <- sample.centralmoments(ais[,"bmi"])
st.cumulants.inversion(cum=c(mom[1:3],mom[4]-3*mom[2]^2))
# parameters of the ST distribution fitted by method of moments
}
\keyword{distribution}