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\name{GammaSupp}
\alias{GammaSupp}
\alias{mgamma}
\alias{levgamma}
\alias{mgfgamma}
\title{Moments and Moment Generating Function of the Gamma Distribution}
\description{
Raw moments, limited moments and moment generating function for the
Gamma distribution with parameters \code{shape} and \code{scale}.
}
\usage{
mgamma(order, shape, rate = 1, scale = 1/rate)
levgamma(limit, shape, rate = 1, scale = 1/rate, order = 1)
mgfgamma(t, shape, rate = 1, scale = 1/rate, log = FALSE)
}
\arguments{
\item{order}{order of the moment.}
\item{limit}{limit of the loss variable.}
\item{rate}{an alternative way to specify the scale.}
\item{shape, scale}{shape and scale parameters. Must be strictly
positive.}
\item{t}{numeric vector.}
\item{log}{logical; if \code{TRUE}, the cumulant generating function
is returned.}
}
\details{
The \eqn{k}th raw moment of the random variable \eqn{X} is
\eqn{E[X^k]}{E[X^k]}, the \eqn{k}th limited moment at some limit
\eqn{d} is \eqn{E[\min(X, d)^k]}{E[min(X, d)^k]} and the moment
generating function is \eqn{E[e^{tX}]}, \eqn{k >
-\alpha}{k > -shape}.
}
\value{
\code{mgamma} gives the \eqn{k}th raw moment,
\code{levgamma} gives the \eqn{k}th moment of the limited loss
variable, and
\code{mgfgamma} gives the moment generating function in \code{t}.
Invalid arguments will result in return value \code{NaN}, with a warning.
}
\seealso{
\code{\link[stats]{GammaDist}}
}
\references{
Johnson, N. L. and Kotz, S. (1970), \emph{Continuous Univariate
Distributions, Volume 1}, Wiley.
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012),
\emph{Loss Models, From Data to Decisions, Fourth Edition}, Wiley.
}
\author{
Vincent Goulet \email{vincent.goulet@act.ulaval.ca},
Christophe Dutang and Mathieu Pigeon
}
\examples{
mgamma(2, 3, 4) - mgamma(1, 3, 4)^2
levgamma(10, 3, 4, order = 2)
mgfgamma(1,3,2)
}
\keyword{distribution}
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