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\name{gh}
\alias{gh}
\alias{dgh}
\alias{pgh}
\alias{qgh}
\alias{rgh}
\title{Generalized Hyperbolic Distribution}
\description{
Calculates moments of the generalized hyperbbolic
distribution function.
}
\usage{
dgh(x, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2, log = FALSE)
pgh(q, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)
qgh(p, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)
rgh(n, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)
}
\arguments{
\item{alpha, beta, delta, mu, lambda}{
numeric values.
\code{alpha} is the first shape parameter;
\code{beta} is the second shape parameter in the range \code{(0, alpha)};
\code{delta} is the scale parameter, must be zero or positive;
\code{mu} is the location parameter, by default 0; and
\code{lambda} defines the sublclass, by default -1/2.
These are the meanings of the parameters in the first
parameterization \code{pm=1} which is the default
parameterization.
In the second parameterization, \code{pm=2} \code{alpha}
and \code{beta} take the meaning of the shape parameters
(usually named) \code{zeta} and \code{rho}.
In the third parameterization, \code{pm=3} \code{alpha}
and \code{beta} take the meaning of the shape parameters
(usually named) \code{xi} and \code{chi}.
In the fourth parameterization, \code{pm=4} \code{alpha}
and \code{beta} take the meaning of the shape parameters
(usually named) \code{a.bar} and \code{b.bar}.
}
\item{log}{
a logical flag by default \code{FALSE}.
Should labels and a main title drawn to the plot?
}
\item{n}{
number of observations.
}
\item{p}{
a numeric vector of probabilities.
}
\item{x, q}{
a numeric vector of quantiles.
}
\item{\dots}{
arguments to be passed to the function \code{integrate}.
}
}
\value{
All values for the \code{*gh} functions are numeric vectors:
\code{d*} returns the density,
\code{p*} returns the distribution function,
\code{q*} returns the quantile function, and
\code{r*} generates random deviates.
All values have attributes named \code{"param"} listing
the values of the distributional parameters.
}
\details{
The generator \code{rgh} is based on the GH algorithm given
by Scott (2004).
}
\references{
Atkinson, A.C. (1982);
\emph{The simulation of generalized inverse Gaussian and hyperbolic
random variables},
SIAM J. Sci. Stat. Comput. 3, 502--515.
Barndorff-Nielsen O. (1977);
\emph{Exponentially decreasing distributions for the logarithm of
particle size},
Proc. Roy. Soc. Lond., A353, 401--419.
Barndorff-Nielsen O., Blaesild, P. (1983);
\emph{Hyperbolic distributions. In Encyclopedia of Statistical
Sciences},
Eds., Johnson N.L., Kotz S. and Read C.B.,
Vol. 3, pp. 700--707. New York: Wiley.
Raible S. (2000);
\emph{Levy Processes in Finance: Theory, Numerics and Empirical Facts},
PhD Thesis, University of Freiburg, Germany, 161 pages.
}
\author{
David Scott for code implemented from \R's
contributed package \code{HyperbolicDist}.
}
\examples{
## rgh -
set.seed(1953)
r = rgh(5000, alpha = 1, beta = 0.3, delta = 1)
plot(r, type = "l", col = "steelblue",
main = "gh: alpha=1 beta=0.3 delta=1")
## dgh -
# Plot empirical density and compare with true density:
hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
x = seq(-5, 5, 0.25)
lines(x, dgh(x, alpha = 1, beta = 0.3, delta = 1))
## pgh -
# Plot df and compare with true df:
plot(sort(r), (1:5000/5000), main = "Probability", col = "steelblue")
lines(x, pgh(x, alpha = 1, beta = 0.3, delta = 1))
## qgh -
# Compute Quantiles:
qgh(pgh(seq(-5, 5, 1), alpha = 1, beta = 0.3, delta = 1),
alpha = 1, beta = 0.3, delta = 1)
}
\keyword{distribution}
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