swh:1:snp:d1587d616651317fdcebcbb237dce82c32266449
Tip revision: dfe6577bb164f53feaedf2da9530457197edfb1a authored by Georgi N. Boshnakov on 20 October 2022, 11:25:10 UTC
version 4021.93
version 4021.93
Tip revision: dfe6577
dist-hyp.Rd
\name{hyp}
\alias{hyp}
\alias{dhyp}
\alias{phyp}
\alias{qhyp}
\alias{rhyp}
\title{Hyperbolic distribution}
\description{
Density, distribution function, quantile function
and random generation for the hyperbolic distribution.
}
\usage{
dhyp(x, alpha = 1, beta = 0, delta = 1, mu = 0,
pm = c("1", "2", "3", "4"), log = FALSE)
phyp(q, alpha = 1, beta = 0, delta = 1, mu = 0,
pm = c("1", "2", "3", "4"), \dots)
qhyp(p, alpha = 1, beta = 0, delta = 1, mu = 0,
pm = c("1", "2", "3", "4"), \dots)
rhyp(n, alpha = 1, beta = 0, delta = 1, mu = 0,
pm = c("1", "2", "3", "4"))
}
\arguments{
\item{alpha, beta, delta, mu}{
shape parameter \code{alpha};
skewness parameter \code{beta}, \code{abs(beta)} is in the
range (0, alpha);
scale parameter \code{delta}, \code{delta} must be zero or
positive;
location parameter \code{mu}, by default 0.
These is the meaning of the parameters in the first
parameterization \code{pm=1} which is the default
parameterization selection.
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{n}{
number of observations.
}
\item{p}{
a numeric vector of probabilities.
}
\item{pm}{
an integer value between \code{1} and \code{4} for the
selection of the parameterization. The default takes the
first parameterization.
}
\item{x, q}{
a numeric vector of quantiles.
}
\item{log}{
a logical, if TRUE, probabilities \code{p} are given as
\code{log(p)}.
}
\item{\dots}{
arguments to be passed to the function \code{integrate}.
}
}
\value{
All values for the \code{*hyp} 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{rhyp} is based on the HYP algorithm given
by Atkinson (1982).
}
\author{
David Scott for code implemented from \R's
contributed package \code{HyperbolicDist}.
}
\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.
}
\examples{
## hyp -
set.seed(1953)
r = rhyp(5000, alpha = 1, beta = 0.3, delta = 1)
plot(r, type = "l", col = "steelblue",
main = "hyp: alpha=1 beta=0.3 delta=1")
## hyp -
# 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, dhyp(x, alpha = 1, beta = 0.3, delta = 1))
## hyp -
# Plot df and compare with true df:
plot(sort(r), (1:5000/5000), main = "Probability", col = "steelblue")
lines(x, phyp(x, alpha = 1, beta = 0.3, delta = 1))
## hyp -
# Compute Quantiles:
qhyp(phyp(seq(-5, 5, 1), alpha = 1, beta = 0.3, delta = 1),
alpha = 1, beta = 0.3, delta = 1)
}
\keyword{distribution}