Density, distribution function, quantile function and random generation for the hyperbolic distribution.

## Usage

dhyp(x, alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1, log = FALSE)
phyp(q, alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1, ...)
qhyp(p, alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1, ...)
rhyp(n, alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1)

## Arguments

x, q

numeric vector of quantiles.

p

numeric vector of probabilities.

n

number of observations.

alpha

shape parameter, a positive number. alpha can also be a vector of length four, containing alpha, beta, delta and mu (in that order).

beta

skewness parameter, abs(beta) is in the range (0, alpha).

delta

scale parameter, must be zero or positive.

mu

location parameter, by default 0.

pm

integer number specifying the parameterisation, one of 1, 2, 3, or 4. The default is the first parameterization.

log

a logical value, if TRUE, probabilities p are given as log(p).

...

arguments to be passed to the function integrate.

## Details

dhyp gives the density, phyp gives the distribution function, qhyp gives the quantile function, and rhyp generates random deviates.

The meaning of the parameters given above corresponds to the first parameterization, pm = 1, which is the default.

In the second parameterization, pm=2, alpha and beta take the meaning of the shape parameters (usually named) zeta and rho.

In the third parameterization, pm=3, alpha and beta take the meaning of the shape parameters (usually named) xi and chi.

In the fourth parameterization, pm=4, alpha and beta take the meaning of the shape parameters (usually named) a.bar and b.bar.

The generator rhyp is based on the HYP algorithm given by Atkinson (1982).

numeric vector

## Author

David Scott for code implemented from R's contributed package HyperbolicDist.

## References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502--515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401--419.

Barndorff-Nielsen O., Blaesild, P. (1983); 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); 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)
#>  [1] -5.000017e+00 -3.999997e+00 -3.000000e+00 -2.000023e+00 -1.000003e+00
#>  [6] -2.344483e-06  1.000000e+00  2.000000e+00  2.999997e+00  3.999989e+00
#> [11]  5.000002e+00