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

## 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

x, q

a numeric vector of quantiles.

p

a numeric vector of probabilities.

n

number of observations.

alpha

first shape parameter.

beta

second shape parameter, should in the range `(0, alpha).`

delta

scale parameter, must be zero or positive.

mu

location parameter, by default 0.

lambda

defines the sublclass, by default \(-1/2\).

log

a logical flag by default `FALSE`. Should labels and a main title drawn to the plot?

## Details

`dgh` gives the density, `pgh` gives the distribution function, `qgh` gives the quantile function, and `rgh` generates random deviates.

The meanings of the parameters correspond to the first parameterization, `pm=1`, which is the default parameterization for this distribution.

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 `rgh` is based on the GH algorithm given by Scott (2004).

numeric vector

## 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.

## Author

David Scott for code implemented from R's contributed package `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)
#>  [1] -5.000001e+00 -4.000000e+00 -3.000006e+00 -1.999996e+00 -1.000010e+00
#>  [6] -8.504243e-06  1.000003e+00  1.999972e+00  3.000000e+00  3.999998e+00
#> [11]  4.999976e+00
``````