# Generalized Hyperbolic Distribution

`dist-gh.Rd`

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).

## 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

```
## 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
```