fBasics (version 3011.87)

gh: 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

alpha, beta, delta, mu, lambda

numeric values. alpha is the first shape parameter; beta is the second shape parameter in the range (0, alpha); delta is the scale parameter, must be zero or positive; mu is the location parameter, by default 0; and lambda defines the sublclass, by default -1/2. These are the meanings of the parameters in the first parameterization pm=1 which is the default parameterization. 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.

log

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

n

number of observations.

p

a numeric vector of probabilities.

x, q

a numeric vector of quantiles.

...

arguments to be passed to the function integrate.

Value

All values for the *gh functions are numeric vectors: d* returns the density, p* returns the distribution function, q* returns the quantile function, and r* generates random deviates.All values have attributes named "param" listing the values of the distributional parameters.

Details

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) 

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