# dghyp: Generalized Hyperbolic Distribution In HyperbolicDist: The Hyperbolic Distribution

 GeneralizedHyperbolic R Documentation

## Generalized Hyperbolic Distribution

### Description

Density function, distribution function, quantiles and random number generation for the generalized hyperbolic distribution with parameter vector `Theta`. Utility routines are included for the derivative of the density function and to find suitable break points for use in determining the distribution function.

### Usage

```dghyp(x, Theta)
pghyp(q, Theta, small = 10^(-6), tiny = 10^(-10),
deriv = 0.3, subdivisions = 100, accuracy = FALSE, ...)
qghyp(p, Theta, small = 10^(-6), tiny = 10^(-10),
deriv = 0.3, nInterpol = 100, subdivisions = 100, ...)
rghyp(n, Theta)
ddghyp(x, Theta)
ghypBreaks(Theta, small = 10^(-6), tiny = 10^(-10), deriv = 0.3, ...)
```

### Arguments

 `x,q` Vector of quantiles. `p` Vector of probabilities. `n` Number of observations to be generated. `Theta` Parameter vector taking the form `c(lambda,alpha,beta,delta,mu)`. `small` Size of a small difference between the distribution function and zero or one. See Details. `tiny` Size of a tiny difference between the distribution function and zero or one. See Details. `deriv` Value between 0 and 1. Determines the point where the derivative becomes substantial, compared to its maximum value. See Details. `accuracy` Uses accuracy calculated by~`integrate` to try and determine the accuracy of the distribution function calculation. `subdivisions` The maximum number of subdivisions used to integrate the density returning the distribution function. `nInterpol` The number of points used in qghyp for cubic spline interpolation (see `splinefun`) of the distribution function. `...` Passes arguments to `uniroot`. See Details.

### Details

The generalized hyperbolic distribution has density

f(x)=c(lambda,alpha,beta,delta) (K_(lambda-1/2)(alpha sqrt(delta^2+(x-mu)^2)))/ ((sqrt(delta^2+(x-mu)^2)/alpha)^(1/2-lambda)) exp(beta(x-mu))

where K_nu() is the modified Bessel function of the third kind with order nu, and

c(lambda,alpha,beta,delta)= (sqrt(alpha^2-beta^2)/delta)^lambda/ (sqrt(2π)K_lambda(delta sqrt(alpha^2-beta^2)))

Use `ghypChangePars` to convert from the (zeta, rho), xi,chi), or (alpha bar, beta bar) parameterisations to the (alpha, beta) parameterisation used above.

`pghyp` breaks the real line into eight regions in order to determine the integral of `dghyp`. The break points determining the regions are found by `ghypBreaks`, based on the values of `small`, `tiny`, and `deriv`. In the extreme tails of the distribution where the probability is `tiny` according to `ghypCalcRange`, the probability is taken to be zero. In the inner part of the distribution, the range is divided in 6 regions, 3 above the mode, and 3 below. On each side of the mode, there are two break points giving the required three regions. The outer break point is where the probability in the tail has the value given by the variable `small`. The inner break point is where the derivative of the density function is `deriv` times the maximum value of the derivative on that side of the mode. In each of the 6 inner regions the numerical integration routine `safeIntegrate` (which is a wrapper for `integrate`) is used to integrate the density `dghyp`.

`qghyp` use the breakup of the real line into the same 8 regions as `pghyp`. For quantiles which fall in the 2 extreme regions, the quantile is returned as `-Inf` or `Inf` as appropriate. In the 6 inner regions `splinefun` is used to fit values of the distribution function generated by `pghyp`. The quantiles are then found using the `uniroot` function.

`pghyp` and `qghyp` may generally be expected to be accurate to 5 decimal places.

The generalized hyperbolic distribution is discussed in Bibby and S<f6>renson (2003). It can be represented as a particular mixture of the normal distribution where the mixing distribution is the generalized inverse Gaussian. `rghyp` uses this representation to generate observations from the generalized hyperbolic distribution. Generalized inverse Gaussian observations are obtained via the algorithm of Dagpunar (1989) which is implemented in `rgig`.

### Value

`dghyp` gives the density, `pghyp` gives the distribution function, `qghyp` gives the quantile function and `rghyp` generates random variates. An estimate of the accuracy of the approximation to the distribution function may be found by setting `accuracy=TRUE` in the call to `pghyp` which then returns a list with components `value` and `error`.

`ddghyp` gives the derivative of `dghyp`.

`ghypBreaks` returns a list with components:

 `xTiny` Value such that probability to the left is less than `tiny`. `xSmall` Value such that probability to the left is less than `small`. `lowBreak` Point to the left of the mode such that the derivative of the density is `deriv` times its maximum value on that side of the mode. `highBreak` Point to the right of the mode such that the derivative of the density is `deriv` times its maximum value on that side of the mode. `xLarge` Value such that probability to the right is less than `small`. `xHuge` Value such that probability to the right is less than `tiny`. `modeDist` The mode of the given generalized hyperbolic distribution.

### Author(s)

David Scott d.scott@auckland.ac.nz, Richard Trendall

### References

Barndorff-Nielsen, O. and Bl<e6>sild, 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.

Bibby, B. M. and S<f6>renson, M. (2003). Hyperbolic processes in finance. In Handbook of Heavy Tailed Distributions in Finance, ed., Rachev, S. T. pp. 212–248. Elsevier Science B.~V.

Dagpunar, J.S. (1989). An easily implemented generalised inverse Gaussian generator Commun. Statist. -Simula., 18, 703–710.

Prause, K. (1999) The generalized hyperbolic models: Estimation, financial derivatives and risk measurement. PhD Thesis, Mathematics Faculty, University of Freiburg.

`dhyperb` for the hyperbolic distribution, `dgig` for the generalized inverse Gaussian distribution `safeIntegrate`, `integrate` for its shortfalls, `splinefun`, `uniroot` and `ghypChangePars` for changing parameters to the (alpha,beta) parameterisation

### Examples

```Theta <- c(1/2,3,1,1,0)
ghypRange <- ghypCalcRange(Theta, tol = 10^(-3))
par(mfrow = c(1,2))
curve(dghyp(x, Theta), from = ghypRange[1], to = ghypRange[2],
n = 1000)
title("Density of the\n Generalized Hyperbolic Distribution")
curve(pghyp(x, Theta), from = ghypRange[1], to = ghypRange[2],
n = 1000)
title("Distribution Function of the\n Generalized Hyperbolic Distribution")
dataVector <- rghyp(500, Theta)
curve(dghyp(x, Theta), range(dataVector)[1], range(dataVector)[2],
n = 500)
hist(dataVector, freq = FALSE, add =TRUE)
title("Density and Histogram of the\n Generalized Hyperbolic Distribution")
logHist(dataVector, main =
"Log-Density and Log-Histogramof the\n Generalized Hyperbolic Distribution")
range(dataVector)[1], range(dataVector)[2], n = 500)
par(mfrow = c(2,1))
curve(dghyp(x, Theta), from = ghypRange[1], to = ghypRange[2],
n = 1000)
title("Density of the\n Generalized Hyperbolic Distribution")
curve(ddghyp(x, Theta), from = ghypRange[1], to = ghypRange[2],
n = 1000)
title("Derivative of the Density of the\n Generalized Hyperbolic Distribution")
par(mfrow = c(1,1))
ghypRange <- ghypCalcRange(Theta, tol = 10^(-6))
curve(dghyp(x, Theta), from = ghypRange[1], to = ghypRange[2],
n = 1000)
bks <- ghypBreaks(Theta)
abline(v = bks)
```

HyperbolicDist documentation built on March 18, 2022, 6:23 p.m.