Last data update: 2014.03.03

R: Moments and Mode of the Generalized Hyperbolic Distribution
 Specific Generalized Hyperbolic Moments and Mode R Documentation

## Moments and Mode of the Generalized Hyperbolic Distribution

### Description

Functions to calculate the mean, variance, skewness, kurtosis and mode of a specific generalized hyperbolic distribution.

### Usage

ghypMean(mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
param = c(mu, delta, alpha, beta, lambda))
ghypVar(mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
param = c(mu, delta, alpha, beta, lambda))
ghypSkew(mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
param = c(mu, delta, alpha, beta, lambda))
ghypKurt(mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
param = c(mu, delta, alpha, beta, lambda))
ghypMode(mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
param = c(mu, delta, alpha, beta, lambda))

### Arguments

 mu mu is the location parameter. By default this is set to 0. delta delta is the scale parameter of the distribution. A default value of 1 has been set. alpha alpha is the tail parameter, with a default value of 1. beta beta is the skewness parameter, by default this is 0. lambda lambda is the shape parameter and dictates the shape that the distribution shall take. Default value is 1. param Parameter vector of the generalized hyperbolic distribution.

### Value

ghypMean gives the mean of the generalized hyperbolic distribution, ghypVar the variance, ghypSkew the skewness, ghypKurt the kurtosis, and ghypMode the mode. The formulae used for the mean is given in Prause (1999). The variance, skewness and kurtosis are obtained using the recursive formula implemented in ghypMom which can calculate moments of all orders about any point.

The mode is found by a numerical optimisation using optim. For the special case of the hyperbolic distribution a formula for the mode is available, see hyperbMode.

The parameterization of the generalized hyperbolic distribution used for these functions is the (alpha, beta) one. See ghypChangePars to transfer between parameterizations.

### Author(s)

David Scott d.scott@auckland.ac.nz, Thomas Tran

### References

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

dghyp, ghypChangePars, besselK, RLambda.

### Examples

param <- c(2, 2, 2, 1, 2)
ghypMean(param = param)
ghypVar(param = param)
ghypSkew(param = param)
ghypKurt(param = param)
ghypMode(param = param)
maxDens <- dghyp(ghypMode(param = param), param = param)
ghypRange <- ghypCalcRange(param = param, tol = 10^(-3) * maxDens)
curve(dghyp(x, param = param), ghypRange[1], ghypRange[2])
abline(v = ghypMode(param = param), col = "blue")
abline(v = ghypMean(param = param), col = "red")

### Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(GeneralizedHyperbolic)
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/GeneralizedHyperbolic/ghypMeanVarMode.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Specific Generalized Hyperbolic Moments and Mode
> ### Title: Moments and Mode of the Generalized Hyperbolic Distribution
> ### Aliases: ghypMean ghypVar ghypSkew ghypKurt ghypMode
> ### Keywords: distribution
>
> ### ** Examples
>
> param <- c(2, 2, 2, 1, 2)
> ghypMean(param = param)
[1] 4.125532
> ghypVar(param = param)
[1] 3.192044
> ghypSkew(param = param)
[1] 0.7928604
> ghypKurt(param = param)
[1] 1.343511
> ghypMode(param = param)
[1] 3.499997
> maxDens <- dghyp(ghypMode(param = param), param = param)
> ghypRange <- ghypCalcRange(param = param, tol = 10^(-3) * maxDens)
> curve(dghyp(x, param = param), ghypRange[1], ghypRange[2])
> abline(v = ghypMode(param = param), col = "blue")
> abline(v = ghypMean(param = param), col = "red")
>
>
>
>
>
> dev.off()
null device
1
>