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

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.

See Also

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)

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Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
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> library(GeneralizedHyperbolic)
Loading required package: DistributionUtils
Loading required package: RUnit
> 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 
>