ghypMeanVarMode: Moments and Mode of the Generalized Hyperbolic Distribution

In GeneralizedHyperbolic: The Generalized Hyperbolic Distribution

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 [e-mail], 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")

GeneralizedHyperbolic documentation built on Nov. 26, 2023, 5:07 p.m.