ghypMeanVarMode function

Moments and Mode of the Generalized Hyperbolic Distribution

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

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.

Returns

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.

References

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

Author(s)

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

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")