skewhypMeanVarMode {SkewHyperbolic}

Moments and Mode of the Skew Hyperbolic Student t-Distribution.

Description

Functions to calculate the mean, variance, skewness, kurtosis and mode of a specified skew hyperbolic t-distribution.

Usage

skewhypMean(mu = 0, delta = 1, beta = 1, nu = 1,
            param = c(mu,delta,beta,nu))
skewhypVar(mu = 0, delta = 1, beta = 1, nu = 1,
           param = c(mu,delta,beta,nu))
skewhypSkew(mu = 0, delta = 1, beta = 1, nu = 1,
            param = c(mu,delta,beta,nu))
skewhypKurt(mu = 0, delta = 1, beta = 1, nu = 1,
            param = c(mu,delta,beta,nu))
skewhypMode(mu = 0, delta = 1, beta = 1, nu = 1,
            param = c(mu,delta,beta,nu),
            tolerance = .Machine$double.eps ^ 0.5)

Arguments

mu

Location parameter \mu, default is 0.

delta

Scale parameter \sigma, default is 1.

beta

Skewness parameter \beta, default is 1.

nu

Shape parameter \nu, default is 1.

param

Specifying the parameters as a vector of the form
c(mu,delta,beta,nu).

tolerance

A difference smaller than this value is taken to be zero.

Details

Users may either specify the values of the parameters individually or as a vector. If both forms are specified, then the values specified by the vector param will overwrite the other ones.In addition the parameter values are examined by calling the function skewhypCheckPars to see if they are valid.

The moments are calculated as per formulae in Aas&Haff(2006) and the mode is calculated by numerical optimisation of the density function using optim.

Note that the mean does not exist when \nu = 2, the variance does not exist for \nu \le 4, the skewness does not exist for \nu \le 6, and the kurtosis does not exist for \nu \le 8.

Value

skewhypMean gives the mean of the skew hyperbolic t-distribution, skewhypVar the variance, skewhypSkew the skewness, skewhypKurt the kurtosis and skewhypMode the mode.

Author(s)

David Scott [e-mail], Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

See Also

dskewhyp, optim, skewhypCheckPars, skewhypMom

Examples

param <- c(10,1,5,9)
skewhypMean(param = param)
skewhypVar(param = param)
skewhypSkew(param = param)
skewhypKurt(param = param)
skewhypMode(param = param)
range <- skewhypCalcRange(param = param)
curve(dskewhyp(x, param = param), range[1], range[2])
abline(v = skewhypMode(param = param), col = "red")
abline(v = skewhypMean(param = param), col = "blue")