GeneralizedHyperbolic (version 0.8-4)

Specific Hyperbolic Distribution Moments and Mode: Moments and Mode of the Hyperbolic Distribution

Description

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

Usage

hyperbMean(mu = 0, delta = 1, alpha = 1, beta = 0,
           param = c(mu, delta, alpha, beta))
hyperbVar(mu = 0, delta = 1, alpha = 1, beta = 0,
          param = c(mu, delta, alpha, beta))
hyperbSkew(mu = 0, delta = 1, alpha = 1, beta = 0,
           param = c(mu, delta, alpha, beta))
hyperbKurt(mu = 0, delta = 1, alpha = 1, beta = 0,
           param = c(mu, delta, alpha, beta))
hyperbMode(mu = 0, delta = 1, alpha = 1, beta = 0,
           param = c(mu, delta, alpha, beta))

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.

param

Parameter vector of the hyperbolic distribution.

Value

hyperbMean gives the mean of the hyperbolic distribution, hyperbVar the variance, hyperbSkew the skewness, hyperbKurt the kurtosis and hyperbMode the mode.

Note that the kurtosis is the standardised fourth cumulant or what is sometimes called the kurtosis excess. (See http://mathworld.wolfram.com/Kurtosis.html for a discussion.)

The parameterization of the hyperbolic distribution used for this and other components of the GeneralizedHyperbolic package is the \((\alpha, \beta)\) one. See hyperbChangePars to transfer between parameterizations.

Details

The formulae used for the mean, variance and mode are as given in Barndorff-Nielsen and Bl<e6>sild (1983), p. 702. The formulae used for the skewness and kurtosis are those of Barndorff-Nielsen and Bl<e6>sild (1981), Appendix 2.

Note that the variance, skewness and kurtosis can be obtained from the functions for the generalized hyperbolic distribution as special cases. Likewise other moments can be obtained from the function ghypMom which implements a recursive method to moments of any desired order. Note that functions for the generalized hyperbolic distribution use a different parameterization, so care is required.

References

Barndorff-Nielsen, O. and Bl<e6>sild, P (1981). Hyperbolic distributions and ramifications: contributions to theory and application. In Statistical Distributions in Scientific Work, eds., Taillie, C., Patil, G. P., and Baldessari, B. A., Vol. 4, pp. 19--44. Dordrecht: Reidel.

Barndorff-Nielsen, O. and Bl<e6>sild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700--707. New York: Wiley.

See Also

dhyperb, hyperbChangePars, besselK, ghypMom, ghypMean, ghypVar, ghypSkew, ghypKurt

Examples

# NOT RUN {
param <- c(2, 2, 2, 1)
hyperbMean(param = param)
hyperbVar(param = param)
hyperbSkew(param = param)
hyperbKurt(param = param)
hyperbMode(param = param)
# }