Hyperbolic Moments and Mode {HyperbolicDist}R Documentation

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(Theta)
hyperbVar(Theta)
hyperbSkew(Theta)
hyperbKurt(Theta)
hyperbMode(Theta)

Arguments

Theta Parameter vector of the hyperbolic distribution.

Value

hyperbMean gives the mean of the hyperbolic distribution, hyperbVar the variance, and hyperbMode the mode. The formulae used are as given in Barndorff-Nielsen and Blęsild (1983), p. 702. hyperbSkew and hyperbKurt are the skewness and kurtosis of the hyperbolic distribution. The formulae used are those of Barndorff-Nielsen and Blęsild (1981), Appendix 2. 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 parameterisation of the hyperbolic distribution used for this and other components of the HyperbolicDist package is the (pi,zeta) one. See hyperbChangePars to transfer between parameterisations.

Author(s)

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

References

Barndorff-Nielsen, O. and Blęsild, 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 Blęsild, 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

Examples

Theta <- c(2,2,2,2)
hyperbMean(Theta)
hyperbVar(Theta)
hyperbSkew(Theta)
hyperbKurt(Theta)
hyperbMode(Theta)

[Package HyperbolicDist version 0.5-3 Index]