hyperbMeanVarMode function

Moments and Mode of the Hyperbolic Distribution

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

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

Arguments

• Theta: Parameter vector of the hyperbolic distribution.

Returns

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 HyperbolicDist package is the $(pi,zeta)$ one. See hyperbChangePars

to transfer between parameterizations.

Details

The formulae used for the mean, variance and mode are as given in Barndorff-Nielsen and (1983), p. 702. The formulae used for the skewness and kurtosis are those of Barndorff-Nielsen and (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 , 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 , 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.

Author(s)

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

See Also

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

Examples

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