# hyperbMeanVarMode: Moments and Mode of the Hyperbolic Distribution In HyperbolicDist: The Hyperbolic Distribution

 Specific Hyperbolic Distribution Moments and Mode 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.

### Details

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

### 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 HyperbolicDist package is the (\pi,\zeta) one. See hyperbChangePars to transfer between parameterizations.

### Author(s)

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

### References

Barndorff-Nielsen, O. and Blæsild, 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 Blæsild, 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.

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)


HyperbolicDist documentation built on Nov. 26, 2023, 9:07 a.m.