hyperbMeanVarMode: Moments and Mode of the Hyperbolic Distribution
In sjp/GeneralizedHyperbolic: The generalized hyperbolic distribution

Description Usage Arguments Details Value Author(s) References See Also Examples

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

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))

`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.

The formulae used for the mean, variance and mode are as given in Barndorff-Nielsen and Bl<c3><a6>sild (1983), p. 702. The formulae used for the skewness and kurtosis are those of Barndorff-Nielsen and Bl<c3><a6>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.

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 (alpha, beta) one. See hyperbChangePars to transfer between parameterizations.

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

Barndorff-Nielsen, O. and Bl<c3><a6>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<c3><a6>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.

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

param <- c(2, 2, 2, 1)
hyperbMean(param = param)
hyperbVar(param = param)
hyperbSkew(param = param)
hyperbKurt(param = param)
hyperbMode(param = param)