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))
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.
\(\mu\) is the location parameter. By default this is set to 0.
\(\delta\) is the scale parameter of the distribution. A default value of 1 has been set.
\(\alpha\) is the tail parameter, with a default value of 1.
\(\beta\) is the skewness parameter, by default this is 0.
Parameter vector of the hyperbolic distribution.
David Scott d.scott@auckland.ac.nz, Richard Trendall, Thomas Tran
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.
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
param <- c(2, 2, 2, 1)
hyperbMean(param = param)
hyperbVar(param = param)
hyperbSkew(param = param)
hyperbKurt(param = param)
hyperbMode(param = param)
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