RLambda function

Functions for Calculating Moments

Functions used to calculate the mean, variance, skewness and kurtosis of a hyperbolic distribution. Not expected to be called directly by users.

RLambda(zeta, lambda = 1)
SLambda(zeta, lambda = 1)
MLambda(zeta, lambda = 1)
WLambda1(zeta, lambda = 1)
WLambda2(zeta, lambda = 1)
WLambda3(zeta, lambda = 1)
WLambda4(zeta, lambda = 1)
gammaLambda1(hyperbPi, zeta, lambda = 1)
gammaLambda1(hyperbPi, zeta, lambda = 1)

Arguments

• hyperbPi: Value of the parameter $pi$ of the hyperbolic distribution.
• zeta: Value of the parameter $zeta$ of the hyperbolic distribution.
• lambda: Parameter related to order of Bessel functions.

Returns

The functions RLambda and SLambda are used in the calculation of the mean and variance. They are functions of the Bessel functions of the third kind, implemented in as besselK. The other functions are used in calculation of higher moments. See Barndorff-Nielsen, O. and , P. (1981) for details of the calculations.

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.

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, hyperbMean,hyperbChangePars, besselK