# RLambda: Functions for Calculating Moments In sjp/GeneralizedHyperbolic: The generalized hyperbolic distribution

## Description

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

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```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.

## Value

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 R as `besselK`. The other functions are used in calculation of higher moments. See Barndorff-Nielsen, O. and Blaesild, P (1981) for details of the calculations.

The parameterisation 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 [email protected], Richard Trendall, Thomas Tran

## References

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`, `hyperbMean`,`hyperbChangePars`, `besselK`