Calculate Moments of the Variance Gamma Distribution

Description

This function can be used to calculate raw moments, mu moments, central moments and moments about any other given location for the variance gamma (VG) distribution.

Usage

  vgMom(order, vgC = 0, sigma = 1, theta = 0, nu = 1,
    param = c(vgC,sigma,theta,nu), momType = "raw", about = 0)

Arguments

Details

For the parameters of the variance gamma distribution, users may either specify the values individually or as a vector. If both forms are specified but with different values, then the values specified by vector param will always overwrite the other ones. In addition, the parameters values are examined by calling the function vgCheckPars to see if they are valid for the VG distribution.

order is also checked by calling the function is.wholenumber in DistributionUtils package to see whether a whole number is given.

momType can be either "raw" (moments about zero), "mu" (moments about vgC), or "central" (moments about mean). If one of these moment types is specified, then there is no need to specify the about value. For moments about any other location, the about value must be specified. In the case that both momType and about are specified and contradicting, the function will always calculate the moments based on about rather than momType.

To calculate moments of the VG distribution, the function first calculates mu moments by the formula defined below and then transforms mu moments to central moments or raw moments or moments about any other location as required by calling momChangeAbout in DistributionUtils package.

To calculate mu moments of the variance gamma distribution, the function first transforms the parameterization of c,sigma,theta,nu to the generalized hyperbolic distribution's parameterization of lambda,alpha,beta,mu (see vgChangePars for details). Then, the mu moments of the variance gamma distribution are given by

sum_{l = [(k+1)/2]}^{k} a_{k, ell} beta^{2l - k} [Gamma(lambda+l)/Gamma(lambda) 2^l/(alpha^2-beta^2)^l]

where k = order and k > 0 and a_{k, l} is the recursive coefficient (see momRecursion for details).

This formula is developed from the mu moments formula of the generalized hyperbolic distribution given in Scott, W<c3><bc>rtz and Tran (2008). Note that the part in [] of this equation is actually equivalent to the formula of raw moments of the gamma distribution. So the function calls gammaRawMom in GeneralizedHyperbolic package when implementing the computations.

Value

The moment specified. In the case of raw moments, Inf is returned if the moment is infinite.

Author(s)

David Scott d.scott@auckland.ac.nz, Christine Yang Dong c.dong@auckland.ac.nz

References

Paolella, Marc S. (2007) Intermediate Probability: A Computational Approach, Chichester: Wiley Scott, D. J., W<c3><bc>rtz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

See Also

vgCheckPars, vgChangePars, vgMean, vgVar, vgSkew, vgKurt, is.wholenumber, momRecursion, momChangeAbout and momIntegrated.

Examples

  ### Raw moments of the VG distribution
  vgMom(3, param=c(2,1,2,1), momType = "raw")

  ### Mu moments of the VG distribution
  vgMom(2, param=c(2,1,2,1), momType = "mu")

  ### Central moments of the VG distribution
  vgMom(4, param=c(2,1,2,1), momType = "central")

  ### Moments about any locations
  vgMom(4, param=c(2,1,2,1), about = 1)

Results