ghypMom function

Calculate Moments of the Generalized Hyperbolic Distribution

Function to calculate raw moments, mu moments, central moments and moments about any other given location for the generalized hyperbolic distribution.

ghypMom(order, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
        param = c(mu, delta, alpha, beta, lambda),
        momType = c("raw", "central", "mu"), about = 0)

Arguments

• order: Numeric. The order of the moment to be calculated. Not permitted to be a vector. Must be a positive whole number except for moments about zero.

• 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.

• lambda: $lambda$ is the shape parameter and dictates the shape that the distribution shall take. Default value is 1.

• param: Numeric. The parameter vector specifying the generalized hyperbolic distribution. Of the form

  c(mu, delta, alpha, beta, lambda)

  (see dghyp).

• momType: Common types of moments to be calculated, default is "raw". See Details .

• about: Numeric. The point around which the moment is to be calculated.

Details

Checking whether order is a whole number is carried out using the function is.wholenumber.

momType can be either "raw" (moments about zero), "mu" (moments about mu), 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 generalized hyperbolic distribution, the function firstly calculates mu moments by formula defined below and then transforms mu moments to central moments or raw moments or moments about any other locations as required by calling momChangeAbout.

The mu moments are obtained from the recursion formula given in Scott, and Tran (2011).

Returns

The moment specified.

References

Scott, D. J., , D., Dong, C. and Tran, T. T. (2011) Moments of the generalized hyperbolic distribution. Comp. Statistics., 26 , 459--476.

Author(s)

David Scott d.scott@auckland.ac.nz

See Also

ghypChangePars and from package list("DistributionUtils"): logHist, is.wholenumber, momChangeAbout, and momIntegrated.

Further, ghypMean, ghypVar, ghypSkew, ghypKurt.

Examples

param <- c(1, 2, 2, 1, 2)
mu <- param[1]
### mu moments
m1 <- ghypMean(param = param)
m1 - mu
ghypMom(1, param = param, momType = "mu")

## Comparison, using momIntegrated from pkg 'DistributionUtils':
momIntegrated <- DistributionUtils :: momIntegrated

momIntegrated("ghyp", order = 1, param = param, about = mu)
ghypMom(2, param = param, momType = "mu")
momIntegrated("ghyp", order = 2, param = param, about = mu)
ghypMom(10, param = param, momType = "mu")
momIntegrated("ghyp", order = 10, param = param, about = mu)

### raw moments
ghypMean(param = param)
ghypMom(1, param = param, momType = "raw")
momIntegrated("ghyp", order = 1, param = param, about = 0)
ghypMom(2, param = param, momType = "raw")
momIntegrated("ghyp", order = 2, param = param, about = 0)
ghypMom(10, param = param, momType = "raw")
momIntegrated("ghyp", order = 10, param = param, about = 0)

### central moments
ghypMom(1, param = param, momType = "central")
momIntegrated("ghyp", order = 1, param = param, about = m1)
ghypVar(param = param)
ghypMom(2, param = param, momType = "central")
momIntegrated("ghyp", order = 2, param = param, about = m1)
ghypMom(10, param = param, momType = "central")
momIntegrated("ghyp", order = 10, param = param, about = m1)