Calculate Moments of the Generalized Hyperbolic Distribution

Description

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

Usage

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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, W<fc>rtz and Tran (2011).

Value

The moment specified.

Author(s)

David Scott d.scott@auckland.ac.nz

References

Scott, D. J., W<fc>rtz, D., Dong, C. and Tran, T. T. (2011) Moments of the generalized hyperbolic distribution. Comp. Statistics., 26, 459–476.

See Also

ghypChangePars, is.wholenumber, momChangeAbout, momIntegrated, ghypMean, ghypVar, ghypSkew, ghypKurt.

Examples

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

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