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
1 2 3 
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 
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  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)
