momIntegrated: Moments Using Integration
In sjp/GeneralizedHyperbolic: The generalized hyperbolic distribution

Description Usage Arguments Details Value Author(s) See Also Examples

Calculates moments and absolute moments about a given location for the generalized hyperbolic and related distributions.

1 2	momIntegrated(densFn = c("ghyp", "hyperb", "gig", "gamma", "invgamma", "vg"), order, param = NULL, about = 0, absolute = FALSE)

`densFn`	Character. The name of the density function whose moments are to be calculated. See Details.
`order`	Numeric. The order of the moment or absolute moment to be calculated.
`param`	Numeric. A vector giving the parameter values for the distribution specified by `densFn`. If no `param` values are specified, then the default parameter values of each distribution are used instead.
`about`	Numeric. The point about which the moment is to be calculated.
`absolute`	Logical. Whether absolute moments or ordinary moments are to be calculated. Default is `FALSE`.

Denote the density function by f. Then if order=k and about=a, momIntegrated calculates

integral_{-infinity}^infinity (x - a)^k f(x) dx

when absolute = FALSE and

integral_{-infinity}^infinity |x - a|^k f(x) dx

when absolute = TRUE.

Only certain density functions are permitted.

When densFn="ghyp" or "generalized hyperbolic" the density used is dghyp. The default value for param is c(0, 1, 1, 0, 1).

When densFn="hyperb" or "hyperbolic" the density used is dhyperb. The default value for param is c(0, 1, 1, 0).

When densFn="gig" or "generalized inverse gaussian" the density used is dgig. The default value for param is c(1, 1, 1).

When densFn="gamma" the density used is dgamma. The default value for param is c(1, 1).

When densFn="invgamma" or "inverse gamma" the density used is the density of the inverse gamma distribution given by

f(x) = u^alpha exp(-u)/(x Gamma(alpha)), u = theta/x

for x > 0, alpha > 0 and theta > 0. The parameter vector param = c(shape, rate) where shape =alpha and rate=1/theta. The default value for param is c(-1, 1).

When densFn = "vg" or "variance gamma" the density used is dvg from the package VarianceGamma. In this case, the package VarianceGamma must be loaded or an error will result. The default value for param is c(0, 1, 0, 1).

The value of the integral as specified in Details.

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

dghyp, dhyperb, dgamma, dgig, VarianceGamma

### Calculate the mean of a generalized hyperbolic distribution
### Compare the use of integration and the formula for the mean
m1 <- momIntegrated("ghyp", param = c(0, 1, 3, 1, 1 / 2), order = 1, about = 0)
m1
ghypMean(param = c(0, 1, 3, 1, 1 / 2))
### The first moment about the mean should be zero
momIntegrated("ghyp", order = 1, param = c(0, 1, 3, 1, 1 / 2), about = m1)
### The variance can be calculated from the raw moments
m2 <- momIntegrated("ghyp", order = 2, param = c(0, 1, 3, 1, 1 / 2), about = 0)
m2
m2 - m1^2
### Compare with direct calculation using integration
momIntegrated("ghyp", order = 2, param = c(0, 1, 3, 1, 1 / 2), about = m1)
momIntegrated("ghyp", param = c(0, 1, 3, 1, 1 / 2), order = 2,
              about = m1)
### Compare with use of the formula for the variance
ghypVar(param = c(0, 1, 3, 1, 1 / 2))