momIntegrated function

Moments Using Integration

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

momIntegrated(densFn, order, param = NULL, about = 0, absolute = FALSE)

Arguments

• 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.

Details

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

$$\int_{-\infty}^\infty (x - a)^k f(x) dx$$

when absolute = FALSE and

$$\int_{-\infty}^\infty |x - a|^k f(x) dx$$

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(1,1,0,1,0).

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

$$f(x) = \frac{u^\alpha e^{-u}}{x \Gamma(\alpha)},$$

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).

Returns

The value of the integral as specified in Details .

Author(s)

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

See Also

dghyp, dhyperb, dgamma, dgig

Examples

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