momIntegrated() R function from [DistributionUtils]

Moments Using Integration

Calculates moments and absolute moments about a given location for any given distribution.


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

Arguments

densFn: Character. The name of the density function whose moments are to be calculated. See Details .
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 the distribution are used instead.
order: Numeric. The order of the moment or absolute moment to be calculated.
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.
...: Passes arguments to integrate. In particular, the parameters of the distribution.

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

when absolute = FALSE and

\int_{-\infty}^\infty |x - a|^k f(x) dx%integral_{-infinity}^infinity |x - a|^k f(x) dx

when absolute = TRUE.

The name of the density function must be supplied as the characters of the root for that density (e.g. norm, ghyp).

When densFn="ghyp", densFn="hyperb", densFn="gig" or densFn = "vg", the relevant package must be loaded or an error will result.

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)}, %\quad u = \theta/x%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).

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 , Xinxing Li xli053@aucklanduni.ac.nz

Examples


require(GeneralizedHyperbolic)
### 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))

DistributionUtils package Read PDF manual

Maintainer: David Scott
License: GPL (>= 2)
Last published: 2025-03-29

Useful links

momIntegrated function