gigMom | R Documentation |

Functions to calculate raw moments and moments about a given location for the generalized inverse Gaussian (GIG) distribution, including the gamma and inverse gamma distributions as special cases.

```
gigRawMom(order, Theta)
gigMom(order, Theta, about = 0)
gammaRawMom(order, shape = 1, rate = 1, scale = 1/rate)
```

`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. |

`Theta` |
Numeric. The parameter vector specifying the GIG
distribution. Of the form |

`about` |
Numeric. The point around which the moment is to be calculated. |

`shape` |
Numeric. The shape parameter, must be non-negative, not permitted to be a vector. |

`scale` |
Numeric. The scale parameter, must be positive, not permitted to be a vector. |

`rate` |
Numeric. The rate parameter, an alternative way to specify the scale. |

The vector `Theta`

of parameters is examined using
`gigCheckPars`

to see if the parameters are valid for the GIG
distribution and if they correspond to the special cases which are the
gamma and inverse gamma distributions. Checking of special cases and
valid parameter vector values is carried out using the function
`gigCheckPars`

. Checking whether `order`

is a whole number is carried out using the function
`is.wholenumber`

.

Raw moments (moments about zero) are calculated using the
functions `gigRawMom`

or `gammaRawMom`

. For moments not
about zero, the function `momChangeAbout`

is used to
derive moments about another point from raw moments. Note that raw moments
of the inverse gamma distribution can be obtained from the raw moments
of the gamma distribution because of the relationship between the two
distributions. An alternative implementation of raw moments of the gamma
and inverse gamma distributions may be found in the package
actuar and these may be faster since they are written in C.

To calculate the raw moments of the GIG distribution it is convenient to
use the alternative parameterization of the GIG in terms of
`\omega`

and `\eta`

, given as parameterization 3
in `gigChangePars`

. Then the raw moment of the GIG
distribution of order `k`

is given by

`\eta^k K_{\lambda+k}(\omega)/K_{\lambda}(\omega)`

where `K_\lambda()`

is the modified Bessel function of
the third kind of order `\lambda`

.

The raw moment of the gamma distribution of order `k`

with
shape parameter `\alpha`

and rate parameter
`\beta`

is given by

`\beta^{-k}\Gamma(\alpha+k)/\Gamma(\alpha)`

The raw moment of order `k`

of the inverse gamma distribution
with shape parameter `\alpha`

and rate parameter
`\beta`

is the raw moment of order `-k`

of the gamma
distribution with shape parameter `\alpha`

and rate parameter
`1/\beta`

.

The moment specified. In the case of raw moments, `Inf`

is
returned if the moment is infinite.

David Scott d.scott@auckland.ac.nz

Paolella, Marc S. (2007) Intermediate Probability: A Computational Approach, Chichester: Wiley

`gigCheckPars`

, `gigChangePars`

,
`is.wholenumber`

, `momChangeAbout`

,
`momIntegrated`

, `gigMean`

,
`gigVar`

, `gigSkew`

, `gigKurt`

.

```
### Raw moments of the generalized inverse Gaussian distribution
Theta <- c(-0.5,5,2.5)
gigRawMom(1, Theta)
momIntegrated("gig", order = 1, param = Theta, about = 0)
gigRawMom(2, Theta)
momIntegrated("gig", order = 2, param = Theta, about = 0)
gigRawMom(10, Theta)
momIntegrated("gig", order = 10, param = Theta, about = 0)
gigRawMom(2.5, Theta)
### Moments of the generalized inverse Gaussian distribution
Theta <- c(-0.5,5,2.5)
(m1 <- gigRawMom(1, Theta))
gigMom(1, Theta)
gigMom(2, Theta, m1)
(m2 <- momIntegrated("gig", order = 2, param = Theta, about = m1))
gigMom(1, Theta, m1)
gigMom(3, Theta, m1)
momIntegrated("gig", order = 3, param = Theta, about = m1)
### Raw moments of the gamma distribution
shape <- 2
rate <- 3
Theta <- c(shape, rate)
gammaRawMom(1, shape, rate)
momIntegrated("gamma", order = 1, param = Theta, about = 0)
gammaRawMom(2, shape, rate)
momIntegrated("gamma", order = 2, param = Theta, about = 0)
gammaRawMom(10, shape, rate)
momIntegrated("gamma", order = 10, param = Theta, about = 0)
### Moments of the inverse gamma distribution
Theta <- c(-0.5,5,0)
gigRawMom(2, Theta) # Inf
gigRawMom(-2, Theta)
momIntegrated("invgamma", order = -2,
param = c(-Theta[1],Theta[2]/2), about = 0)
### An example where the moment is infinite: inverse gamma
Theta <- c(-0.5,5,0)
gigMom(1, Theta)
gigMom(2, Theta)
```

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