R: The Inverse Gamma Distribution

InverseGamma {actuar} | R Documentation

## The Inverse Gamma Distribution

### Description

Density function, distribution function, quantile function, random generation, raw moments, and limited moments for the Inverse Gamma distribution with parameters `shape` and `scale`.

### Usage

    dinvgamma(x, shape, rate = 1, scale = 1/rate, log = FALSE)
    pinvgamma(q, shape, rate = 1, scale = 1/rate,
              lower.tail = TRUE, log.p = FALSE)
    qinvgamma(p, shape, rate = 1, scale = 1/rate,
              lower.tail = TRUE, log.p = FALSE)
    rinvgamma(n, shape, rate = 1, scale = 1/rate)
    minvgamma(order, shape, rate = 1, scale = 1/rate)
    levinvgamma(limit, shape, rate = 1, scale = 1/rate,
                order = 1)
    mgfinvgamma(t, shape, rate =1, scale = 1/rate, log =FALSE)

### Arguments

`x, q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1`, the length is taken to be the number required.
`shape, scale` | parameters. Must be strictly positive.
`rate` | an alternative way to specify the scale.
`log, log.p` | logical; if `TRUE`, probabilities/densities p are returned as log(p).
`lower.tail` | logical; if `TRUE` (default), probabilities are P[X <= x], otherwise, P[X > x].
`order` | order of the moment.
`limit` | limit of the loss variable.
`t` | numeric vector.

### Details

The inverse gamma distribution with parameters `shape` = a and `scale` = s has density:

f(x) = u^a exp(-u)/(x Gamma(a)), u = s/x

for x > 0, a > 0 and s > 0. (Here Gamma(a) is the function implemented by R's `gamma()` and defined in its help.)

The special case `shape == 1` is an Inverse Exponential distribution.

The kth raw moment of the random variable X is E[X^k], k < shape, and the kth limited moment at some limit d is E[min(X, d)^k], all k.

The moment generating function is given by E[e^{tX}].

### Value

`dinvgamma` gives the density, `pinvgamma` gives the distribution function, `qinvgamma` gives the quantile function, `rinvgamma` generates random deviates, `minvgamma` gives the kth raw moment, `levinvgamma` gives the kth moment of the limited loss variable, and `mgfinvgamma` gives the moment generating function in `t`.

Invalid arguments will result in return value `NaN`, with a warning.

### Note

`levinvgamma` computes the limited expected value using `gammainc` from package expint.

Also known as the Vinci distribution. See also Kleiber and Kotz (2003) for alternative names and parametrizations.

The `"distributions"` package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

### References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

### Examples

    exp(dinvgamma(2, 3, 4, log = TRUE))
    p <- (1:10)/10
    pinvgamma(qinvgamma(p, 2, 3), 2, 3)
    minvgamma(-1, 2, 2) ^ 2
    levinvgamma(10, 2, 2, order = 1)
    mgfinvgamma(-1, 3, 2)

[Package actuar version 3.1-2 Index]