InverseGamma: The Inverse Gamma Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

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 \le 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 = \alpha and scale = \theta has density:

f(x) = \frac{u^\alpha e^{-u}}{x \Gamma(\alpha)}, % \quad u = \theta/x

for x > 0, \alpha > 0 and \theta > 0. (Here \Gamma(\alpha) 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 < \alpha, 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 [e-mail] 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)

actuar documentation built on Nov. 8, 2023, 9:06 a.m.