actuar (version 3.3-4)

InverseGamma: 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)

Value

dinvgamma gives the density,

pinvgamma gives the distribution function,

qinvgamma gives the quantile function,

rinvgamma generates random deviates,

minvgamma gives the \(k\)th raw moment,

levinvgamma gives the \(k\)th 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.

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.

Author

Vincent Goulet and Mathieu Pigeon

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

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

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)