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(x, 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.

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] and the kth limited moment at some limit d is E[min(X, d)^k].

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

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, and levinvgamma gives the kth moment of the limited loss variable, mgfinvgamma gives the moment generating function in x.
Invalid arguments will result in return value NaN, with a warning.

Note

Also known as the Vinci distribution.

Author(s)

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

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second 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 1.0-2 Index]