extraDistr (version 1.9.1)

InvGamma: Inverse-gamma distribution

Description

Density, distribution function and random generation for the inverse-gamma distribution.

Usage

dinvgamma(x, alpha, beta = 1, log = FALSE)
pinvgamma(q, alpha, beta = 1, lower.tail = TRUE, log.p = FALSE)
qinvgamma(p, alpha, beta = 1, lower.tail = TRUE, log.p = FALSE)
rinvgamma(n, alpha, beta = 1)

Arguments

x, q

vector of quantiles.

alpha, beta

positive valued shape and scale parameters.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\) otherwise, \(P[X > x]\).

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function $$ f(x) = \frac{\beta^\alpha x^{-\alpha-1} \exp(-\frac{\beta}{x})}{\Gamma(\alpha)} $$

Cumulative distribution function $$ F(x) = \frac{\gamma(\alpha, \frac{\beta}{x})}{\Gamma(\alpha)} $$

References

Witkovsky, V. (2001). Computing the distribution of a linear combination of inverted gamma variables. Kybernetika 37(1), 79-90.

Leemis, L.M. and McQueston, L.T. (2008). Univariate Distribution Relationships. American Statistician 62(1): 45-53.

See Also

GammaDist

Examples

x <- rinvgamma(1e5, 20, 3)
hist(x, 100, freq = FALSE)
curve(dinvgamma(x, 20, 3), 0, 1, col = "red", add = TRUE, n = 5000)
hist(pinvgamma(x, 20, 3))
plot(ecdf(x))
curve(pinvgamma(x, 20, 3), 0, 1, col = "red", lwd = 2, add = TRUE, n = 5000)