invgamma: The Inverse Gamma Distribution

In dkahle/invgamma: The Inverse Gamma Distribution

The Inverse Gamma Distribution

Description

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

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)

Arguments

x, q	vector of quantiles.
shape	inverse gamma shape parameter
rate	inverse gamma rate parameter
scale	alternative to rate; scale = 1/rate
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X \leq x]; if FALSE 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

The inverse gamma distribution with parameters shape and rate has density

f(x) = \frac{rate^{shape}}{\Gamma(shape)} x^{-1-shape} e^{-rate/x}

it is the inverse of the standard gamma parameterization in R. If X \sim InvGamma(shape, rate),

E[X] = \frac{rate}{shape-1}

when shape > 1 and

Var(X) = \frac{rate^2}{(shape - 1)^2(shape - 2)}

for shape > 2.

The functions ⁠(d/p/q/r)invgamma()⁠ simply wrap those of the standard ⁠(d/p/q/r)gamma()⁠ R implementation, so look at, say, stats::dgamma() for details.

See Also

stats::dgamma(); these functions just wrap the ⁠(d/p/q/r)gamma()⁠ functions.

Examples

s <- seq(0, 5, .01)
plot(s, dinvgamma(s, 7, 10), type = 'l')

f <- function(x) dinvgamma(x, 7, 10)
q <- 2
integrate(f, 0, q)
(p <- pinvgamma(q, 7, 10))
qinvgamma(p, 7, 10) # = q
mean(rinvgamma(1e5, 7, 10) <= q)

shape <- 3; rate <- 7
x <- rinvgamma(1e6, shape, rate)
mean(x) # = rate / (shape - 1)
var(x)  # = rate^2 / ( (shape - 1)^2 * (shape - 2) )

shape <- 7; rate <- 2.01
x <- rinvgamma(1e6, shape, rate)
mean(x) # = rate / (shape - 1)
var(x)  # = rate^2 / ( (shape - 1)^2 * (shape - 2) )

dkahle/invgamma documentation built on June 12, 2025, 2:47 a.m.