The Inverse Gamma Distribution

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, rate, scale

shape, rate, and scale parameters of corresponding gamma distribution. In particular, rate and scale are not the rate and scale of the inverse gamma distribution, but of the gamma distribution.

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