Inverse Gamma distribution — invgamma • RTMBdist

Density, distribution function, and random generation for the inverse Gamma distribution.

Usage

dinvgamma(x, shape, rate, scale = 1/rate, log = FALSE)

pinvgamma(q, shape, rate, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

qinvgamma(p, shape, rate, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

rinvgamma(n, shape, rate, scale = 1/rate)

Arguments

x, q: vector of quantiles, must be positive.
shape, rate, scale: positive parameters of corresponding gamma distribution
log, log.p: logical; if TRUE, probabilities/ densities $p$ are returned as $\log(p)$.
lower.tail: logical; if TRUE, probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
p: vector of probabilities
n: number of random values to return

Value

dinvgamma gives the density, pinvgamma gives the distribution function, qinvgamma gives the quantile function, and rinvgamma generates random deviates.

Details

This implementation of dinvgamma, pinvgamma, and qinvgamma allows for automatic differentiation with RTMB.

If $X \sim \Gamma(\alpha, \beta)$, then $1/X \sim \text{InvGamma}(\alpha, \beta)$.

$$f(x;\,\alpha,s) = \frac{s^\alpha}{\Gamma(\alpha)}\, x^{-(\alpha+1)}\exp\!\left(-\frac{s}{x}\right), \quad x > 0,$$ where $s = \text{scale} = 1/\text{rate}$.

Examples

x <- rinvgamma(1, 1, 0.5)
d <- dinvgamma(x, 1, 0.5)
p <- pinvgamma(x, 1, 0.5)
q <- qinvgamma(p, 1, 0.5)