LaplacesDemon (version 16.1.1)

dist.Inverse.Gamma: Inverse Gamma Distribution

Description

This is the density function and random generation from the inverse gamma distribution.

Usage

dinvgamma(x, shape=1, scale=1, log=FALSE)
rinvgamma(n, shape=1, scale=1)

Arguments

n

This is the number of draws from the distribution.

x

This is the scalar location to evaluate density.

shape

This is the scalar shape parameter \(\alpha\), which defaults to one.

scale

This is the scalar scale parameter \(\beta\), which defaults to one.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Value

dinvgamma gives the density and rinvgamma generates random deviates. The parameterization is consistent with the Gamma Distribution in the stats package.

Details

• Application: Continuous Univariate

• Density: \(p(\theta) = \frac{\beta^\alpha}{\Gamma(\alpha)} \theta^{-(\alpha + 1)} \exp(-\frac{\beta}{\theta}), \quad \theta > 0\)

• Inventor: Unknown (to me, anyway)

• Notation 1: \(\theta \sim \mathcal{G}^{-1}(\alpha, \beta)\)

• Notation 2: \(p(\theta) = \mathcal{G}^{-1}(\theta | \alpha, \beta)\)

• Parameter 1: shape \(\alpha > 0\)

• Parameter 2: scale \(\beta > 0\)

• Mean: \(E(\theta) = \frac{\beta}{\alpha - 1}\), for \(\alpha > 1\)

• Variance: \(var(\theta) = \frac{\beta^2}{(\alpha - 1)^2 (\alpha - 2)}, \alpha > 2\)

• Mode: \(mode(\theta) = \frac{\beta}{\alpha + 1}\)

The inverse-gamma is the conjugate prior distribution for the normal or Gaussian variance, and has been traditionally specified as a vague prior in that application. The density is always finite; its integral is finite if \(\alpha > 0\). Prior information decreases as \(\alpha, \beta \rightarrow 0\).

These functions are similar to those in the MCMCpack package.

See Also

dgamma, dnorm, dnormp, and dnormv.

Examples

# NOT RUN {
library(LaplacesDemon)
x <- dinvgamma(4.3, 1.1)
x <- rinvgamma(10, 3.3)

#Plot Probability Functions
x <- seq(from=0.1, to=20, by=0.1)
plot(x, dinvgamma(x,1,1), ylim=c(0,1), type="l", main="Probability Function",
     ylab="density", col="red")
lines(x, dinvgamma(x,1,0.6), type="l", col="green")
lines(x, dinvgamma(x,0.6,1), type="l", col="blue")
legend(2, 0.9, expression(paste(alpha==1, ", ", beta==1),
     paste(alpha==1, ", ", beta==0.6), paste(alpha==0.6, ", ", beta==1)),
     lty=c(1,1,1), col=c("red","green","blue"))
# }