R: Inverse Gamma Distribution

dist.Inverse.Gamma {LaplacesDemon} | R Documentation

## 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.

### Details

- Application: Continuous Univariate

- Density: p(theta) = (beta^alpha / Gamma(alpha)) * theta^(-(alpha + 1)) * exp(-beta / theta), theta > 0

- Inventor: Unknown (to me, anyway)

- Notation 1: theta ~ G^-1(alpha, beta)

- Notation 2: p(theta) = G^-1(theta | alpha, beta)

- Parameter 1: shape alpha > 0

- Parameter 2: scale beta > 0

- Mean: E(theta) = beta / (alpha - 1), for alpha > 1

- Variance: var(theta) = beta^2 / ((alpha - 1)^2 * (alpha - 2)), alpha > 2

- Mode: mode(theta) = 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 -> 0.

These functions are similar to those in the `MCMCpack` package.

### Value

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

### See Also

`dgamma`, `dnorm`, `dnormp`, and `dnormv`.

### Examples

    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"))

[Package LaplacesDemon version 16.1.1 Index]