R: Inverse Beta Distribution

dist.Inverse.Beta {LaplacesDemon} | R Documentation

## Inverse Beta Distribution

### Description

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

### Usage

    dinvbeta(x, a, b, log=FALSE)
    rinvbeta(n, a, b)

### Arguments

`n` | This is the number of draws from the distribution.
`x` | This is a location vector at which to evaluate density.
`a` | This is the scalar shape parameter alpha.
`b` | This is the scalar shape parameter beta
`log` | Logical. If `log=TRUE`, then the logarithm of the density is returned.

### Details

- Application: Continuous Univariate

- Density: (theta^(alpha - 1) * (1 + theta)^(-alpha - beta)) / beta(alpha, beta)

- Inventor: Dubey (1970)

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

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

- Parameter 1: shape alpha > 0

- Parameter 2: shape beta > 0

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

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

- Mode: mode(theta) = (alpha - 1) / (beta + 1)

The inverse-beta, also called the beta prime distribution, applies to variables that are continuous and positive. The inverse beta is the conjugate prior distribution of a parameter of a Bernoulli distribution expressed in odds.

The inverse-beta distribution has also been extended to the generalized beta prime distribution, though it is not (yet) included here.

### Value

`dinvbeta` gives the density and `rinvbeta` generates random deviates.

### References

Dubey, S.D. (1970). "Compound Gamma, Beta and F Distributions". Metrika, 16, p. 27–31.

### See Also

`dbeta`

### Examples

    library(LaplacesDemon)
    x <- dinvbeta(5:10, 2, 3)
    x <- rinvbeta(10, 2, 3)

    #Plot Probability Functions
    x <- seq(from=0.1, to=20, by=0.1)
    plot(x, dinvbeta(x,2,2), ylim=c(0,1), type="l", main="Probability Function",
         ylab="density", col="red")
    lines(x, dinvbeta(x,2,3), type="l", col="green")
    lines(x, dinvbeta(x,3,2), type="l", col="blue")
    legend(2, 0.9, expression(paste(alpha==2, ", ", beta==2),
         paste(alpha==2, ", ", beta==3), paste(alpha==3, ", ", beta==2)),
         lty=c(1,1,1), col=c("red","green","blue"))

[Package LaplacesDemon version 16.1.1 Index]