dist.Inverse.Beta: Inverse Beta Distribution
In LaplacesDemonR/LaplacesDemonCpp: C++ Extension for LaplacesDemon

Description Usage Arguments Details Value References See Also Examples

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

1 2	dinvbeta(x, a, b, log=FALSE) rinvbeta(n, a, b)

`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.

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.

dinvbeta gives the density and rinvbeta generates random deviates.

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

dbeta

library(LaplacesDemonCpp)
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"))