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

Description

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

Usage

 ```1 2``` ```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.

`dbeta`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```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")) ```