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 |
b |
This is the scalar shape parameter |
log |
Logical. If |
Details
Application: Continuous Univariate
Density:
p(\theta) = \frac{\theta^{\alpha - 1} (1 + \theta)^{-\alpha - \beta}}{\beta(\alpha, \beta)}
Inventor: Dubey (1970)
Notation 1:
\theta \sim \mathcal{B}^{-1}(\alpha, \beta)
Notation 2:
p(\theta) = \mathcal{B}^{-1}(\theta | \alpha, \beta)
Parameter 1: shape
\alpha > 0
Parameter 2: shape
\beta > 0
Mean:
E(\theta) = \frac{\alpha}{\beta - 1}
, for\beta > 1
Variance:
var(\theta) = \frac{\alpha(\alpha + \beta - 1)}{(\beta - 1)^2 (\beta - 2)}
Mode:
mode(\theta) = \frac{\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
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"))