dist.HalfCauchy: Half-Cauchy Distribution

In benmarwick/LaplacesDemon: Complete Environment for Bayesian Inference

Description

These functions provide the density, distribution function, quantile function, and random generation for the half-Cauchy distribution.

Usage

dhalfcauchy(x, scale=25, log=FALSE)
phalfcauchy(q, scale=25)
qhalfcauchy(p, scale=25)
rhalfcauchy(n, scale=25)

Arguments

x,q	These are each a vector of quantiles.
p	This is a vector of probabilities.
n	This is the number of observations, which must be a positive integer that has length 1.
scale	This is the scale parameter alpha, which must be positive.
log	Logical. If log=TRUE, then the logarithm of the density is returned.

Details

• Application: Continuous Univariate

• Density: p(theta) = 2alpha / pi(theta^2 + alpha^2), theta >= 0

• Inventor: Derived from Cauchy

• Notation 1: theta ~ HC(alpha)

• Notation 2: p(theta) = HC(theta | alpha)

• Parameter 1: scale parameter alpha > 0

• Mean: E(theta) = does not exist

• Variance: var(theta) = does not exist

• Mode: mode(theta) = 0

The half-Cauchy distribution with scale alpha=25 is a recommended, default, weakly informative prior distribution for a scale parameter. Otherwise, the scale, alpha, is recommended to be set to be just a little larger than the expected standard deviation, as a weakly informative prior distribution on a standard deviation parameter.

Value

dhalfcauchy gives the density, phalfcauchy gives the distribution function, qhalfcauchy gives the quantile function, and rhalfcauchy generates random deviates.

See Also

dcauchy

Examples

library(LaplacesDemon)
x <- dhalfcauchy(1,25)
x <- phalfcauchy(1,25)
x <- qhalfcauchy(0.5,25)
x <- rhalfcauchy(1,25)

#Plot Probability Functions
x <- seq(from=0, to=20, by=0.1)
plot(x, dhalfcauchy(x,1), ylim=c(0,1), type="l", main="Probability Function",
     ylab="density", col="red")
lines(x, dhalfcauchy(x,5), type="l", col="green")
lines(x, dhalfcauchy(x,10), type="l", col="blue")
legend(2, 0.9, expression(alpha==1, alpha==5, alpha==10),
     lty=c(1,1,1), col=c("red","green","blue"))

benmarwick/LaplacesDemon documentation built on May 12, 2019, 12:59 p.m.