LaplacesDemon (version 16.1.1)

dist.HalfCauchy: Half-Cauchy Distribution

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.

Value

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

Details

• Application: Continuous Univariate

• Density: \(p(\theta) = \frac{2 \alpha}{\pi(\theta^2 + \alpha^2)}, \quad \theta > 0\)

• Inventor: Derived from Cauchy

• Notation 1: \(\theta \sim \mathcal{HC}(\alpha)\)

• Notation 2: \(p(\theta) = \mathcal{HC}(\theta | \alpha)\)

• 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.

The Cauchy distribution is known as a pathological distribution because its mean and variance are undefined, and it does not satisfy the central limit theorem.

See Also

dcauchy

Examples

# NOT RUN {
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"))
# }