dist.HalfCauchy {LaplacesDemon} R Documentation

## 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.

### 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.

### Value

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

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"))


[Package LaplacesDemon version 16.1.6 Index]