LaplacesDemon: dist.HalfCauchy – R documentation

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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.

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.

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.

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"))

LaplacesDemon

Complete Environment for Bayesian Inference

v16.1.4

MIT + file LICENSE

Authors

Byron Hall [aut], Martina Hall [aut], Statisticat, LLC [aut], Eric Brown [ctb], Richard Hermanson [ctb], Emmanuel Charpentier [ctb], Daniel Heck [ctb], Stephane Laurent [ctb], Quentin F. Gronau [ctb], Henrik Singmann [cre]

Initial release