R: Half-Cauchy Distribution

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) = 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.

### 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"))

[Package LaplacesDemon version 16.1.1 Index]