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 |

`log` |
Logical. If |

### 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 existVariance:

`var(\theta)`

= does not existMode:

`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

### 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*version 16.1.6 Index]