dist.Halft {LaplacesDemon} | R Documentation |

## Half-t Distribution

### Description

These functions provide the density, distribution function, quantile function, and random generation for the half-t distribution.

### Usage

```
dhalft(x, scale=25, nu=1, log=FALSE)
phalft(q, scale=25, nu=1)
qhalft(p, scale=25, nu=1)
rhalft(n, scale=25, nu=1)
```

### 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 |

`nu` |
This is the scalar degrees of freedom parameter, which is
usually represented as |

`log` |
Logical. If |

### Details

Application: Continuous Univariate

Density:

`p(\theta) = (1 + \frac{1}{\nu} (\theta / \alpha)^2)^{-(\nu+1)/2}, \quad \theta \ge 0`

Inventor: Derived from the Student t

Notation 1:

`\theta \sim \mathcal{HT}(\alpha, \nu)`

Notation 2:

`p(\theta) = \mathcal{HT}(\theta | \alpha, \nu)`

Parameter 1: scale parameter

`\alpha > 0`

Parameter 2: degrees of freedom parameter

`\nu`

Mean:

`E(\theta)`

= unknownVariance:

`var(\theta)`

= unknownMode:

`mode(\theta) = 0`

The half-t distribution is derived from the Student t distribution, and
is useful as a weakly informative prior distribution for a scale
parameter. It is more adaptable than the default recommended
half-Cauchy, though it may also be more difficult to estimate due to its
additional degrees of freedom parameter, `\nu`

. When
`\nu=1`

, the density is proportional to a proper half-Cauchy
distribution. When `\nu=-1`

, the density becomes an improper,
uniform prior distribution. For more information on propriety, see
`is.proper`

.

Wand et al. (2011) demonstrated that the half-t distribution may be represented as a scale mixture of inverse-gamma distributions. This representation is useful for conjugacy.

### Value

`dhalft`

gives the density,
`phalft`

gives the distribution function,
`qhalft`

gives the quantile function, and
`rhalft`

generates random deviates.

### References

Wand, M.P., Ormerod, J.T., Padoan, S.A., and Fruhwirth, R. (2011).
"Mean Field Variational Bayes for Elaborate Distributions".
*Bayesian Analysis*, 6: p. 847–900.

### See Also

`dhalfcauchy`

,
`dst`

,
`dt`

,
`dunif`

, and
`is.proper`

.

### Examples

```
library(LaplacesDemon)
x <- dhalft(1,25,1)
x <- phalft(1,25,1)
x <- qhalft(0.5,25,1)
x <- rhalft(10,25,1)
#Plot Probability Functions
x <- seq(from=0.1, to=20, by=0.1)
plot(x, dhalft(x,1,-1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dhalft(x,1,0.5), type="l", col="green")
lines(x, dhalft(x,1,500), type="l", col="blue")
legend(2, 0.9, expression(paste(alpha==1, ", ", nu==-1),
paste(alpha==1, ", ", nu==0.5), paste(alpha==1, ", ", nu==500)),
lty=c(1,1,1), col=c("red","green","blue"))
```

*LaplacesDemon*version 16.1.6 Index]