R: Half-t Distribution

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 alpha, which must be positive.
`nu` | This is the scalar degrees of freedom parameter, which is usually represented as nu.
`log` | Logical. If `log=TRUE` then the logarithm of the density is returned.

### Details

- Application: Continuous Univariate

- Density: p(theta) = (1 + (1/nu)*(theta/alpha)^2)^(-(nu+1)/2), theta >= 0

- Inventor: Derived from the Student t

- Notation 1: theta ~ HT(alpha,nu)

- Notation 2: p(theta) = HT(theta | alpha,nu)

- Parameter 1: scale parameter alpha > 0

- Parameter 2: degrees of freedom parameter nu

- Mean: E(theta) = unknown

- Variance: var(theta) = unknown

- Mode: 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"))

[Package LaplacesDemon version 16.1.1 Index]