# dist.Halft: Half-t Distribution In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

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

## Usage

 ```1 2 3 4``` ```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.

`dhalfcauchy`, `dst`, `dt`, `dunif`, and `is.proper`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```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")) ```

### Example output

```Warning message:
In log(1 + (1/nu) * (x/scale) * (x/scale)) : NaNs produced
```

LaplacesDemon documentation built on Dec. 23, 2017, 5:13 p.m.