# dhalfnormal: Half-normal, half-Student-t and half-Cauchy distributions. In bayesmeta: Bayesian Random-Effects Meta-Analysis

## Description

Half-normal, half-Student-t and half-Cauchy density, distribution, quantile functions and random number generation.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ``` dhalfnormal(x, scale=1, log=FALSE) phalfnormal(q, scale=1) qhalfnormal(p, scale=1) rhalfnormal(n, scale=1) dhalft(x, scale=1, df, log=FALSE) phalft(q, scale=1, df) qhalft(p, scale=1, df) rhalft(n, scale=1, df) dhalfcauchy(x, scale=1, log=FALSE) phalfcauchy(q, scale=1) qhalfcauchy(p, scale=1) rhalfcauchy(n, scale=1) ```

## Arguments

 `x, q` quantile. `p` probability. `n` number of observations. `scale` scale parameter (>0). `df` degrees-of-freedom parameter (>0). `log` logical; if `TRUE`, logarithmic density will be returned.

## Details

The half-normal distribution is simply a zero-mean normal distribution that is restricted to take only positive values. The scale parameter σ here corresponds to the underlying normal distribution's standard deviation: if X ~ Normal(0,sigma), then |X| ~ halfNormal(scale=sigma). Its mean is sigma*sqrt(2/pi), and its variance is sigma^2*(1-2/pi). Analogously, the half-t distribution is a truncated Student-t distribution with `df` degrees-of-freedom, and the half-Cauchy distribution is again a special case of the half-t distribution with `df=1` degrees of freedom.

Note that (half-) Student-t and Cauchy distributions arise as continuous mixture distributions of (half-) normal distributions. If

Y|sigma ~ Normal(0,sigma^2)

where the standard deviation is sigma=sqrt(k/X) and X is drawn from a χ^2-distribution with k degrees of freedom, then the marginal distribution of Y is Student-t with k degrees of freedom.

## Value

`dhalfnormal()`’ gives the density function, ‘`phalfnormal()`’ gives the cumulative distribution function (CDF), ‘`qhalfnormal()`’ gives the quantile function (inverse CDF), and ‘`rhalfnormal()`’ generates random deviates. For the ‘`dhalft()`’, ‘`dhalfcauchy()`’ and related function it works analogously.

## Author(s)

Christian Roever [email protected]

## References

A. Gelman. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1(3):515-534, 2006.

F. C. Leone, L. S. Nelson, R. B. Nottingham. The folded normal distribution. Technometrics, 3(4):543-550, 1961.

S. Psarakis, J. Panaretos. The folded t distribution. Communications in Statistics - Theory and Methods, 19(7):2717-2734, 1990.

`dnorm`, `dt`, `dcauchy`, `dlomax`, `drayleigh`, `TurnerEtAlPrior`, `RhodesEtAlPrior`, `bayesmeta`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```####################### # illustrate densities: x <- seq(0,6,le=200) plot(x, dhalfnormal(x), type="l", col="red", ylim=c(0,1), xlab=expression(tau), ylab=expression("probability density "*f(tau))) lines(x, dhalft(x, df=3), col="green") lines(x, dhalfcauchy(x), col="blue") lines(x, dexp(x), col="cyan") abline(h=0, v=0, col="grey") # show log-densities (note the differing tail behaviour): plot(x, dhalfnormal(x), type="l", col="red", ylim=c(0.001,1), log="y", xlab=expression(tau), ylab=expression("probability density "*f(tau))) lines(x, dhalft(x, df=3), col="green") lines(x, dhalfcauchy(x), col="blue") lines(x, dexp(x), col="cyan") abline(v=0, col="grey") ```

### Example output

```Loading required package: forestplot