Half-Cauchy distribution

Density, distribution function, quantile function, and random generation for the half-Cauchy distribution.

Usage

dhalfcauchy(x, sigma = 1, log = FALSE)

phalfcauchy(q, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qhalfcauchy(p, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rhalfcauchy(n, sigma = 1)

Arguments

x, q: vector of quantiles.
sigma: scale parameter, must be positive.
log, log.p: logical; if TRUE, probabilities/ densities $p$ are returned as $\log(p)$.
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise $P[X > x]$.
p: vector of probabilities.
n: number of random values to return.

Value

dhalfcauchy gives the density, phalfcauchy gives the distribution function, qhalfcauchy gives the quantile function, and rhalfcauchy generates random deviates.

Details

dhalfcauchy and phalfcauchy allow for automatic differentiation with RTMB.

The half-Cauchy is the distribution of $|Y|$ for $Y$ Cauchy with scale $\sigma$: $$f(x;\,\sigma) = \frac{2}{\pi\sigma\bigl(1 + (x/\sigma)^2\bigr)}, \quad x \ge 0,$$ with distribution function $F(x) = \frac{2}{\pi}\arctan(x/\sigma)$.

It is the standard weakly informative prior for the standard deviation of a hierarchical model, recommended by Gelman (2006) in place of the inverse-gamma: the density is flat and non-zero at the origin, so it does not force the variance component away from zero, while the tail is heavy enough to leave large values unpenalised. That makes it a natural companion to models fitted by the Laplace approximation, where the variance components are exactly the parameters at issue.

It has no moments of any order. It is the half-t with df = 1, and the folded normal with mu = 0 is the corresponding half-normal, which the half-t approaches as df grows.

References

Gelman, A. (2006) Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1, 515-534, doi:10.1214/06-BA117A.

Examples

set.seed(123)
x <- rhalfcauchy(5, sigma = 2)
d <- dhalfcauchy(x, sigma = 2)
p <- phalfcauchy(x, sigma = 2)
q <- qhalfcauchy(p, sigma = 2)