HalfCauchy: Half-Cauchy distribution

Description

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

Usage

dhcauchy(x, sigma = 1, log = FALSE)
phcauchy(q, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qhcauchy(p, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rhcauchy(n, sigma = 1)

Arguments

x, q

vector of quantiles.

sigma

positive valued scale parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.

vector of probabilities.

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If $X$ follows Cauchy centered at 0 and parametrized by scale $\sigma$, then $|X|$ follows half-Cauchy distribution parametrized by scale $\sigma$. Half-Cauchy distribution is a special case of half-t distribution with $\nu=1$ degrees of freedom.

References

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian analysis, 1(3), 515-534.

Jacob, E. and Jayakumar, K. (2012). On Half-Cauchy Distribution and Process. International Journal of Statistika and Mathematika, 3(2), 77-81.

Examples

Run this code


x <- rhcauchy(1e5, 2)
xx <- seq(-1, 100, by = 0.01)
hist(x, 2e5, freq = FALSE, xlim = c(0, 100))
lines(xx, dhcauchy(xx, 2), col = "red")
hist(phcauchy(x, 2))
plot(ecdf(x), xlim = c(0, 100))
lines(xx, phcauchy(xx, 2), col = "red", lwd = 2)