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, \(P[X > x]\).
p: vector of probabilities.
n: 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)
hist(x, 2e5, freq = FALSE, xlim = c(0, 100))
curve(dhcauchy(x, 2), 0, 100, col = "red", add = TRUE)
hist(phcauchy(x, 2))
plot(ecdf(x), xlim = c(0, 100))
curve(phcauchy(x, 2), col = "red", lwd = 2, add = TRUE)