extraDistr (version 1.9.1)

Slash: Slash distribution

Description

Probability mass function, distribution function and random generation for slash distribution.

Usage

dslash(x, mu = 0, sigma = 1, log = FALSE)
pslash(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rslash(n, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles.

mu

vector of locations

sigma

vector of positive valued scale parameters.

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, \(P[X > x]\).

n

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

Details

If \(Z \sim \mathrm{Normal}(0, 1)\) and \(U \sim \mathrm{Uniform}(0, 1)\), then \(Z/U\) follows slash distribution.

Probability density function

$$ f(x) = \left\{\begin{array}{ll} \frac{\phi(0) - \phi(x)}{x^2} & x \ne 0 \\ \frac{1}{2\sqrt{2\pi}} & x = 0 \end{array}\right. $$

Cumulative distribution function

$$ F(x) = \left\{\begin{array}{ll} \Phi(x) - \frac{\phi(0)-\phi(x)}{x} & x \neq 0 \\ \frac{1}{2} & x = 0 \end{array}\right. $$

Examples

x <- rslash(1e5, 5, 3)
hist(x, 1e5, freq = FALSE, xlim = c(-100, 100))
curve(dslash(x, 5, 3), -100, 100, col = "red", n = 500, add = TRUE)
hist(pslash(x, 5, 3))
plot(ecdf(x), xlim = c(-100, 100))
curve(pslash(x, 5, 3), -100, 100, col = "red", lwd = 2, n = 500, add = TRUE)