Reparameterised skew normal distribution

Density, distribution function, quantile function and random generation for the skew normal distribution reparameterised in terms of mean, standard deviation and skew magnitude

Usage

dskewnorm2(x, mean = 0, sd = 1, alpha = 0, log = FALSE)

pskewnorm2(q, mean = 0, sd = 1, alpha = 0, lower.tail = TRUE, log.p = FALSE)

qskewnorm2(p, mean = 0, sd = 1, alpha = 0, lower.tail = TRUE, log.p = FALSE)

rskewnorm2(n, mean = 0, sd = 1, alpha = 0)

Arguments

x, q: vector of quantiles
mean: mean parameter
sd: standard deviation, must be positive.
alpha: skewness parameter, +/- Inf is allowed.
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

dskewnorm2 gives the density, pskewnorm2 gives the distribution function, qskewnorm2 gives the quantile function, and rskewnorm2 generates random deviates.

Details

This implementation of dskewnorm2 allows for automatic differentiation with RTMB while the other functions are imported from the sn package.

Uses the same density as skewnorm with location $\xi$, scale $\omega$, and shape $\alpha$ reparameterised from the mean $m$, standard deviation $s$, and skewness parameter $\alpha$: $$\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}, \quad \omega = \frac{s}{\sqrt{1 - 2\delta^2/\pi}}, \quad \xi = m - \omega\delta\sqrt{2/\pi}.$$

Examples

# alpha is skew parameter
x <- rskewnorm2(1, alpha = 1)
d <- dskewnorm2(x, alpha = 1)
p <- pskewnorm2(x, alpha = 1)
q <- qskewnorm2(p, alpha = 1)