Moment-parameterised skew t distribution

Density, distribution function, quantile function, and random generation for the skew t distribution reparameterised so that mean and sd correspond to the true mean and standard deviation.

Usage

dskewt2(x, mean = 0, sd = 1, skew = 0, df = 100, log = FALSE)

pskewt2(q, mean = 0, sd = 1, skew = 0, df = 100,
        method = 0, lower.tail = TRUE, log.p = FALSE)

qskewt2(p, mean = 0, sd = 1, skew = 0, df = 100, tol = 1e-08, method = 0)

rskewt2(n, mean = 0, sd = 1, skew = 0, df = 100)

Arguments

x, q: vector of quantiles
mean: mean parameter
sd: standard deviation parameter, must be positive.
skew: skewness parameter, can be positive or negative.
df: degrees of freedom, must be positive.
log, log.p: logical; if TRUE, probabilities/ densities $p$ are returned as $\log(p)$.
method: an integer value between 0 and 5 which selects the computing method; see ‘Details’ in the pst documentation below for the meaning of these values. If method=0 (default value), an automatic choice is made among the four actual computing methods, depending on the other arguments.
lower.tail: logical; if TRUE, probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
p: vector of probabilities
tol: a scalar value which regulates the accuracy of the result of qsn, measured on the probability scale.
n: number of random values to return.

Value

dskewt2 gives the density, pskewt2 gives the distribution function, qskewt2 gives the quantile function, and rskewt2 generates random deviates.

Details

This corresponds to the skew t type 2 distribution in GAMLSS (ST2), see pp. 411-412 of Rigby et al. (2019) and the version implemented in the sn package. However, it is reparameterised in terms of a standard deviation parameter sd rather than just a scale parameter sigma. Details of this reparameterisation are given below. This implementation of dskewt allows for automatic differentiation with RTMB while the other functions are imported from the sn package. See sn::dst for more details.

Caution: In a numerial optimisation, the skew parameter should NEVER be initialised with exactly zero. This will cause the initial and all subsequent derivatives to be exactly zero and hence the parameter will remain at its initial value.

For given skew $= \alpha$ and df = $\nu$, define $$ \delta = \alpha / \sqrt{1 + \alpha^2}, \qquad b_\nu = \sqrt{\nu / \pi}\, \Gamma((\nu-1)/2)/\Gamma(\nu/2), $$ then $$ E(X) = \mu + \sigma \delta b_\nu,\quad Var(X) = \sigma^2 \left( \frac{\nu}{\nu-2} - \delta^2 b_\nu^2 \right). $$

Examples

x <- rskewt2(1, 1, 2, 5, 5)
d <- dskewt2(x, 1, 2, 5, 5)
p <- pskewt2(x, 1, 2, 5, 5)
q <- qskewt2(p, 1, 2, 5, 5)