Johnson SU distribution (JSU)

Density, distribution function, quantile function, and random generation for the Johnson SU distribution, in the original and in the moment parameterisation.

Usage

djsu(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)

pjsu(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)

qjsu(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)

rjsu(n, mu = 0, sigma = 1, nu = 0, tau = 1)

djsu2(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)

pjsu2(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)

qjsu2(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)

rjsu2(n, mu = 0, sigma = 1, nu = 0, tau = 1)

Arguments

x, q: vector of quantiles
mu: location parameter for djsu; the mean for djsu2.
sigma: scale parameter for djsu; the standard deviation for djsu2. Must be positive.
nu: skewness parameter (real). Positive $\nu$ gives left skewness in djsu and right skewness in djsu2.
tau: kurtosis parameter, must be positive. Large $\tau$ approaches the normal distribution.
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

djsu gives the density, pjsu gives the distribution function, qjsu gives the quantile function, and rjsu generates random deviates. djsu2, pjsu2, qjsu2 and rjsu2 are the corresponding functions for the moment parameterisation.

Details

The Johnson SU distribution is a four-parameter continuous distribution on the whole real line, obtained by applying the transformation $$Z = \nu + \tau \,\mathrm{asinh}\!\left(\frac{x-\mu}{\sigma}\right) \sim N(0,1)$$ to a standard normal variable. It covers a wide range of skewness and kurtosis combinations and is a common alternative to the Box-Cox families for data that are not restricted to be positive.

djsu uses the original parameterisation, in which $\mu$ and $\sigma$ are a location and a scale parameter, $\nu$ controls skewness and $\tau$ controls kurtosis. The density is $$f(x; \mu, \sigma, \nu, \tau) = \frac{\tau}{\sigma \sqrt{2\pi}} \frac{1}{\sqrt{z^2 + 1}} \exp\!\left(-\frac{r^2}{2}\right),$$ where $z = (x-\mu)/\sigma$ and $r = \nu + \tau\,\mathrm{asinh}(z)$. Here $\mu$ is not the mean and $\sigma$ is not the standard deviation.

djsu2 uses the moment parameterisation, in which $\mu$ is the mean and $\sigma$ is the standard deviation of the distribution, for any admissible $\nu$ and $\tau$. This is usually the more convenient parameterisation for regression modelling, because the location and scale parameters keep their interpretation as $\nu$ and $\tau$ change. Writing $\omega = -\nu/\tau$ and $w = \exp(\tau^{-2})$, it is obtained from the original parameterisation by the reparameterisation $$c = \left[\tfrac{1}{2}(w-1)\left(w \cosh(2\omega) + 1\right)\right]^{-1/2}, \qquad \sigma^* = c\,\sigma, \qquad \mu^* = \mu + c\,\sigma\sqrt{w}\,\sinh(\omega),$$ so that djsu2(x, mu, sigma, nu, tau) equals djsu(x, $\mu^*$, $\sigma^*$, -nu, tau).

These correspond to the JSUo and JSU families of the gamlss.dist package respectively; see Chapter 18 of Rigby et al. (2019). Note that the sign convention for $\nu$ differs between the two parameterisations, exactly as it does in gamlss.dist.

All four d and p functions are compatible with automatic differentiation by RTMB, so both simulation and one-step-ahead residuals are supported.

References

Johnson, N. L. (1954). Systems of frequency curves derived from the first law of Laplace. Trabajos de Estadistica, 5, 283-291.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

set.seed(123)
# original parameterisation
x <- rjsu(5, mu = 0, sigma = 1, nu = -1, tau = 2)
d <- djsu(x, mu = 0, sigma = 1, nu = -1, tau = 2)
p <- pjsu(x, mu = 0, sigma = 1, nu = -1, tau = 2)
q <- qjsu(p, mu = 0, sigma = 1, nu = -1, tau = 2)

# moment parameterisation: mu is the mean, sigma the standard deviation
y <- rjsu2(1000, mu = 3, sigma = 2, nu = 1, tau = 3)
c(mean = mean(y), sd = sd(y))
#>     mean       sd 
#> 2.972965 1.979446