gamlss.family object to be used for a
GAMLSS fitting using the function gamlss(). The functions dJSU,
pJSU, qJSU and rJSU define the density, distribution function, quantile function and random
generation for the the Johnson's Su distribution.JSU(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link = "log")
dJSU(x, mu = 0, sigma = 1, nu = 1, tau = 0.5, log = FALSE)
pJSU(q, mu = 0, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qJSU(p, mu = 0, sigma = 1, nu = 0, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rJSU(n, mu = 0, sigma = 1, nu = 0, tau = 0.5)mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse" "log" ans "own"sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" ans "own"nu.link, with "identity" link as the default for the nu parameter. Other links are "onverse", "log" and "own"tau.link, with "log" link as the default for the tau parameter. Other links are "onverse", "identity" ans "own"nu parameter valuestau parameter valueslength(n) > 1, the length is
taken to be the number requiredJSU() returns a gamlss.family object which can be used to fit a Johnson's Su distribution in the gamlss() function.
dJSU() gives the density, pJSU() gives the distribution
function, qJSU() gives the quantile function, and rJSU()
generates random deviates.JSU uses first derivatives square in the fitting procedure so
standard errors should be interpreted with cautionJSU), is defined as
$$f(y|n,\mu,\sigma\,\nu,\tau)==\frac{1}{c \sigma} \frac{1}{\tau(z^2+1)^{\frac{1}{2}}} \frac{1}{\sqrt{2\pi}} \hspace{1mm} \exp{\left[ -\frac{1}{2} r^2 \right]}$$
for $-\infty < y < \infty$, $\mu=(-\infty,+\infty)$,
$\sigma>0$, $\nu=(-\infty,+\infty)$ and $\tau>0$.
where $r=-\nu+\frac{1}{\tau} \sinh^{-1}(z)$, $z = \frac{y-(\mu+c\sigma w^{\frac{1}{2}}\sinh{\Omega})}{c\sigma}$,
$c = [ \frac{1}{2}(w-1)(w \cosh{2 \Omega} +1) ]^{\frac{1}{2}}$,
$w=e^{\tau^2}$ and $\Omega = -\nu\tau$.
This is a reparameterization of the original Johnson Su distribution, Johnson (1954),
so the parameters mu and sigma are the mean and the standard deviation of the distribution.
The parameter nu determines the skewness of the distribution with nu>0 indicating positive skewness and
nu<0< code=""> negative. The parameter tau determines the kurtosis of the distribution. tau should be positive and most likely in the region
from zero to 1. As tau goes to 0 (and for nu=0) the distribution approaches the the Normal density function.
The distribution is appropriate for leptokurtic data that is data with kurtosis larger that the Normal distribution one.gamlss.family, JSUo, BCTJSU()
plot(function(x)dJSU(x, mu=0,sigma=1,nu=-1, tau=.5), -4, 4,
main = "The JSU density mu=0,sigma=1,nu=-1, tau=.5")
plot(function(x) pJSU(x, mu=0,sigma=1,nu=-1, tau=.5), -4, 4,
main = "The JSU cdf mu=0, sigma=1, nu=-1, tau=.5")
# library(gamlss)
# data(abdom)
# h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=JSU, data=abdom)Run the code above in your browser using DataLab