This function defines the , a four parameter distribution, for a gamlss.family object to be used for a
GAMLSS fitting using the function gamlss(). The functions dJSUo,
pJSUo, qJSUo and rJSUo define the density, distribution function, quantile function and random
generation for the the Johnson's Su distribution.
JSUo(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link = "log")
dJSUo(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)
pJSUo(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)
qJSUo(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)
rJSUo(n, mu = 0, sigma = 1, nu = 0, tau = 1)JSUo() returns a gamlss.family object which can be used to fit a Johnson's Su distribution in the gamlss() function.
dJSUo() gives the density, pJSUo() gives the distribution
function, qJSUo() gives the quantile function, and rJSUo()
generates random deviates.
Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse", "log" and "own"
Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" and "own"
Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "inverse", "log" ans "own"
Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "inverse", "identity" and "own"
vector of quantiles
vector of location parameter values
vector of scale parameter values
vector of skewness nu parameter values
vector of kurtosis tau parameter values
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
vector of probabilities.
number of observations. If length(n) > 1, the length is
taken to be the number required
Mikis Stasinopoulos and Bob Rigby
The function JSU uses first derivatives square in the fitting procedure so
standard errors should be interpreted with caution. It is recomented to be used only with method=mixed(2,20)
The probability density function of the orininal Jonhson's SU distribution, (JSUo), is defined as
$$f(y|\mu,\sigma\,\nu,\tau)=\frac{\tau}{\sigma (z^{2}+1)^{\frac{1}{2}} \sqrt{2\pi}}\hspace{1mm} \exp{\left[ -\frac{1}{2} r^2 \right]}$$
for \( -\infty < y < \infty \), \(\mu=(-\infty,+\infty)\), \(\sigma>0\), \(\nu=(-\infty,+\infty)\) and \(\tau>0\). where \(z = \frac{(y-\mu)}{\sigma}\), \(r = \nu + \tau \sinh^{-1}(z)\), see pp. 389-390 of Rigby et al. (2019).
Johnson, N. L. (1954). Systems of frequency curves derived from the first law of Laplace., Trabajos de Estadistica, 5, 283-291.
Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.
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, tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://www.gamlss.com/.
Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, tools:::Rd_expr_doi("10.18637/jss.v023.i07").
Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. tools:::Rd_expr_doi("10.1201/b21973")
(see also https://www.gamlss.com/).
gamlss.family, JSU, BCT
JSU()
plot(function(x)dJSUo(x, mu=0,sigma=1,nu=-1, tau=.5), -4, 15,
main = "The JSUo density mu=0,sigma=1,nu=-1, tau=.5")
plot(function(x) pJSUo(x, mu=0,sigma=1,nu=-1, tau=.5), -4, 15,
main = "The JSUo 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=JSUo,
# data=abdom, method=mixed(2,20))
# plot(h)
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