JSUoriginal: The original Johnson's Su distribution for fitting a GAMLSS

In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

The original Johnson's Su distribution for fitting a GAMLSS

Description

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.

Usage

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)

Arguments

mu.link	Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse", "log" and "own"
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" and "own"
nu.link	Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "inverse", "log" ans "own"
tau.link	Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "inverse", "identity" and "own"
x,q	vector of quantiles
mu	vector of location parameter values
sigma	vector of scale parameter values
nu	vector of skewness nu parameter values
tau	vector of kurtosis tau parameter values
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required

Details

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).

Value

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.

Warning

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)

Author(s)

Mikis Stasinopoulos and Bob Rigby

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. 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, \Sexpr[results=rd]{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, \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

(see also https://www.gamlss.com/).

See Also

gamlss.family, JSU, BCT

Examples

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)

mstasinopoulos/GAMLSS-Distibutions documentation built on Nov. 3, 2023, 10:33 a.m.