JSU: The Johnson's Su distribution for fitting a GAMLSS

In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

The 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 dJSU, pJSU, qJSU and rJSU define the density, distribution function, quantile function and random generation for the the Johnson's Su distribution.

Usage

JSU(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link = "log")
dJSU(x, mu = 0, sigma = 1, nu = 1, tau = 1, log = FALSE)
pJSU(q, mu = 0, sigma = 1, nu = 1, tau = 1, lower.tail = TRUE, log.p = FALSE)
qJSU(p, mu = 0, sigma = 1, nu = 1, tau = 1, lower.tail = TRUE, log.p = FALSE)
rJSU(n, mu = 0, sigma = 1, nu = 1, tau = 1)

Arguments

mu.link	Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse" "log" ans "own"
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" ans "own"
nu.link	Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "onverse", "log" and "own"
tau.link	Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "onverse", "identity" ans "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 Jonhson's SU distribution, (JSU), is defined as

f(y|\mu,\sigma\,\nu,\tau)=\frac{\tau}{c \sigma(s^2+1)^{\frac{1}{2}}\sqrt{2\pi}} \hspace{1mm} \exp{\left[ -\frac{1}{2} z^2 \right]}

for -\infty < y < \infty , -\infty<\mu< \infty, \sigma>0, -\infty<\nu<\infty) and \tau>0 and where

z=-\nu+\tau \sinh^{-1}(s)

,

s = \frac{y-\mu+c \sigma w^{\frac{1}{2}} \sinh(\nu/\tau)}{c\sigma}

,

c = \left\{ \frac{1}{2} (w-1) \left[w \cosh(2\nu/\tau) +1) \right] \right\}^{-\frac{1}{2}}

and

w=e^{1/\tau^2}

see pp. 393-394 of Rigby et al. (2019).

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 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.

Value

JSU() 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.

Warning

The function JSU uses first derivatives square in the fitting procedure so standard errors should be interpreted with caution

Author(s)

Bob Rigby and Mikis Stasinopoulos

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. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also 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, JSUo, BCT

Examples

JSU()   
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) 

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.