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

JSU {gamlss.dist} | R Documentation

## 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)=1/(c*sigma*2*pi)*(1/(tau*(z^2+1)^.5))exp(-.5r^2)

for 0<y<0, mu=(-Inf,+Inf), sigma>0, nu=(-Inf,+Inf) and tau>0. where r=-nu+sinh^-1(z)/tau, z=(y-(mu+c*sigma*w^(.5) sinh(omega)))/(c*sigma), c=(.5*(w-1)*(w*cosh(2*omega)+1))^(-.5), w=exp{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` 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. 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, https://www.jstatsoft.org/v23/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.

### 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)

[Package gamlss.dist version 5.3-2 Index]