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

JSUo {gamlss.dist} | R Documentation

## 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, (`JSU`), is defined as

f(y|mu,sigma,nu,tau)=tau/(sigma)*(1/(z^2+1)^.5)*(1/2*Pi)^(.5)exp(-.5r^2)

for 0<y<0, mu=(-Inf,+Inf), sigma>0, nu=(-Inf,+Inf) and tau>0. where z=(y-mu)/sigma, nu + tau* asinh(z).

### 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. 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, 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`, `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)

[Package gamlss.dist version 5.3-2 Index]