R: The simplex distribution for fitting a GAMLSS

SIMPLEX {gamlss.dist} | R Documentation

## The simplex distribution for fitting a GAMLSS

### Description

The functions `SIMPLEX()` define the simplex distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. `SIMPLEX()` has mean equal to the parameter `mu` and `sigma` as scale parameter, see below. The functions `dSIMPLEX`, `pSIMPLEX` `qSIMPLEX` and `rSIMPLEX` define the density, comulative distribution function, quantile function and random generation for the simplex distribution.

### Usage

    SIMPLEX(mu.link = "logit", sigma.link = "log")
    dSIMPLEX(x, mu = 0.5, sigma = 1, log = FALSE)
    pSIMPLEX(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    qSIMPLEX(p, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    rSIMPLEX(n = 1, mu = 0.5, sigma = 1)

### Arguments

`mu.link` | the `mu` link function with default `logit`
`sigma.link` | the `sigma` link function with default `log`
`x,q` | vector of quantiles
`mu` | vector of location parameter values
`sigma` | vector of scale 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 simplex distribution is given as

f(y|mu,sigma) = 1/(2*pi*sigma^2*(y*(1-y))^3)^(1/2) exp(-(1/(2*sigma^2)) * ((y-mu)^2)/(y*(1-y)*mu^2*(1-mu)^2))

for y in (0,1), 0<μ<1 and σ>0.

### Value

`SIMPLEX()` returns a `gamlss.family` object which can be used to fit a simplex distribution in the `gamlss()` function.

### Author(s)

Bob Rigby, Mikis Stasinopoulos and Fernanda De Bastiani

### References

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.

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.

### Examples

    SIMPLEX()#  default links for the simplex distribution
    plot(function(y) dSIMPLEX(y, mu=.5 ,sigma=1), 0.001, .999)
    plot(function(y) pSIMPLEX(y, mu=.5 ,sigma=1), 0.001, 0.999)
    plot(function(y) qSIMPLEX(y, mu=.5 ,sigma=1), 0.001, 0.999)
    plot(function(y) qSIMPLEX(y, mu=.5 ,sigma=1, lower.tail=FALSE), 0.001, .999)

[Package gamlss.dist version 5.3-2 Index]