SIMPLEX {gamlss.dist}

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, SIMPLEX, is given as

f(y|\mu, \sigma) = \frac{1}{\left( 2 \pi \sigma^2 y^3 (1-y)^3\right)^{1/2}} \exp(-\frac{1}{2\sigma^2} \frac{(y-\mu)^2}{y(1-y) \mu^2 (1-\mu)^2 } )

for 0<y<1, 0<\mu<1 and \sigma>0 see p 464 of Rigby et al. (2019).

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.

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, 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, 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. doi:10.1201/b21973

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

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)