exGAUS: The ex-Gaussian distribution

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

The ex-Gaussian distribution

Description

The ex-Gaussian distribution is often used by psychologists to model response time (RT). It is defined by adding two random variables, one from a normal distribution and the other from an exponential. The parameters mu and sigma are the mean and standard deviation from the normal distribution variable while the parameter nu is the mean of the exponential variable. The functions dexGAUS, pexGAUS, qexGAUS and rexGAUS define the density, distribution function, quantile function and random generation for the ex-Gaussian distribution.

Usage

exGAUS(mu.link = "identity", sigma.link = "log", nu.link = "log")
dexGAUS(x, mu = 5, sigma = 1, nu = 1, log = FALSE)
pexGAUS(q, mu = 5, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qexGAUS(p, mu = 5, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rexGAUS(n, mu = 5, sigma = 1, nu = 1, ...)

Arguments

mu.link	Defines the mu.link, with "identity" link as the default for the mu parameter.
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter.
nu.link	Defines the nu.link, with "log" link as the default for the nu parameter. Other links are "inverse", "identity", "logshifted" (shifted from one) and "own"
x,q	vector of quantiles
mu	vector of mu parameter values
sigma	vector of scale parameter values
nu	vector of nu 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
...	for extra arguments

Details

The probability density function of the ex-Gaussian distribution, (exGAUS), is defined as

f(y|\mu,\sigma,\nu)=\frac{1}{\nu} e^{\frac{\mu-y}{\nu}+\frac{\sigma^2}{2 \nu^2}} \Phi(\frac{y-\mu}{\sigma}-\frac{\sigma}{\nu})

where \Phi is the cdf of the standard normal distribution, for -\infty<y<\infty, -\infty<\mu<\infty, \sigma>0 and \nu>0 see pp. 372-373 of Rigby et al. (2019).

Value

exGAUS() returns a gamlss.family object which can be used to fit ex-Gaussian distribution in the gamlss() function. dexGAUS() gives the density, pexGAUS() gives the distribution function, qexGAUS() gives the quantile function, and rexGAUS() generates random deviates.

Note

The mean of the ex-Gaussian is \mu+\nu and the variance is \sigma^2+\nu^2.

Author(s)

Mikis Stasinopoulos and Bob Rigby

References

Cousineau, D. Brown, S. and Heathecote A. (2004) Fitting distributions using maximum likelihood: Methods and packages, Behavior Research Methods, Instruments and Computers, 46, 742-756.

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. (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, BCCG, GA, IG LNO

Examples

exGAUS()   # 
y<- rexGAUS(100, mu=300, nu=100, sigma=35)
hist(y)
# library(gamlss)
# m1<-gamlss(y~1, family=exGAUS) 
# plot(m1)
curve(dexGAUS(x, mu=300 ,sigma=35,nu=100), 100, 600, 
 main = "The ex-GAUS  density mu=300 ,sigma=35,nu=100")
plot(function(x) pexGAUS(x, mu=300,sigma=35,nu=100), 100, 600, 
 main = "The ex-GAUS  cdf mu=300, sigma=35, nu=100")

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