gamlss.dist (version 5.1-6)

exGAUS: 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

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.

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 \(\nu>0\).

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. An older version can be found in http://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, http://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, BCCG, GA, IG LNO

Examples

# NOT RUN {
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")
# }