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.
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, ...)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.
Defines the mu.link, with "identity" link as the default for the mu parameter.
Defines the sigma.link, with "log" link as the default for the sigma parameter.
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"
vector of quantiles
vector of mu parameter values
vector of scale parameter values
vector of nu parameter values
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
vector of probabilities.
number of observations. If length(n) > 1, the length is
taken to be the number required
for extra arguments
Mikis Stasinopoulos and Bob Rigby
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).
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,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, 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. tools:::Rd_expr_doi("10.1201/b21973")
(see also https://www.gamlss.com/).
gamlss.family, BCCG, GA,
IG LNO
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")
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