exGAUS {gamlss.dist} | R Documentation |
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(y, 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, ...)
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" |
y,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 |
The probability density function of the ex-Gaussian distribution, (exGAUS
), is defined as
f(y|mu,sigma,nu)=(1/nu)*exp(((mu-y)/nu)+(sigma^2/(2*nu^2)))*Phi(((y-mu)/sigma)+(sigma/tau))
where Phi is the cdf of the standard normal distribution, for -Inf<y<Inf, -Inf<mu<Inf, σ>0 and nu>0.
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.
The mean of the ex-Gaussian is mu+nu and the variance is sigma^2+nu^2.
Mikis Stasinopoulos and Bob Rigby
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.
Stasinopoulos D. M. Rigby R. A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also 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.
gamlss
, gamlss.family
, BCCG
, GA
, IG
LNO
y<- rexGAUS(100, mu=300, nu=100, sigma=35) hist(y) m1<-gamlss(y~1, family=exGAUS) plot(m1) curve(dexGAUS(y=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")