R: The ex-Gaussian distribution

exGAUS {gamlss.dist} | R Documentation

## 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)=(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 ν>0.

### 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. 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, https://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

    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")

[Package gamlss.dist version 5.3-2 Index]