GU {gamlss.dist}

The Gumbel distribution for fitting a GAMLSS

Description

The function GU defines the Gumbel distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dGU, pGU, qGU and rGU define the density, distribution function, quantile function and random generation for the specific parameterization of the Gumbel distribution.

Usage

GU(mu.link = "identity", sigma.link = "log")
dGU(x, mu = 0, sigma = 1, log = FALSE)
pGU(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qGU(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rGU(n, mu = 0, sigma = 1)

Arguments

mu.link

Defines the mu.link, with "identity" link as the default for the mu parameter. other available link is "inverse", "log" and "own")

sigma.link

Defines the sigma.link, with "log" link as the default for the sigma parameter, other links are the "inverse", "identity" and "own"

x, q

vector of quantiles

mu

vector of location parameter values

sigma

vector of scale 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

Details

The specific parameterization of the Gumbel distribution used in GU is

f(y|\mu,\sigma)=\frac{1}{\sigma} \hspace{1mm} \exp\left\{\left(\frac{y-\mu}{\sigma}\right)-\exp\left(\frac{y-\mu}{\sigma}\right)\right\}

for y=(-\infty,\infty), \mu=(-\infty,+\infty) and \sigma>0, see pp. 366-367 of Rigby et al. (2019).

Value

GU() returns a gamlss.family object which can be used to fit a Gumbel distribution in the gamlss() function. dGU() gives the density, pGU() gives the distribution function, qGU() gives the quantile function, and rGU() generates random deviates.

Note

The mean of the distribution is \mu-0.57722 \sigma and the variance is \pi^2 \sigma^2/6.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

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, 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, \doi10.18637/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. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, RG

Examples

plot(function(x) dGU(x, mu=0,sigma=1), -6, 3, 
 main = "{Gumbel  density mu=0,sigma=1}")
GU()# gives information about the default links for the Gumbel distribution      
dat<-rGU(100, mu=10, sigma=2) # generates 100 random observations 
hist(dat)
# library(gamlss)
# gamlss(dat~1,family=GU) # fits a constant for each parameter mu and sigma