The function RGE defines the reverse generalized extreme family distribution, a three parameter distribution,
for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().
The functions dRGE, pRGE, qRGE and rRGE define the density, distribution function, quantile function and random
generation for the specific parameterization of the reverse generalized extreme distribution given in details below.
RGE(mu.link = "identity", sigma.link = "log", nu.link = "log")
dRGE(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
pRGE(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qRGE(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rRGE(n, mu = 1, sigma = 0.1, nu = 1)RGE() returns a gamlss.family object which can be used to fit a reverse generalized extreme distribution in the gamlss() function.
dRGE() gives the density, pRGE() gives the distribution
function, qRGE() gives the quantile function, and rRGE()
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
vector of quantiles
vector of location parameter values
vector of scale parameter values
vector of the shape 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
Bob Rigby, Mikis Stasinopoulos and Kalliope Akantziliotou
Definition file for reverse generalized extreme family distribution.
The probability density function of the generalized extreme value distribution is obtained from Johnson et al. (1995), Volume 2, p76, equation (22.184) [where \((\xi, \theta, \gamma) \longrightarrow (\mu, \sigma, \nu)\)].
The probability density function of the reverse generalized extreme value distribution is then obtained by replacing y by -y and \(\mu\) by \(-\mu\).
Hence the probability density function of the reverse generalized extreme value distribution with \(\nu>0\) is given by
$$f(y|\mu,\sigma, \nu)=\frac{1}{\sigma}\left[1+\frac{\nu(y-\mu)}{\sigma}\right]^{\frac{1}{\nu}-1}S_1(y|\mu,\sigma,\nu)$$
for $$\mu-\frac{\sigma}{\nu}<y<\infty$$
where
$$S_1(y|\mu,\sigma,\nu)=\exp\left\{-\left[1+\frac{\nu(y-\mu)}{\sigma}\right]^\frac{1}{\nu}\right\}$$
and where \(-\infty<\mu<y+\frac{\sigma}{\nu}\), \(\sigma>0\) and \(\nu>0\). Note that only the case \(\nu>0\) is allowed here. The reverse generalized extreme value distribution is denoted as RGE(\(\mu,\sigma,\nu\)) or as Reverse Generalized.Extreme.Family(\(\mu,\sigma,\nu\)).
Note the the above distribution is a reparameterization of the three parameter Weibull distribution given by
$$f(y|\alpha_1,\alpha_2,\alpha_3)=\frac{\alpha_3}{\alpha_2}\left[\frac{y-\alpha_1}{\alpha_2}\right]^{\alpha_3-1} \exp\left[ -\left(\frac{y-\alpha_1}{\alpha_2} \right)^{\alpha_3} \right]$$
given by setting \(\alpha_1=\mu-\sigma/\nu\), \(\alpha_2=\sigma/\nu\), \(\alpha_3=1/\nu\).
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
RGE()# default links for the reverse generalized extreme family distribution
newdata<-rRGE(100,mu=0,sigma=1,nu=5) # generates 100 random observations
# library(gamlss)
# gamlss(newdata~1, family=RGE, method=mixed(5,50)) # difficult to converse
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