RGE: Reverse generalized extreme family distribution for fitting a...

In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

Reverse generalized extreme family distribution for fitting a GAMLSS

Description

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.

Usage

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)

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
x,q	vector of quantiles
mu	vector of location parameter values
sigma	vector of scale parameter values
nu	vector of the shape 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

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.

Value

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.

Note

This distribution is very difficult to fit because the y values depends on the parameter values. The RS() and CG() algorithms are not appropriate for this type of problem.

Author(s)

Bob Rigby, Mikis Stasinopoulos and Kalliope 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, \Sexpr[results=rd]{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, \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

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

See Also

gamlss.family

Examples

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 

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.