RGE {gamlss.dist} | R Documentation |
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(y, 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)
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 |
y,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 |
Definition file for reverse generalized extreme family distribution.
The probability density function of the generalized extreme value distribution is obtained from Johnson {it et al.} (1995), Volume 2, p76, equation (22.184) [where (xi,theta,gamma)->(mu,sigma, nu)].
The probability density function of the reverse generalized extreme value distribution is then obtained by replacing y by -y and μ by -μ.
Hence the probability density function of the reverse generalized extreme value distribution with nu>0 is given by
f(y|mu,sigma,nu)=(1/sigma)(1+(nu*(y-mu))/(sigma))^(1/(nu-1))*S1(y|mu,sigma,nu)
for
μ-frac{σ}{nu}<y<infty
where
S1(y|mu,sigma,nu)=exp(-[1+(nu*(y-mu))/(sigms)]^(1/nu))
and where -infty<μ<y+frac{σ}{nu}, σ>0 and nu>0. Note that only the case nu>0 is allowed here. The reverse generalized extreme value distribution is denoted as {bf RGE}(μ,σ,nu) or as {bf Reverse Generalized.Extreme. Family}(μ,σ,nu).
Note the the above distribution is a reparameterization of the three parameter Weibull distribution given by
f(y|mu,sigma,nu)=(a3/a2)*((y-a1)/a2)^(a3-1)exp(-((y-a1)/a2)^a3)
given by setting a1=mu-(sigma/nu), a2=sigma/nu, 1/nu.
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.
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.
Bob Rigby r.rigby@londonmet.ac.uk, Mikis Stasinopoulos d.stasinopoulos@londonmet.ac.uk and Kalliope Akantziliotou
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.
RGE()# gives information about the default links for the reverse generalized extreme family distribution newdata<-rRGE(100,mu=0,sigma=1,nu=5) # generates 100 random observations gamlss(newdata~1, family=RGE, method=mixed(5,50)) # difficult to converse rm(newdata)