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.
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/).
RGE()# gives information about the default links for the reverse generalized extreme family distribution newdata<-rRGE(1000,mu=0,sigma=1,nu=5) # generates 1000 random observations histDist(newdata) rm(h,newdata)