GB1 {gamlss.dist} | R Documentation |
This function defines the generalized t distribution, a four parameter distribution,
for a gamlss.family
object to be used for a
GAMLSS fitting using the function gamlss()
. Note the range of the response variable is from zero to one.
The functions dGB1
,
GB1
, qGB1
and rGB1
define the density,
distribution function, quantile function and random
generation for the generalized beta type 1 distribution.
GB1(mu.link = "logit", sigma.link = "logit", nu.link = "log", tau.link = "log") dGB1(y, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, log = FALSE) pGB1(q, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, lower.tail = TRUE, log.p = FALSE) qGB1(p, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, lower.tail = TRUE, log.p = FALSE) rGB1(n, mu = 0.5, sigma = 0.4, nu = 1, tau = 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. |
tau.link |
Defines the tau.link , with "log" link as the default for the tau parameter. |
y,q |
vector of quantiles |
mu |
vector of location parameter values |
sigma |
vector of scale parameter values |
nu |
vector of skewness nu parameter values |
tau |
vector of kurtosis tau 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 |
The probability density function of the Generalized Beta type 1, (GB1
), is defined as
f(y|mu,sigma,nu,tau)=(tau*nu^beta*y^(tau*alpha-1)(1-y^tau))^(beta-1)/(Beta(alpha,beta)*(nu+(1-nu)*y^tau))^(alpha*beta))
where 0<y<1, alpha=mu*(1-sigma^2)/sigma^2 and (1-mu)*(1-sigma^2)/sigma^2, and alpha>0, beta>0. Note the alpha/(alpha+beta), sigma=(alpha+beta+1)^(-1/2). .
GB1()
returns a gamlss.family
object which can be used to fit the GB1 distribution in the
gamlss()
function.
dGB1()
gives the density, pGB1()
gives the distribution
function, qGB1()
gives the quantile function, and rGB1()
generates random deviates.
The qSHASH and rSHASH are slow since they are relying on golden section for finding the quantiles
Bob Rigby r.rigby@londonmet.ac.uk and Mikis Stasinopoulos d.stasinopoulos@londonmet.ac.uk
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.
gamlss
, gamlss.family
, JSU
, BCT
GB1() # y<- rGB1(200, mu=.1, sigma=.6, nu=1, tau=4) hist(y) histDist(y, family=GB1, n.cyc=60) curve(dGB1(y=x, mu=.1 ,sigma=.6, nu=1, tau=4), 0.01, 0.99, main = "The GB1 density mu=0.1, sigma=.6, nu=1, tau=4")