The generalized Beta type 2 and generalized Pareto distributions for fitting a GAMLSS

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Description

This function defines the generalized beta type 2 distribution, a four parameter distribution. The function GB2 creates a gamlss.family object which can be used to fit the distribution using the function gamlss(). The response variable is in the range from zero to infinity. The functions dGB2, GB2, qGB2 and rGB2 define the density, distribution function, quantile function and random generation for the generalized beta type 2 distribution. The generalised Pareto GP distribution is defined by setting the parameters sigma and nu of the GB2 distribution to 1.

Usage

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GB2(mu.link = "log", sigma.link = "log", nu.link = "log", 
     tau.link = "log")
dGB2(x, mu = 1, sigma = 1, nu = 1, tau = 0.5, log = FALSE)
pGB2(q, mu = 1, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE, 
     log.p = FALSE)
qGB2(p, mu = 1, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE, 
     log.p = FALSE)
rGB2(n, mu = 1, sigma = 1, nu = 1, tau = 0.5)

GP(mu.link = "log", sigma.link = "log")
dGP(x, mu = 1, sigma = 1, log = FALSE)
pGP(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qGP(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rGP(n, mu = 1, sigma = 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.

tau.link

Defines the tau.link, with "log" link as the default for the tau parameter.

x,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

Details

The probability density function of the Generalized Beta type 2, (GB2), is defined as

f(y|mu,sigma,nu,tau)=abs(sigma)*y^{sigma*nu-1}(mu^(sigma*nu)*Beta(nu,tau)(1+(y/mu)^sigma)^(nu+tau))^-1

where y>0, mu>0, -Inf<sigma<Inf, nu>0 and tau>0. .

Value

GB2() returns a gamlss.family object which can be used to fit the GB2 distribution in the gamlss() function. dGB2() gives the density, pGB2() gives the distribution function, qGB2() gives the quantile function, and rGB2() generates random deviates.

Warning

The qSHASH and rSHASH are slow since they are relying on golden section for finding the quantiles

Author(s)

Bob Rigby and Mikis Stasinopoulos mikis.stasinopoulos@gamlss.org

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.

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.org/).

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.

See Also

gamlss.family, JSU, BCT

Examples

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GB2()   # 
y<- rGB2(200, mu=5, sigma=2, nu=1, tau=1)
library(MASS)
truehist(y)
fx<-dGB2(seq(0.01, 20, length=200), mu=5 ,sigma=2, nu=1, tau=1)
lines(seq(0.01,20,length=200),fx)
integrate(function(x) x*dGB2(x=x, mu=5, sigma=2, nu=1, tau=1), 0, Inf)
mean(y)
curve(dGB2(x, mu=5 ,sigma=2, nu=1, tau=1), 0.01, 20, 
            main = "The GB2  density mu=5, sigma=2, nu=1, tau=4")

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