EGB2: The exponential generalized Beta type 2 distribution for...

In gamlss.dist: Distributions to be used for GAMLSS modelling.

Description

This function defines the generalized t distribution, a four parameter distribution. The response variable is in the range from minus infinity to plus infinity. The functions dEGB2, pEGB2, qEGB2 and rEGB2 define the density, distribution function, quantile function and random generation for the generalized beta type 2 distribution.

Usage

EGB2(mu.link = "identity", sigma.link = "identity", nu.link = "log", 
      tau.link = "log")
dEGB2(x, mu = 0, sigma = 1, nu = 1, tau = 0.5, log = FALSE)
pEGB2(q, mu = 0, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE, 
      log.p = FALSE)
qEGB2(p, mu = 0, sigma = 1, nu = 0, tau = 0.5, lower.tail = TRUE, 
      log.p = FALSE)
rEGB2(n, mu = 0, sigma = 1, nu = 0, tau = 0.5)

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)=exp{nu*z}(abs(sigma)*Beta(nu.tau)*(1+exp(z))^(nu+tau) )^(-1)

for -Inf<y<Inf, where z=(y-mu)/sigma and -Inf<mu<Inf, -Inf<sigma<Inf, nu>0 and tau>0, McDonald and Xu (1995).

Value

EGB2() returns a gamlss.family object which can be used to fit the EGB2 distribution in the gamlss() function. dEGB2() gives the density, pEGB2() gives the distribution function, qEGB2() gives the quantile function, and rEGB2() generates random deviates.

Author(s)

Bob Rigby [e-mail] and Mikis Stasinopoulos [e-mail]

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.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.

See Also

gamlss.family, JSU, BCT

Examples

EGB2()   # 
y<- rEGB2(200, mu=5, sigma=2, nu=1, tau=4)
library(MASS)
truehist(y)
fx<-dEGB2(seq(min(y), 20, length=200), mu=5 ,sigma=2, nu=1, tau=4)
lines(seq(min(y),20,length=200),fx)
# something funny here
# library(gamlss)
# histDist(y, family=EGB2, n.cyc=60)
integrate(function(x) x*dEGB2(x=x, mu=5, sigma=2, nu=1, tau=4), -Inf, Inf)
curve(dEGB2(x, mu=5 ,sigma=2, nu=1, tau=4), -10, 10, main = "The EGB2  density mu=5, sigma=2, nu=1, tau=4")

Example output

Loading required package: MASS

GAMLSS Family: EGB2 Exponential generalized beta 2 (i.e. of the second kind) 
Link function for mu   : identity 
Link function for sigma: log 
Link function for nu   : log 
Link function for tau  : log 
1.333333 with absolute error < 7.3e-05

gamlss.dist documentation built on May 2, 2019, 5:20 p.m.