gamlss.dist (version 6.1-1)

EGB2: The exponential generalized Beta type 2 distribution for fitting a GAMLSS

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 = "log", 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 = 1, tau = 0.5, lower.tail = TRUE, 
      log.p = FALSE)
rEGB2(n, mu = 0, sigma = 1, nu = 1, tau = 0.5)

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.

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

Author

Bob Rigby and Mikis Stasinopoulos

Details

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

$$f(y|\mu,\sigma\,\nu,\tau) = e^{\nu z }\left[|\sigma| B(\nu,\tau) \left(1+ e^z\right)^{\nu+\tau} \right]^{-1}$$

for \(-\infty<y<\infty\), where \(z=(y-\mu)/\sigma\) and \(-\infty<\mu<\infty\), \(-\infty<\sigma<\infty\), \(\nu>0\) and \(\tau>0\), McDonald and Xu (1995), see also pp. 385-386 of Rigby et al. (2019).

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.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://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, tools:::Rd_expr_doi("10.18637/jss.v023.i07").

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. tools:::Rd_expr_doi("10.1201/b21973")

(see also https://www.gamlss.com/).

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")