GT {gamlss.dist}

The generalized t distribution for fitting a GAMLSS

Description

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(). The functions dGT, pGT, qGT and rGT define the density, distribution function, quantile function and random generation for the generalized t distribution.

Usage

GT(mu.link = "identity", sigma.link = "log", nu.link = "log", 
   tau.link = "log")
dGT(x, mu = 0, sigma = 1, nu = 3, tau = 1.5, log = FALSE)
pGT(q, mu = 0, sigma = 1, nu = 3, tau = 1.5, lower.tail = TRUE, 
   log.p = FALSE)
qGT(p, mu = 0, sigma = 1, nu = 3, tau = 1.5, lower.tail = TRUE, 
   log.p = FALSE)
rGT(n, mu = 0, sigma = 1, nu = 3, tau = 1.5)

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 t distribution, (GT), , is defined as

f(y|\mu,\sigma\,\nu,\tau)= \tau \left\{2\sigma \nu^{1/\tau} B\left(\frac{1}{\tau},\nu\right)[1+|z|^{\tau}/\nu]^{\nu+1/\tau} \right\}^{-1}

where -\infty < y < \infty , z=(y-\mu)/\sigma \mu=(-\infty,+\infty), \sigma>0, \nu>0 and \tau>0, see pp. 387-388 of Rigby et al. (2019).

Value

GT() returns a gamlss.family object which can be used to fit the GT distribution in the gamlss() function. dGT() gives the density, pGT() gives the distribution function, qGT() gives the quantile function, and rGT() generates random deviates.

Warning

The qGT and rGT are slow since they are relying on optimization for finding the quantiles

Author(s)

Bob Rigby and Mikis Stasinopoulos

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, 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, doi:10.18637/jss.v023.i07.

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

See Also

gamlss.family, JSU, BCT

Examples

GT()   # 
y<- rGT(200, mu=5, sigma=1, nu=1, tau=4)
hist(y)
curve(dGT(x, mu=5 ,sigma=2,nu=1, tau=4), -2, 11, 
      main = "The GT  density mu=5 ,sigma=1, nu=1, tau=4")
# library(gamlss)
# m1<-gamlss(y~1, family=GT)