R: The generalized t distribution for fitting a GAMLSS

GT {gamlss.dist} | R Documentation

## 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)=

where 0<y<0, z=(y-mu)/sigma mu=(-Inf,+Inf), sigma>0, nu>0 and tau>0.

### 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 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.

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.

### 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)

[Package gamlss.dist version 5.0-2 Index]