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

### 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. 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, https://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 5.3-2 Index]