R: t family distribution for fitting a GAMLSS

TF {gamlss.dist} | R Documentation

## t family distribution for fitting a GAMLSS

### Description

The function `TF` defines the t-family distribution, a three parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dTF`, `pTF`, `qTF` and `rTF` define the density, distribution function, quantile function and random generation for the specific parameterization of the t distribution given in details below, with mean equal to mu and standard deviation equal to sigma*(nu/(nu-2))^0.5 with the degrees of freedom nu The function `TF2` is a different parametrization where `sigma` is the standard deviation.

### Usage

    TF(mu.link = "identity", sigma.link = "log", nu.link = "log")
    dTF(x, mu = 0, sigma = 1, nu = 10, log = FALSE)
    pTF(q, mu = 0, sigma = 1, nu = 10, lower.tail = TRUE, log.p = FALSE)
    qTF(p, mu = 0, sigma = 1, nu = 10, lower.tail = TRUE, log.p = FALSE)
    rTF(n, mu = 0, sigma = 1, nu = 10)

    TF2(mu.link = "identity", sigma.link = "log", nu.link = "logshiftto2")
    dTF2(x, mu = 0, sigma = 1, nu = 10, log = FALSE)
    pTF2(q, mu = 0, sigma = 1, nu = 10, lower.tail = TRUE, log.p = FALSE)
    qTF2(p, mu = 0, sigma = 1, nu = 10, lower.tail = TRUE, log.p = FALSE)
    rTF2(n, mu = 0, sigma = 1, nu = 10)

### 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
`x,q` | vector of quantiles
`mu` | vector of location parameter values
`sigma` | vector of scale parameter values
`nu` | vector of the degrees of freedom 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

Definition file for t family distribution.

f(y|mu,sigma,nu)=((Gamma((nu+1)/2)/(sigma*Gamma(1/2)*Gamma(nu/2))*nu^0.5 )*(1+(y-mu)^2/(nu*sigma^2))^(-(nu+1)/2)

y=(-Inf,+Inf), μ=(-Inf,+Inf), σ>0 and ν>0. Note that z=(y-mu)/sigma has a standard t distribution with degrees of freedom nu.

### Value

`TF()` returns a `gamlss.family` object which can be used to fit a t distribution in the `gamlss()` function. `dTF()` gives the density, `pTF()` gives the distribution function, `qTF()` gives the quantile function, and `rTF()` generates random deviates. The latest functions are based on the equivalent `R` functions for gamma distribution.

### Note

mu is the mean and sigma*(nu/(nu-2))^0.5 is the standard deviation of the t family distribution. nu>0 is a positive real valued parameter.

### Author(s)

Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou

### 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`

### Examples

    TF()# gives information about the default links for the t-family distribution
    # library(gamlss)
    #data(abdom)
    #h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=TF, data=abdom) # fits
    #plot(h)
    newdata<-rTF(1000,mu=0,sigma=1,nu=5) # generates 1000 random observations
    hist(newdata)

[Package gamlss.dist version 5.3-2 Index]