# TF: t family distribution for fitting a GAMLSS In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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 [email protected], 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.

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.

`gamlss.family`

## Examples

 ```1 2 3 4 5 6 7``` ```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) ```

### Example output

```Loading required package: MASS

GAMLSS Family: TF t Family
Link function for mu   : identity