# 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

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 |

`sigma.link` |
Defines the |

`nu.link` |
Defines the |

`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 |

### 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 mikis.stasinopoulos@gamlss.org, 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.

### See Also

`gamlss.family`

### Examples

1 2 3 4 5 6 7 |