# Weibull distribution for fitting a GAMLSS

### Description

The function `WEI`

can be used to define the Weibull distribution, a two parameter distribution, for a
`gamlss.family`

object to be used in GAMLSS fitting using the function `gamlss()`

. [Note that the GAMLSS function `WEI2`

uses a
different parameterization for fitting the Weibull distribution.]
The functions `dWEI`

, `pWEI`

, `qWEI`

and `rWEI`

define the density, distribution function, quantile function and random
generation for the specific parameterization of the Weibul distribution.

### Usage

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### Arguments

`mu.link` |
Defines the |

`sigma.link` |
Defines the |

`x,q` |
vector of quantiles |

`mu` |
vector of the mu parameter |

`sigma` |
vector of sigma parameter |

`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

The parameterization of the function `WEI`

is given by

*f(y|mu,sigma)= (sigma*y^(sigma-1)/mu^sigma)exp(-(y/mu)^sigma)*

for *y>0*, *μ>0* and *σ>0*.
The GAMLSS functions `dWEI`

, `pWEI`

, `qWEI`

, and `rWEI`

can be used to provide the pdf, the cdf, the quantiles and
random generated numbers for the Weibull distribution with argument `mu`

, and `sigma`

.
[See the GAMLSS function `WEI2`

for a different parameterization of the Weibull.]

### Value

`WEI()`

returns a `gamlss.family`

object which can be used to fit a Weibull distribution in the `gamlss()`

function.
`dWEI()`

gives the density, `pWEI()`

gives the distribution
function, `qWEI()`

gives the quantile function, and `rWEI()`

generates random deviates. The latest functions are based on the equivalent `R`

functions for Weibull distribution.

### Note

The mean in `WEI`

is given by *mu*gamma((1/sigma)+1)* and the variance
*(mu^2)*(gamma((2/sigma)+1)-gamma((1/sigma)+1)^2)*

### Author(s)

Mikis Stasinopoulos mikis.stasinopoulos@gamlss.org, Bob Rigby and Calliope 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`

, `WEI2`

, `WEI3`

### Examples

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