# Reverse generalized extreme family distribution for fitting a GAMLSS

### Description

The function `RGE`

defines the reverse generalized extreme family distribution, a three parameter distribution,
for a `gamlss.family`

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

.
The functions `dRGE`

, `pRGE`

, `qRGE`

and `rRGE`

define the density, distribution function, quantile function and random
generation for the specific parameterization of the reverse generalized extreme distribution given in details below.

### Usage

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### 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 shape 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 reverse generalized extreme family distribution.

The probability density function of the generalized extreme value
distribution is obtained from Johnson *et al.* (1995), Volume 2,
p76, equation (22.184) [where *(xi,theta,gamma)->(mu,sigma, nu)*].

The probability density function of the reverse generalized extreme value distribution
is then obtained by replacing y by -y and *μ* by *-μ*.

Hence the probability density function of the reverse generalized extreme value distribution
with *ν>0* is given by

*f(y|mu,sigma,nu)=(1/sigma)(1+(nu*(y-mu))/(sigma))^(1/(nu-1))*S1(y|mu,sigma,nu)*

for

*μ-\frac{σ}{ν}<y<∞*

where

*S1(y|mu,sigma,nu)=exp(-[1+(nu*(y-mu))/(sigms)]^(1/nu))*

and where *-∞<μ<y+\frac{σ}{ν}*, *σ>0*
and *ν>0*. Note that only the case *nu>0* is allowed here. The reverse generalized extreme value distribution is denoted
as RGE(*μ,σ,ν*) or as Reverse Generalized.Extreme.Family(*μ,σ,ν*).

Note the the above distribution is a reparameterization of the three parameter Weibull distribution given by

*f(y|mu,sigma,nu)=(a3/a2)*((y-a1)/a2)^(a3-1)exp(-((y-a1)/a2)^a3)*

given by setting *a1=mu-(sigma/nu)*, *a2=sigma/nu*, *1/nu*.

### Value

`RGE()`

returns a `gamlss.family`

object which can be used to fit a reverse generalized extreme distribution in the `gamlss()`

function.
`dRGE()`

gives the density, `pRGE()`

gives the distribution
function, `qRGE()`

gives the quantile function, and `rRGE()`

generates random deviates.

### Note

This distribution is very difficult to fit because the y values depends
on the parameter values. The `RS()`

and `CG()`

algorithms are not appropriate for this type of problem.

### Author(s)

Bob Rigby, Mikis Stasinopoulos mikis.stasinopoulos@gamlss.org 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

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