R: Reverse generalized extreme family distribution for fitting a...

RGE {gamlss.dist} | R Documentation

## 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

    RGE(mu.link = "identity", sigma.link = "log", nu.link = "log")
    dRGE(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
    pRGE(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
    qRGE(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
    rRGE(n, mu = 1, sigma = 0.1, nu = 1)

### 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 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 `length(n) > 1`, the length is taken to be the number required

### 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.

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

    RGE()# default links for the reverse generalized extreme family distribution
    newdata<-rRGE(100,mu=0,sigma=1,nu=5) # generates 100 random observations
    # library(gamlss)
    # gamlss(newdata~1, family=RGE, method=mixed(5,50)) # difficult to converse

[Package gamlss.dist version 5.0-2 Index]