R: The Reverse Gumbel distribution for fitting a GAMLSS

RG {gamlss.dist} | R Documentation

## The Reverse Gumbel distribution for fitting a GAMLSS

### Description

The function `RG` defines the reverse Gumbel distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dRG`, `pRG`, `qRG` and `rRG` define the density, distribution function, quantile function and random generation for the specific parameterization of the reverse Gumbel distribution.

### Usage

    RG(mu.link = "identity", sigma.link = "log")
    dRG(x, mu = 0, sigma = 1, log = FALSE)
    pRG(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    qRG(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    rRG(n, mu = 0, sigma = 1)

### Arguments

`mu.link` | Defines the `mu.link`, with "identity" link as the default for the mu parameter. other available link is "inverse", "log" and "own"
`sigma.link` | Defines the `sigma.link`, with "log" link as the default for the sigma parameter, other links are the "inverse", "identity" and "own"
`x,q` | vector of quantiles
`mu` | vector of location parameter values
`sigma` | vector of scale 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

The specific parameterization of the reverse Gumbel distribution used in `RG` is

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

for y=(-Inf,+Inf), μ=(-Inf,+Inf) and σ>0.

### Value

`RG()` returns a `gamlss.family` object which can be used to fit a Gumbel distribution in the `gamlss()` function. `dRG()` gives the density, `pGU()` gives the distribution function, `qRG()` gives the quantile function, and `rRG()` generates random deviates.

### Note

The mean of the distribution is mu-0.57722*sigma and the variance is pi^2 sigma^2/6.

### Author(s)

Mikis Stasinopoulos, 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.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

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, https://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

    plot(function(x) dRG(x, mu=0,sigma=1), -3, 6,
     main = "{Reverse Gumbel  density mu=0,sigma=1}")
    RG()# gives information about the default links for the Gumbel distribution
    dat<-rRG(100, mu=10, sigma=2) # generates 100 random observations
    # library(gamlss)
    # gamlss(dat~1,family=RG) # fits a constant for each parameter mu and sigma

[Package gamlss.dist version 5.3-2 Index]