R: The generalized Beta type 1 distribution for fitting a GAMLSS

GB1 {gamlss.dist} | R Documentation

## The generalized Beta type 1 distribution for fitting a GAMLSS

### Description

This function defines the generalized beta type 1 distribution, a four parameter distribution. The function `GB1` creates a `gamlss.family` object which can be used to fit the distribution using the function `gamlss()`. Note the range of the response variable is from zero to one. The functions `dGB1`, `GB1`, `qGB1` and `rGB1` define the density, distribution function, quantile function and random generation for the generalized beta type 1 distribution.

### Usage

    GB1(mu.link = "logit", sigma.link = "logit", nu.link = "log",
          tau.link = "log")
    dGB1(x, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, log = FALSE)
    pGB1(q, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, lower.tail = TRUE,
           log.p = FALSE)
    qGB1(p, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, lower.tail = TRUE,
           log.p = FALSE)
    rGB1(n, mu = 0.5, sigma = 0.4, nu = 1, tau = 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.
`tau.link` | Defines the `tau.link`, with "log" link as the default for the `tau` parameter.
`x,q` | vector of quantiles
`mu` | vector of location parameter values
`sigma` | vector of scale parameter values
`nu` | vector of skewness `nu` parameter values
`tau` | vector of kurtosis `tau` 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 probability density function of the Generalized Beta type 1, (`GB1`), is defined as

f(y|mu,sigma,nu,tau)=(tau*nu^beta*y^(tau*alpha-1)(1-y^tau))^(beta-1)/(Beta(alpha,beta)*(nu+(1-nu)*y^tau))^(alpha*beta))

where 0<y<1, alpha=mu*(1-sigma^2)/sigma^2 and (1-mu)*(1-sigma^2)/sigma^2, and alpha>0, beta>0. Note the alpha/(alpha+beta), sigma=(alpha+beta+1)^(-1/2). .

### Value

`GB1()` returns a `gamlss.family` object which can be used to fit the GB1 distribution in the `gamlss()` function. `dGB1()` gives the density, `pGB1()` gives the distribution function, `qGB1()` gives the quantile function, and `rGB1()` generates random deviates.

### Warning

The qSHASH and rSHASH are slow since they are relying on golden section for finding the quantiles

### Author(s)

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

### 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`, `JSU`, `BCT`

### Examples

    GB1()   #
    y<- rGB1(200, mu=.1, sigma=.6, nu=1, tau=4)
    hist(y)
    # library(gamlss)
    # histDist(y, family=GB1, n.cyc=60)
    curve(dGB1(x, mu=.1 ,sigma=.6, nu=1, tau=4), 0.01, 0.99, main = "The GB1
               density mu=0.1, sigma=.6, nu=1, tau=4")

[Package gamlss.dist version 5.0-2 Index]