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

### 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`, `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 5.3-2 Index]