R: The generalized Beta type 2 and generalized Pareto...

GB2 {gamlss.dist} | R Documentation

## The generalized Beta type 2 and generalized Pareto distributions for fitting a GAMLSS

### Description

This function defines the generalized beta type 2 distribution, a four parameter distribution. The function `GB2` creates a `gamlss.family` object which can be used to fit the distribution using the function `gamlss()`. The response variable is in the range from zero to infinity. The functions `dGB2`, `GB2`, `qGB2` and `rGB2` define the density, distribution function, quantile function and random generation for the generalized beta type 2 distribution. The generalised Pareto `GP` distribution is defined by setting the parameters `sigma` and `nu` of the `GB2` distribution to 1.

### Usage

    GB2(mu.link = "log", sigma.link = "log", nu.link = "log",
         tau.link = "log")
    dGB2(x, mu = 1, sigma = 1, nu = 1, tau = 0.5, log = FALSE)
    pGB2(q, mu = 1, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE,
         log.p = FALSE)
    qGB2(p, mu = 1, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE,
         log.p = FALSE)
    rGB2(n, mu = 1, sigma = 1, nu = 1, tau = 0.5)

    GP(mu.link = "log", sigma.link = "log")
    dGP(x, mu = 1, sigma = 1, log = FALSE)
    pGP(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    qGP(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    rGP(n, mu = 1, sigma = 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 2, (`GB2`), is defined as

f(y|mu,sigma,nu,tau)=abs(sigma)*y^{sigma*nu-1}(mu^(sigma*nu)*Beta(nu,tau)(1+(y/mu)^sigma)^(nu+tau))^-1

where y>0, mu>0, -Inf<sigma<Inf, nu>0 and tau>0. .

### Value

`GB2()` returns a `gamlss.family` object which can be used to fit the GB2 distribution in the `gamlss()` function. `dGB2()` gives the density, `pGB2()` gives the distribution function, `qGB2()` gives the quantile function, and `rGB2()` 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

    GB2()   #
    y<- rGB2(200, mu=5, sigma=2, nu=1, tau=1)
    library(MASS)
    truehist(y)
    fx<-dGB2(seq(0.01, 20, length=200), mu=5 ,sigma=2, nu=1, tau=1)
    lines(seq(0.01,20,length=200),fx)
    integrate(function(x) x*dGB2(x=x, mu=5, sigma=2, nu=1, tau=1), 0, Inf)
    mean(y)
    curve(dGB2(x, mu=5 ,sigma=2, nu=1, tau=1), 0.01, 20,
                main = "The GB2  density mu=5, sigma=2, nu=1, tau=4")

[Package gamlss.dist version 5.0-2 Index]