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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
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)
``` |

`mu.link` |
Defines the |

`sigma.link` |
Defines the |

`nu.link` |
Defines the |

`tau.link` |
Defines the |

`x,q` |
vector of quantiles |

`mu` |
vector of location parameter values |

`sigma` |
vector of scale parameter values |

`nu` |
vector of skewness |

`tau` |
vector of kurtosis |

`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 |

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*.
.

`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.

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

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

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.

1 2 3 4 5 6 7 8 9 10 | ```
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")
``` |

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