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

### 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 https://www.gamlss.com/).

### 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 5.3-2 Index]