R: The beta distribution for fitting a GAMLSS

BE {gamlss.dist} | R Documentation

## The beta distribution for fitting a GAMLSS

### Description

The functions `BE()` and `BEo()` define the beta distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. `BE()` has mean equal to the parameter `mu` and `sigma` as scale parameter, see below. `BE()` is the original parameterizations of the beta distribution as in `dbeta()` with `shape1`=mu and `shape2`=sigma. The functions `dBE` and `dBEo`, `pBE` and `pBEo`, `qBE` and `qBEo` and finally `rBE` and `rBE` define the density, distribution function, quantile function and random generation for the `BE` and `BEo` parameterizations respectively of the beta distribution.

### Usage

    BE(mu.link = "logit", sigma.link = "logit")
    dBE(x, mu = 0.5, sigma = 0.2, log = FALSE)
    pBE(q, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
    qBE(p, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
    rBE(n, mu = 0.5, sigma = 0.2)
    BEo(mu.link = "log", sigma.link = "log")
    dBEo(x, mu = 0.5, sigma = 0.2, log = FALSE)
    pBEo(q, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
    qBEo(p, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)

### Arguments

`mu.link` | the `mu` link function with default `logit`
`sigma.link` | the `sigma` link function with default `logit`
`x,q` | vector of quantiles
`mu` | vector of location parameter values
`sigma` | vector of scale 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 original beta distributions distribution is given as

f(y|a,b)=1/(Beta(a,b)) y^(a-1)(1-y)^(b-1)

for y=(0,1), α>0 and β>0. In the `gamlss` implementation of `BEo` α=μ and β>σ. The reparametrization in the function `BE()` is mu=a/(a+b) and sigma=1/(a+b+1) for mu=(0,1) and sigma=(0,1). The expected value of y is mu and the variance is sigma^2*mu*(1-mu).

### Value

returns a `gamlss.family` object which can be used to fit a normal distribution in the `gamlss()` function.

### Note

Note that for `BE`, `mu` is the mean and `sigma` a scale parameter contributing to the variance of y

### 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.

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`, `BEINF`

### Examples

    BE()# gives information about the default links for the normal distribution
    dat1<-rBE(100, mu=.3, sigma=.5)
    hist(dat1)
    #library(gamlss)
    # mod1<-gamlss(dat1~1,family=BE) # fits a constant for mu and sigma
    #fitted(mod1)[1]
    #fitted(mod1,"sigma")[1]
    plot(function(y) dBE(y, mu=.1 ,sigma=.5), 0.001, .999)
    plot(function(y) pBE(y, mu=.1 ,sigma=.5), 0.001, 0.999)
    plot(function(y) qBE(y, mu=.1 ,sigma=.5), 0.001, 0.999)
    plot(function(y) qBE(y, mu=.1 ,sigma=.5, lower.tail=FALSE), 0.001, .999)
    dat2<-rBEo(100, mu=1, sigma=2)
    #mod2<-gamlss(dat2~1,family=BEo) # fits a constant for mu and sigma
    #fitted(mod2)[1]
    #fitted(mod2,"sigma")[1]

[Package gamlss.dist version 5.0-2 Index]