BE: The beta distribution for fitting a GAMLSS In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

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

 ```1 2 3 4 5 6 7 8 9``` ```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.

`gamlss.family`, `BEINF`

Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```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] ```

Example output

```Loading required package: MASS

GAMLSS Family: BE Beta
Link function for mu   : logit