BE: The beta distribution for fitting a GAMLSS

In Stan125/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. BEo() 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 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))^0.5 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

BE() and BEo() return a gamlss.family object which can be used to fit a beta 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, BE, LOGITNO, GB1, BEINF

Examples

BE()# gives information about the default links for the beta 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]

Stan125/gamlss.dist documentation built on May 12, 2019, 7:38 a.m.