The beta distribution for fitting a GAMLSS
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.
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)
mu.link |
the |
sigma.link |
the |
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 |
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).
BE() and BEo() return a gamlss.family object which can be used to fit a beta distribution in the gamlss() function.
Note that for BE, mu is the mean and sigma a scale parameter contributing to the variance of y
Bob Rigby and Mikis Stasinopoulos
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. An older version can be found in https://www.gamlss.com/.
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 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.
gamlss.family, BE, LOGITNO, GB1, BEINF
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]
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