BEOI: The one-inflated beta distribution for fitting a GAMLSS

Description

The function BEOI() defines the one-inflated beta distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The one-inflated beta is similar to the beta distribution but allows ones as y values. This distribution is an extension of the beta distribution using a parameterization of the beta law that is indexed by mean and precision parameters (Ferrari and Cribari-Neto, 2004). The extra parameter models the probability at one. The functions dBEOI, pBEOI, qBEOI and rBEOI define the density, distribution function, quantile function and random generation for the BEOI parameterization of the one-inflated beta distribution. plotBEOI can be used to plot the distribution. meanBEOI calculates the expected value of the response for a fitted model.

Usage

BEOI(mu.link = "logit", sigma.link = "log", nu.link = "logit")
dBEOI(x, mu = 0.5, sigma = 1, nu = 0.1, log = FALSE)
pBEOI(q, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)
qBEOI(p, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE,
        log.p = FALSE)
        
rBEOI(n, mu = 0.5, sigma = 1, nu = 0.1)
plotBEOI(mu = .5, sigma = 1, nu = 0.1, from = 0.001, to = 1, n = 101, 
    ...)
    
meanBEOI(obj)

Arguments

mu.link

the mu link function with default logit

sigma.link

the sigma link function with default log

nu.link

the nu link function with default logit

x,q

vector of quantiles

vector of location parameter values

sigma

vector of precision parameter values

vector of parameter values modelling the probability at one

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]

vector of probabilities.

number of observations. If length(n) > 1, the length is taken to be the number required

from

where to start plotting the distribution from

up to where to plot the distribution

obj

a fitted BEOI object

…

other graphical parameters for plotting

Value

returns a gamlss.family object which can be used to fit a one-inflated beta distribution in the gamlss() function.

Details

The one-inflated beta distribution is given as $$f(y)=\nu$$ if $(y=1)$ $$f(y|\mu,\sigma)=(1-\nu)\frac{\Gamma(\sigma)}{\Gamma(\mu \sigma)\Gamma((1-\mu) \sigma)} y^{\mu \sigma}(1-y)^{((1-\mu)\sigma)-1}$$ if $y=(0,1)$. The parameters satisfy $0<\mu<0$, $\sigma>0$ and $0<\nu< 1$.

Here $E(y)=\nu+(1-\nu)\mu$ and $Var(y)=(1-\nu)\frac{\mu(1-\mu)}{\sigma+1}+\nu(1-\nu)(1-\mu)^2$.

References

Ferrari, S.L.P., Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31 (1), 799-815.

Ospina R. and Ferrari S. L. P. (2010) Inflated beta distributions, Statistical Papers, 23, 111-126.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape (with discussion). Applied Statistics, 54 (3), 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 http://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, 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.

Examples

Run this code

# NOT RUN {
BEOI()# gives information about the default links for the BEOI distribution
# plotting the distribution
plotBEOI( mu =0.5 , sigma=5, nu = 0.1, from = 0.001, to=1, n = 101)
# plotting the cdf
plot(function(y) pBEOI(y, mu=.5 ,sigma=5, nu=0.1), 0.001, 0.999)
# plotting the inverse cdf
plot(function(y) qBEOI(y, mu=.5 ,sigma=5, nu=0.1), 0.001, 0.999)
# generate random numbers
dat<-rBEOI(100, mu=.5, sigma=5, nu=0.1)
# fit a model to the data. 
# library(gamlss)
#mod1<-gamlss(dat~1,sigma.formula=~1, nu.formula=~1, family=BEOI) 
#fitted(mod1)[1]
#summary(mod1)
#fitted(mod1,"mu")[1]        #fitted mu
#fitted(mod1,"sigma")[1]     #fitted sigma
#fitted(mod1,"nu")[1]        #fitted nu
#meanBEOI(mod1)[1] # expected value of the response
# }

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