The zero-inflated beta distribution for fitting a GAMLSS

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Description

The function BEZI() defines the zero-inflated beta distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The zero-inflated beta is similar to the beta distribution but allows zeros 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 zero. The functions dBEZI, pBEZI, qBEZI and rBEZI define the density, distribution function, quantile function and random generation for the BEZI parameterization of the zero-inflated beta distribution. plotBEZI can be used to plot the distribution. meanBEZI calculates the expected value of the response for a fitted model.

Usage

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BEZI(mu.link = "logit", sigma.link = "log", nu.link = "logit")

dBEZI(x, mu = 0.5, sigma = 1, nu = 0.1, log = FALSE)

pBEZI(q, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qBEZI(p, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE,
        log.p = FALSE)
        
rBEZI(n, mu = 0.5, sigma = 1, nu = 0.1)

plotBEZI(mu = .5, sigma = 1, nu = 0.1, from = 0, to = 0.999, n = 101, 
    ...)
    
meanBEZI(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

mu

vector of location parameter values

sigma

vector of precision parameter values

nu

vector of parameter values modelling the probability at zero

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

from

where to start plotting the distribution from

to

up to where to plot the distribution

obj

a fitted BEZI object

...

other graphical parameters for plotting

Details

The zero-inflated beta distribution is given as

f(y)=nu

if (y=0)

f(y|mu,sigma)=(1-nu)*(Gamma(sigma)/Gamma(mu*sigma)*Gamma((1-mu)*sigma))*y^(mu*sigma-1)*(1-y)^(((1-mu)*sigma)-1)

if y=(0,1). The parameters satisfy 0<mu<1, sigma>0 and 0<nu<1.

Here E(y)=(1-nu)*mu and Var(y)=(1-nu)*(mu*(1-mu))/(sigma+1) + nu*(1-nu)*mu^2.

Value

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

Note

This work is part of my PhD project at the University of Sao Paulo under the supervion of Professor Silvia Ferrari. My thesis is concerned with regression modelling of rates and proportions with excess of zeros and/or ones

Author(s)

Raydonal Ospina, Department of Statistics, University of Sao Paulo, Brazil.

rospina@ime.usp.br

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.

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.

See Also

gamlss.family, BEZI

Examples

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 BEZI()# gives information about the default links for the BEZI distribution
# plotting the distribution
plotBEZI( mu =0.5 , sigma=5, nu = 0.1, from = 0, to=0.99, n = 101)
# plotting the cdf
plot(function(y) pBEZI(y, mu=.5 ,sigma=5, nu=0.1), 0, 0.999)
# plotting the inverse cdf
plot(function(y) qBEZI(y, mu=.5 ,sigma=5, nu=0.1), 0, 0.999)
# generate random numbers
dat<-rBEZI(100, mu=.5, sigma=5, nu=0.1)
# fit a model to the data. Tits a constant for mu, sigma and nu
# library(gamlss)
#mod1<-gamlss(dat~1,sigma.formula=~1, nu.formula=~1, family=BEZI) 
#fitted(mod1)[1]
#summary(mod1)
#fitted(mod1,"mu")[1]         #fitted mu   
#fitted(mod1,"sigma")[1]      #fitted sigma 
#fitted(mod1,"nu")[1]         #fitted nu  
#meanBEZI(mod1)[1] # expected value of the response

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