# BEZI: The zero-inflated beta distribution for fitting a GAMLSS In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## 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

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

## 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.

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`, `BEZI`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ``` 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 ```