# ZAGA: The zero adjusted Gamma distribution for fitting a GAMLSS... In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The function `ZAGA()` defines the zero adjusted Gamma distribution, a three parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The zero adjusted Gamma distribution is similar to the Gamma distribution but allows zeros as y values. The extra parameter `nu` models the probabilities at zero. The functions `dZAGA`, `pZAGA`, `qZAGA` and `rZAGA` define the density, distribution function, quartile function and random generation for the `ZAGA` parameterization of the zero adjusted Gamma distribution. `plotZAGA` can be used to plot the distribution. `meanZAGA` calculates the expected value of the response for a fitted model.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```ZAGA(mu.link = "log", sigma.link = "log", nu.link = "logit") dZAGA(x, mu = 1, sigma = 1, nu = 0.1, log = FALSE) pZAGA(q, mu = 1, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE) qZAGA(p, mu = 1, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE) rZAGA(n, mu = 1, sigma = 1, nu = 0.1, ...) plotZAGA(mu = 5, sigma = 1, nu = 0.1, from = 0, to = 10, n = 101, main=NULL, ...) meanZAGA(obj) ```

## Arguments

 `mu.link` Defines the `mu.link`, with "log" link as the default for the mu parameter `sigma.link` Defines the `sigma.link`, with "log" link as the default for the sigma parameter `nu.link` Defines the `nu.link`, with "logit" link as the default for the sigma parameter `x,q` vector of quantiles `mu` vector of location parameter values `sigma` vector of scale parameter values `nu` vector of probability at zero 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 `from` where to start plotting the distribution from `to` up to where to plot the distribution `obj` a fitted `gamlss` object `main` for title in the plot `...` `...` can be used to pass the uppr.limit argument to `qIG`

## Details

The Zero adjusted GA distribution is given as

f(y|mu,sigma,nu)=nu

if (y=0)

f(y|mu,sigma,nu)=(1-nu)*(y^((1/sigma^2)-1)*exp[-y/((sigma^2)*mu)])/((sigma^2*mu)^(1/sigma^2) Gamma(1/sigma^2))

otherwise

for y=(0,Inf), mu>0, sigma>0 and 0<nu<1. E(y)=(1-nu)*mu and Var(y)=(1-nu)*mu^2*(nu+sigma^2).

## Value

The function `ZAGA` returns a `gamlss.family` object which can be used to fit a zero adjusted Gamma distribution in the `gamlss()` function.

## Author(s)

Bob Rigby, Mikis Stasinopoulos and Almond Stocker

## 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.

`gamlss.family`, `GA`, `ZAIG`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```ZAGA()# gives information about the default links for the ZAGA distribution # plotting the function PPP <- par(mfrow=c(2,2)) plotZAGA(mu=1, sigma=.5, nu=.2, from=0,to=3) #curve(dZAGA(x,mu=1, sigma=.5, nu=.2), 0,3) # pdf curve(pZAGA(x,mu=1, sigma=.5, nu=.2), 0,3, ylim=c(0,1)) # cdf curve(qZAGA(x,mu=1, sigma=.5, nu=.2), 0,.99) # inverse cdf y<-rZAGA(100, mu=1, sigma=.5, nu=.2) # randomly generated values hist(y) par(PPP) # check that the positive part sums up to .8 (since nu=0.2) integrate(function(x) dZAGA(x,mu=1, sigma=.5, nu=.2), 0,Inf) ```

### Example output

```Loading required package: MASS

GAMLSS Family: ZAGA Zero adjusted GA
Link function for mu   : log