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.
1 2 3 4 5 6 7 8 9 10 11  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,
upper.limit = mu + 10 * sqrt(sigma^2 * mu^2))
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)

mu.link 
Defines the 
sigma.link 
Defines the 
nu.link 
Defines the 
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] 
upper.limit 
the argument 
p 
vector of probabilities. 
n 
number of observations. If 
from 
where to start plotting the distribution from 
to 
up to where to plot the distribution 
obj 
a fitted 
main 
for title in the plot 
... 

The Zero adjusted GA distribution is given as
f(ymu,sigma,nu)=nu
if (y=0)
f(ymu,sigma,nu)=(1nu)*(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)=(1nu)*mu and Var(y)=(1nu)*mu^2*(nu+sigma^2).
The function ZAGA
returns a gamlss.family
object which can be used to fit a
zero adjusted Gamma distribution in the gamlss()
function.
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 507554.
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.
gamlss.family
, GA
, ZAIG
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)

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