ZAGA: The zero adjusted Gamma distribution for fitting a GAMLSS model

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

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)

Value

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

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

Author

Bob Rigby, Mikis Stasinopoulos and Almond Stocker

Details

The Zero adjusted GA distribution is given as $$f(y|\mu,\sigma\,\nu)=\nu$$ if (y=0) $$f(y|\mu,\sigma,\nu)=(1-\nu)\left[\frac{1}{(\sigma^2 \mu)^{1/\sigma^2}}\hspace{1mm}\frac{y^{\frac{1}{\sigma^2}-1}\hspace{1mm} e^{-y/(\sigma^2 \mu)}}{\Gamma{(1/\sigma^2)}} \right]$$ otherwise

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

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.

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, tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://www.gamlss.com/.

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 https://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, tools:::Rd_expr_doi("10.18637/jss.v023.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. tools:::Rd_expr_doi("10.1201/b21973")

(see also https://www.gamlss.com/).

Examples

Run this code

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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