gamlss.dist (version 5.0-0)

PIG: The Poisson-inverse Gaussian distribution for fitting a GAMLSS model

Description

The PIG() function defines the Poisson-inverse Gaussian distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dPIG, pPIG, qPIG and rPIG define the density, distribution function, quantile function and random generation for the Poisson-inverse Gaussian PIG(), distribution.

Usage

PIG(mu.link = "log", sigma.link = "log")
dPIG(x, mu = 0.5, sigma = 0.02, log = FALSE)
pPIG(q, mu = 0.5, sigma = 0.02, lower.tail = TRUE, log.p = FALSE)
qPIG(p, mu = 0.5, sigma = 0.02, lower.tail = TRUE, log.p = FALSE, 
     max.value = 10000)
rPIG(n, mu = 0.5, sigma = 0.02)

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

x

vector of (non-negative integer) quantiles

mu

vector of positive means

sigma

vector of positive despersion parameter

p

vector of probabilities

q

vector of quantiles

n

number of random values to return

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]

max.value

a constant, set to the default value of 10000 for how far the algorithm should look for q

Value

Returns a gamlss.family object which can be used to fit a Poisson-inverse Gaussian distribution in the gamlss() function.

Details

The probability function of the Poisson-inverse Gaussian distribution, is given by

$$f(y|\mu,\sigma)=\left( \frac{2 \alpha}{\pi}^{\frac{1}{2}}\right)\frac{\mu^y e^{\frac{1}{\sigma}} K_{y-\frac{1}{2}}(\alpha)}{(\alpha \sigma)^y y!}$$ where \(\alpha^2=\frac{1}{\sigma^2}+\frac{2\mu}{\sigma}\), for \(y=0,1,2,...,\infty\) where \(\mu>0\) and \(\sigma>0\) and \( K_{\lambda}(t)=\frac{1}{2}\int_0^{\infty} x^{\lambda-1} \exp\{-\frac{1}{2}t(x+x^{-1})\}dx\) is the modified Bessel function of the third kind. [Note that the above parameterization was used by Dean, Lawless and Willmot(1989). It is also a special case of the Sichel distribution SI() when \(\nu=-\frac{1}{2}\).]

References

Dean, C., Lawless, J. F. and Willmot, G. E., A mixed poisson-inverse-Gaussian regression model, Canadian J. Statist., 17, 2, pp 171-181

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.

See Also

gamlss.family, NBI, NBII, SI, SICHEL

Examples

PIG()# gives information about the default links for the  Poisson-inverse Gaussian distribution 
#plot the pdf using plot 
plot(function(y) dPIG(y, mu=10, sigma = 1 ), from=0, to=50, n=50+1, type="h") # pdf
# plot the cdf
plot(seq(from=0,to=50),pPIG(seq(from=0,to=50), mu=10, sigma=1), type="h")   # cdf
# generate random sample
tN <- table(Ni <- rPIG(100, mu=5, sigma=1))
r <- barplot(tN, col='lightblue')
# fit a model to the data 
# library(gamlss)
# gamlss(Ni~1,family=PIG)