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.
The functions ZAPIG() and ZIPIG() are the zero adjusted (hurdle) and zero inflated versions of the Poisson-inverse Gaussian distribution, respectively. That is three parameter distributions.
The functions dZAPIG, dZIPIG, pZAPIG,pZIPIG, qZAPIG qZIPIG rZAPIG and rZIPIG define the probability, cumulative, quantile and random
generation functions for the zero adjusted and zero inflated beta negative binomial distributions, ZAPIG(), ZIPIG(), respectively.
PIG(mu.link = "log", sigma.link = "log")
dPIG(x, mu = 1, sigma = 1, log = FALSE)
pPIG(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qPIG(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE,
max.value = 10000)
rPIG(n, mu = 1, sigma = 1, max.value = 10000)ZIPIG(mu.link = "log", sigma.link = "log", nu.link = "logit")
dZIPIG(x, mu = 1, sigma = 1, nu = 0.3, log = FALSE)
pZIPIG(q, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE)
qZIPIG(p, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE,
max.value = 10000)
rZIPIG(n, mu = 1, sigma = 1, nu = 0.3, max.value = 10000)
ZAPIG(mu.link = "log", sigma.link = "log", nu.link = "logit")
dZAPIG(x, mu = 1, sigma = 1, nu = 0.3, log = FALSE)
pZAPIG(q, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE)
qZAPIG(p, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE,
max.value = 10000)
rZAPIG(n, mu = 1, sigma = 1, nu = 0.3, max.value = 10000)
Defines the mu.link, with "log" link as the default for the mu parameter
Defines the sigma.link, with "log" link as the default for the sigma parameter
Defines the mu.link, with "logit" link as the default for the nu parameter
vector of (non-negative integer) quantiles
vector of positive means
vector of positive despersion parameter
vector of zero probability parameter
vector of probabilities
vector of quantiles
number of random values to return
logical; if TRUE, probabilities p are given as log(p)
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
a constant, set to the default value of 10000 for how far the algorithm should look for q
Returns a gamlss.family object which can be used to fit a Poisson-inverse Gaussian distribution in the gamlss() function.
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 \(\nu=-\frac{1}{2}\).]
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.
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, NBI, NBII,
SI, SICHEL
# NOT RUN {
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)
ZIPIG()
ZAPIG()
# }
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