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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