| PIG {gamlss.dist} | R Documentation |
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.
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)
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 |
The probability function of the Poisson-inverse Gaussian distribution, is given by
f(y|mu,sigma)=(2*alpha/pi)^.5 mu^y e^(1/sigma) K(alpha)/(alpha*sigma)^y y!
where α^2=frac{1}{σ^2}+frac{2μ}{σ}, for y=0,1,2,...,infty where μ>0 and σ>0 and
K_{λ}(t)=frac{1}{2}int_0^{infty} x^{λ-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}.]
Returns a gamlss.family object which can be used to fit a Poisson-inverse Gaussian distribution in the gamlss() function.
Mikis Stasinopoulos ans Bob Rigby
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.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, http://www.jstatsoft.org/v23/i07.
gamlss.family, NBI, NBII,
SI, SICHEL
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)