The Poisson-inverse Gaussian distribution for fitting a GAMLSS model
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 PIG2() function is a repametrization of PIG() where mu and sigma are orthogonal see Heller et al. (2018).
The functions dPIG, pPIG, qPIG and rPIG define the density, distribution function, quantile function and random
generation for the Poisson-inverse Gaussian PIG(), distribution. Also codedPIG2, pPIG2, qPIG2 and rPIG2 are the equivalent functions for codePIG2()
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 Poisson Inverse Gaussian 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)
PIG2(mu.link = "log", sigma.link = "log")
dPIG2(x, mu=0.5, sigma=0.02, log = FALSE)
pPIG2(q, mu=0.5, sigma=0.02, lower.tail = TRUE, log.p = FALSE)
qPIG2(p, mu=0.5, sigma=0.02, lower.tail = TRUE, log.p = FALSE,
max.value = 10000)
rPIG2(n, mu=0.5, sigma=0.02)
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)mu.link |
Defines the |
sigma.link |
Defines the |
nu.link |
Defines the |
x |
vector of (non-negative integer) quantiles |
mu |
vector of positive means |
sigma |
vector of positive dispersion parameter |
nu |
vector of zero probability 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,...,∞ where μ>0 and σ>0 and
K_{λ}(t)=\frac{1}{2}\int_0^{∞} 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 ν=-\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, Bob Rigby and Marco Enea
Dean, C., Lawless, J. F. and Willmot, G. E., A mixed poisson-inverse-Gaussian regression model, Canadian J. Statist., 17, 2, pp 171-181
Heller, G. Z., Couturier, D.L. and Heritier, S. R. (2018) Beyond mean modelling: Bias due to misspecification of dispersion in Poisson-inverse Gaussian regression Biometrical Journal, 2, pp 333-342.
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. 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, https://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
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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